The Ultimate Blackjack Book: Playing Blackjack With Multiple Decks [1 ed.] 1569800243, 9781569800249

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m ULTIMATE BLACKJACK BOOK mmsmmsm

Playing Blackjack With Multiple Decks k

Digitized by the Internet Archive in 2018 with funding from Kahle/Austin Foundation

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THE ULTIMATE BLACKJACK BOOK

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ULTIMATE BLACKJACK VBOOK+ Playing Blackjack with Multiple Decks

EDWARD EARLY BARRICADE BOOKS / New York, N.Y.

Published by Barricade Books Inc. 150 Fifth Avenue New York, NY 10011 Copyright © 1995 by Edward Early All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form, by any means, including mechanical, electronic, photocopying, recording, or otherwise, without the prior written permission of the publisher, except by a reviewer who wishes to quote brief passages in connection with a review written for inclusion in a magazine, newspaper, or broadcast. Printed in the United States of America

Library of Congress Cataloging-in-Publication Data Early, Edward The ultimate blackjack book / by Edward Early, p.

cm.

ISBN 1-56980-024-3 : $12.00 1. Blackjack (Game) I. Title. GV1295.B55E27 795.4*2—dc20

1994 94-25590 CIP

First printing

TABLE OF CONTENTS Introduction.,.11 I. Definitions.15 II. Rules of Play.23 III. Basic Strategy Decisions (In Logical Order) .... 27 IV. Casinos Advantage.31 V. Counting Cards.35 VI. Some Useful Facts.41 VII. Some Good Advice.45 VIII. Some Common Fallacies.49 IX. Reasoning Behind the Proper Decisions.55 X. Mathematics Behind the Proper Decisions.63 XI. Analysis of the Super Sevens Side Bet.71 APPENDIX A:

Analysis of the Under-Over Side Bet.75 1 DECK:

Effect of Adding One Card of Each Value.77 Effect of Adding Two Cards of Each Value.78 Effect of Adding Three Cards of Each Value.78 Effect of Adding Four Cards of Each Value.

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2 DECKS: Effect of Adding One Card of Each Value.79 Effect of Adding Three Cards of Each Value.80 Effect of Adding Six Cards of Each Value . . :.80 3 DECKS: Effect of Adding One Card of Each Value.81 Effect of Adding Two Cards of Each Value.81 Effect of Adding Eight Cards of Each Value.82 4 DECKS: Effect of Adding One Card of Each Value.82 Effect of Adding Eleven Cards of Each Value.83 5 DECKS: Effect of Adding One Card of Each Value.83 Effect of Adding Thirteen Cards of Each Value

. . 84

6 DECKS: Effect of Adding One Card of Each Value.84 Effect of Adding Sixteen Cards of Each Value.85 7 DECKS: Effect of Removing One Card of Each Value.85 Effect of Adding One Card of Each Value.86 Effect of Adding Eighteen Cards of Each Value ... 86 8 DECKS: Effect of Adding One Card of Each Value.87 Effect of Adding Twenty Cards of Each Value.87 Three Counting Methods for Under-Over: 1. Ace thru 5 = +1; 9,10 = -1. 2. Ace, 2 = +2; 9,10

88

.89

=-2

3. Ace = +3; 10 = -1.89

THE ULTIMATE

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APPENDIX B:

Number of Distinct Dealer Hands, 17 thru Bust.91 APPENDIX C:

The Cost of Deviating from Basic Strategy.93 APPENDIX D:

Player Expectation With Certain Cards Removed .... 95 appendix E:

Dealer Probability Tables.97

APPENDIX F:

Multi-Action Blackjack.101

APPENDIX G:

Some Personal Experiences.105

APPENDIX H:

10 Million Hands of Blackjack.119

BIBLIOGRAPHY.147 ABOUT THE AUTHOR.149

THE ULTIMATE BLACKJACK BOOK

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INTRODUCTION

W

hy

another

book

on

Blackjack?

The question is a good one. After all, so very many books have been written on this game. Some are excellent. Some are terrible. The Theory of Blackjack, by Peter Griffin, is an example of an excellent book, and well worth the cost, if you like theory and mathematical insight into strategy. Also, Richard Epstein’s The Theory of Gambling and Statistical Logic has an excellent chapter and appen¬ dix on the mathematics of Blackjack. Those books which in my opinion fall into a ‘terrible’ category are not worth mentioning, and are better left to the obscurity which they deserve. Others, worth reading, are mentioned in the Bibliography. Most books confine themselves to an analysis of the sin¬ gle-deck game, with Las Vegas rules. This book is written

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for the player who plays in casinos which primarily use six to eight decks in order to foil card counters who make a liv¬ ing at the tables. I decided to write this book after years of play at Atlantic City watching hapless players applying obscure logic to decision making at the ‘21’ tables. For example, most players who stand with a hard count of 16 against the dealers 7 feel that they should hit on a 16 against the deal¬ er s 10. In both cases basic strategy demands taking a hit, but hitting against the seven will lose less in the long run than hitting against a ten. A detailed explanation is given in the chapter on the logic behind proper decisions. I believe that the player who understands the why’ of certain plays has an advantage over the one who memo¬ rizes a strategy table. Any form of gambling has a built-in risk of potential loss but for those reckless enough to risk all on the turn of a card it carries the risk of ruin. This book strongly advises against the latter course and seeks to minimize loss. If you must gamble, be smart. Do not play those games in which the odds are heavily in favor of the casino—at least not on a regular basis. Blackjack, when played prop¬ erly, will have the least detrimental effect on your wallet. Played with skill (counting cards), blackjack can make you a consistent winner over the long run. In February, 1992 the Mashantucket Pequot Indian tribe opened their casino in southeastern Connecticut. In addition to using Blackjack rules which are as favorable as those in Atlantic City, the casino offers certain side bets at the Blackjack tables. Super Sevens may be found in

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Chapter XI. Under-Over is covered in Appendix A, MultiAction Blackjack in Appendix F. Many of the statistical values used herein were gener¬ ated on IBM mainframe 4300 series computers. In addi¬ tion, PC spreadsheets, e.g. Lotus, were used for the Under-Over analyses ✓ Data values in this book were verified with Monte Carlo methods in which millions of hands were generated for each specific case with hundreds of hours of computer time dedicated to this task. Appendix H is a detailed breakdown of 10,000,000 hands of Blackjack. The student of the game can learn a great deal from the analysis thus presented. Programs were written in PL/1 and Cobol, primarily for eight-deck play. A modification for one deck was intro¬ duced in order to check results against prior values gener¬ ated by other authors. Appendix G, “Some Personal Experiences,” contains recollections of incidences which have enriched (or damp¬ ened) the authors playing days in various casinos. The reader should find it entertaining as well as informative.

_I._ DEFINITIONS

NOTE:

Words such as 'dealer’, 'player’, 'casino’ are not

defined. Such definitions may be found in any dictionary. ABBREVIATIONS:

ACTION

A = ace; T = 10 (ten, jack, queen, or king)

The total amount wagered over a period of time.

For example, if a player risks $1000, his action would nor¬ mally be many times that amount, depending upon his luck.

ANCHOR MAN

The last player to make a decision. Usually

sits at dealer’s extreme right.

BLACKJACK

The hand contains only ace and ten. Also

called 'natural’.

BURN CARD

The first card of the shoe, placed in the dis¬

card pile.

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BUST, BREAK

BLACKJACK BOOK

The total of the hand, players or dealers, is

over 21.

CARD COUNTER

A player who keeps track of the cards

which have been played to determine whether the remain¬ ing shoe is favorable to the player or the house. See chap¬ ter on card counting.

CHIP

A unit of play, ranging in value from $1 to $5000,

depending upon the color of the chip. Players use chips to place bets, the chips being purchased prior to play.

COMPLIMENTARY

Normally referred to as ‘comp’, an

authorization from the pit boss to a casino restaurant for free food and drink for any player whose ‘action’ warrants being treated this way. Comps are not limited to food and drink, but include lodging, shows, transportation, gift-shop items, and anything else to attract so-called high rollers. In some casinos, the amount of the comp is computed by taking the average bet size times the number of hours of play, times one half. For example, a $20 average bet for four hours would warrant $40 in comps. Some casinos are more liberal, some not.

CREDIT The amount of money a player may borrow via the issuance of one or more ‘markers’ (see ‘Marker’). This money must be used at the casino games and is generally not available as an interest free loan. Credit may be revoked if the casino determines that a player has abused the privilege in this manner.

THE

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CUT THE DECK

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One of the players inserts a blank plastic

card into the deck after the shuffle. Those cards which are located in back of the plastic card are moved as one to the front of the deck.

DISCARD PILE

The cards which are collected by the deal¬

er after each round of play are placed to the side into the discard pile.

DOUBLE DOWN

On first 2 cards, double your bet, get only

1 additional card. Normally, the additional bet equals the original bet. However one may 'double with less’. The additional bet may not exceed the original bet.

EXPECTATION When expressed as a percentage, if positive, the amount which the player will win per $100 of action in the long run. Negative expectation indicates the amount lost in the long run. For example, -0.5% indicates 50 cents lost per $100 risked.

HARD TOTAL

An ace in the hand has been counted as one

and not eleven. Also, see 'soft total’. HARD 12:

5

6

A

* FIVE

♦ SIX

♦ ACE

HIGH ROLLER A casino s most desirable customer, the high roller makes bets which most players consider exorbitant. Usually, those earning the appellation of high roller can afford high losses.

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HOLE CARD

BLACKJACK

BOOK

The dealers face-down card. Also, see

‘upcardf

INSURANCE

If dealer shows ace, you may make a side bet

equal to one half your original bet that the hole card is a ten. If it is a ten, insurance pays two to one, i.e. an amount equal to the amount lost on the original bet, thus breaking even. The player with a natural, who buys insurance, will receive a net amount equal to his original bet. Casinos permit an insurance bet less than the original bet, in which case the payoff is still 2:1 if dealer has a natural.

MARKER

An IOU signed by the player whenever he

requests credit at the table, and receives the amount requested in the form of chips. At the time credit is estab¬ lished, player agrees to a maximum time limit—usually one or two weeks,sometimes a month—in which any cred¬ it extended will be repaid. The procedure is as follows: 1. Player requests credit at the table. 2. Pit boss calls the credit office, to determine whether there is enough available credit. If so, player signs a marker application, setting forth the amount and his birthdate (which serves as identification). 3. A clerk comes to the table, with a filled-in mark¬ er to be signed by the player. 4. Player is given chips in return for the IOU. 5. If player wins, the marker may be paid off with winning chips and given to the player to do with as he wishes.

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6. If player loses he buys the marker with a check in the amount of the loss. 7. Player may ask the casino to hold the check, up to the limit of time available, and player may then pur¬ chase the check any time prior to deposit. Until such time, available credit is reduced accordingly, but may be re¬ established within one week after the check clears.

PIT

An area of play, consisting of several gaming tables,

e.g. the Craps pit, the Blackjack pit, the Roulette pit.

PIT BOSS The person in charge of a pit’s activities and per¬ sonnel. The pit boss will determine whether a player gets a comp, and will be the final arbiter in any dispute.

PLASTIC CARD

A blank, usually yellow, card, used as a

delimiter within the shoe. When this card is reached, play continues to finish the deal, after which cards are reshuf¬ fled. This plastic card is normally inserted by the dealer after the shuffle and the cut, and located about one fourth of the complete shoe from the end. Its purpose is to pre¬ vent card counter advantage if dealer had to deal to the end of the deck. Sometimes, if the pit boss suspects card-counting activ¬ ity at the table, he may order the dealer to insert this plas¬ tic card further toward the front of the shoe, thus cutting more cards out of play.

PRESS THE BET inal bet.

Letting winnings ride, along with the orig¬

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PUSH, OR TIE

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BOOK

Player and dealer have the same total, and

neither one wins,

exception:

blackjack always beats 3-or-

more-card total of 21.

SHOE A box into which the shuffled cards are placed from which the dealer is able to slide the forward-most card onto the table to a player or to himself.

SOFT TOTAL

An ace in the hand is being counted as

eleven, instead of one. Also, see 'hard total’. SOFT 17:

SPLIT

6

A

♦ SIX

♦ ACE

If first two cards have equal value, you play two

hands, each hand starting with one of those cards. This requires placing an additional bet equal to the original bet on the second of the two hands. If aces are split, player will get only one additional card on each ace; if the second card is a ten, the total is counted as 21, but is not a Blackjack.

STAND

Do not deal any more cards to the hand.

STIFF HAND

A hand with a hard total of 12 thru 16.

STIFF HANDS:

13

7

8

* SEVEN

♦ EIGHT

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THE

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TABLE LIMIT OR MAXIMUM

BLACKJACK

BOOK

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21

The maximum amount which

may be wagered on one hand excluding the extra bet on insurance, doubling, or splitting.

TABLE MINIMUM

The minimum amount which must be

wagered on one hand.

TOTAL

The sum of the card values in a hand. Jack, queen,

or king is ten. Ace is counted as either 1 or 11 at player’s option. See hard total and soft total.

A

6

A

5

6

♦ ACE

♦ SIX

♦

* FIVE

♦ SIX

7 OR 17

ACE

12

3 JACK

8 * EIGHT

18

(player WOULD NEVER CHOOSE

TRAY

A compartmented tray, containing dealer’s chips,

arranged by value. Each value has its own distinguishing color.

UPCARD

The dealer’s card which is visible to players.

Also, see 'hole card’.

'

■

.

_IL_ RULES OF PLAY

P

rior to each deal, players

place bets on the table within a designated spot. Bets placed after the first card is dealt to any player are invalid. No player may alter or retrieve his bet after the first card has been dealt. The maximum number of players is usually seven. In most casi¬ nos, a player may play at more than one spot. (In some casinos, the minimum bet for a multiple-spot player is twice the posted minimum.) Cards are dealt out of a shoe that contains eight shuf¬ fled decks. Players and dealer are each dealt two cards. Player cards are normally face up. Only one of the dealer cards is face up. Players are not allowed to touch the cards. If the dealers upcard is an ace, he invites insurance wagers. Those players washing to take insurance place their insurance bet in a designated area in front of their

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cards.The player can insure up to but not more than an amount equal to half his wager. Each player, in turn, makes his decision(s) (see below). Any player who draws a card or cards causing his hard total to exceed 21 loses. His wager is immediately collect¬ ed by the dealer, and cards go into the discard pile. In the event player has doubled, split, and/or taken insurance and the dealer has an upcard of ace or ten, the lost chips are placed on top of the lost hand until dealer exposes his hole card. If dealer does not have Blackjack, lost chips are collected and the players cards discarded. If dealer has Blackjack, the extra chips placed for doubling and/or splitting are pushed back to the player and the orig¬ inal chips collected by dealer. The chips used for insurance are returned to the player, along with his original bet. In other words, he breaks even. No pain, no gain. If a player has Blackjack, and the dealers upcard is not ten or ace, the player is paid one and one half times his bet and his cards are collected and placed in the discard pile. After all players have been dealt to, the dealer turns over his hole card. If the dealers total is more than sixteen, he must stand. Otherwise, he continues to draw cards until his total exceeds sixteen. In some casinos, a rule unfavor¬ able to players requires the dealer hit a soft 17. If the dealer has Blackjack, he collects all bets from those players who do not have Blackjack, and pays off any insurance. If dealer does not have Blackjack, insurance bets are lost. If the dealer breaks, those players remaining in the game are paid an amount equal to their bet, including any doubles or splits.

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If the players total is higher than the dealer’s total, the player wins. If the dealers total exceeds the player’s total, the dealer collects the bet. In the event of a push, the deal¬ er will signify the fact by rapping the table with the back of his palm, and will leave the bet where it was. All cards are collected and placed into the discard pile, and betting begins for the next deal.

■

III. BASK STRATEGY DECISIONS (IN LOGICAL ORDER)

Values shown following Vs’ refer to Dealer Upcard.

NOTE:

1. You have Blackjack. Smile and enjoy the moment.

2.

Dealer shows ace. Do I take Insurance:

If you are a competent card counter and the shoe is ten-heavy, yes; otherwise, never.

3.

Should I Split (if the first two cards have equal

value): Always split if you have 8-8 or A-A; Never split if you have 5-5 or T-T; Split 2-2, 3-3, or 7-7 vs 2 thru 7; Split 4-4 vs 5,6, if doubling after split is permitted; Split 6-6 vs 3 thru 6; Split 9-9 vs 2 thru 9 but not vs 7.

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If you could not split, or after receiving second card on split: 4. Should I double: Do not double if your total is less than 9 or over 18. Double if your total is 11, and upcard is not ace. Double if your total is 10 vs any upcard except T,A; Double if total is 9 vs upcard of 4,5,6; Double A-(4 thru 7) vs 4; Double A-(3 thru 7) vs 5; Double A-(2 thru 7) vs 6; If you could not double:

5.

Should I hit:

Always hit if hard total is less than 12; Always hit if soft total is less than 18; Hit vs 7,8,9,T,A if total is less than 17; Hit vs 2,3 if total is less than 13; Hit soft total of 18 vs 9,T,A. Otherwise, stand.

SUMMARY OF BASIC STRATEGY FOR 8 DECKS D E AL. E R HARD

UPC AR D

2

3

4

5

6

7

8

9

T

A

5-8

Hit

Hit

Hit

Hit

Hit

Hit

Hit

Hit

Hit

Hit

9

Hit

Hit

Dbl

Dbl

Dbl

Hit

Hit

Hit

Hit

Hit

10

Dbl

Dbl

Dbl

Dbl

Dbl

Dbl

Dbl

Dbl

Hit

Hit

11

Dbl

Dbl

Dbl

Dbl

Dbl

Dbl

Dbl

Dbl

Dbl

Hit

12

Hit

Hit

Sta

Sta

Sta

Hit

Hit

Hit

Hit

Hit

13-16

Sta

Sta

Sta

Sta

Sta

Hit

Hit

Hit

Hit

Hit

17-21

Sta

Sta

Sta

Sta

Sta

Sta

Sta

Sta

Sta

Sta

D E AL, E R

UPC AR D

2

3

4

5

6

7

8

9

T

A

13

Hit

Hit

Hit

Hit

Dbl

Hit

Hit

Hit

Hit

Hit

14-17

Hit

Hit

Dbl* *Dbl **

Dbl

Hit

Hit

Hit

Hit

Hit

18

Sta

Sta

Dbl

Dbl

Sta

Sta

Hit

Hit

Hit

9

T

A

SOFT

Dbl

D E AL, E R

UPC AR D

3

4

5

6

7

8

Spl

Spl

Spl

Spl

Spl

Spl

Spl

Spl

Spl

Spl

22,33

Spl

Spl

Spl

Spl

Spl

Spl

Hit

Hit

Hit

Hit

44

Hit

Hit

Hit

Spl*

Spl*

Hit

Hit

Hit

Hit

Hit

55

Dbl

Dbl

Dbl

Dbl

Dbl

Dbl

Dbl

Dbl

Hit

Hit

66

Hit

Spl

Spl

Spl

Spl

Hit

Hit

Hit

Hit

Hit

77

Spl

Spl

Spl

Spl

Spl

Spl

Hit

Hit

Hit

Hit

99

Spl

Spl

Spl

Spl

Spl

Sta

Spl

Spl

Sta

Sta

TT

Sta

Sta

Sta

Sta

Sta

Sta

Sta

Sta

Sta

Sta

PAIR

2

AA,88

*If doubling after split is permitted; otherwise, hit. **Do not double soft 14 vs 4

_IV._ CASINOS ADVANTAGE

he casino’s advantage in Blackjack is based on the fact that the player may have lost the game before the dealer has completed his own hand. If player and dealer both bust (their hands exceed 21), the player loses. To mitigate this advantage, player has the following options: He may stand with any total; He may split cards; He may double on any two cards except Blackjack; He may take insurance (advantageous to a card counter);

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He receives 3:2 odds if he gets Blackjack; He may vary his bet; He may leave the table at any time; He may study to become a proficient card counter. Beat the Dealer (Edward O. Thorp s) is the classic work. It showed for the first time that the house loses its advantage when proper strategy is followed with a single deck. The casinos introduced the multiple-deck game, dealt from a shoe instead of the dealers hand. Although this provided some element of player confidence, by eliminating the dealer’s opportunity to cheat via sleight-of-hand, the house gained a positive advantage even against the best players, if they didn’t card-count. This advantage is as much as one half of one percent with eight decks, using basic strategy. With fewer decks, the advantage decreases to zero. (See chapter VI entitled ‘Some Useful Facts’.) Too little has been written about the casino’s principal advantage. This stems from player greed and impatience, combined with limited funds. Thus, casinos typically real¬ ize a return of seven percent and up at the Blackjack table due to the undisciplined play of their customers. For eveiy Blackjack player whose excessive bets have resulted in a one-time windfall, there are dozens of others who have gone home broke. However, player greed does not match that of the casi¬ no owners, who still begrudge the skilled player an oppor¬ tunity to beat the house. In order to play in Fas Vegas, skilled card counters must outwit trained casino personnel,

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whose job it is to recognize an obvious professional, and keep him from winning at their tables. Even the simply competent player, who sticks to basic strategy, and is at a disadvantage of a fraction of one per¬ cent, is plied with free liquor in order to affect his concen¬ tration. When the casino is so crowded that players find it diffi¬ cult even to get a seat at a table, the table minimum is raised periodically so as to increase the house return.

*

.

_V._ COUNTING (ARDS S

T

he casino industry owes

l much of its good fortune to the fact that players rightly believe that the game of Blackjack can be beaten if one 'counts cards’, a belief launched with the publication of Professor Thorp s Beat the Dealer in 1966. While there are a number of professional card coun¬ ters, not many make their living from playing Blackjack, and the vast number of would-be card counters provide the casinos with a windfall. The cause of their downfall is the belief that they are superior in skill and therefore able to place bets far beyond the amount suggested for their bankroll. The most proficient card counter will lose his stake if he overbets. It does not help to have a long-run edge when

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36 / THE ULTIMATE BLACKJACK BOOK

the long run was not reached because the player had to quit when his bankroll was depleted. There is a widely-shared belief that players cannot count when a shoe contains multiple decks. That is a falla¬ cy. Card-counting methods, including the one shown below, utilize a single number for a running count. The more sophisticated players will refine the count by an addi¬ tional count of particular cards and compute the count per deck (true count) using the number of decks in the discard pile. Each method simply increments one number.

A SAMPLE COUNTING METHOD Assign the number 1 to card values 2 thru 7 and -1 for 9, T, A. Value 8 is neutral (0). Therefore, for any given num¬ ber of decks a count of zero prevails at the start: 2 thru 7 occurs 24 times, 9, T, A occurs 24 times, using the basic 13 values in each deck. The player Adjusts the count’, adding 1 whenever 2 thru 7 shows and subtracting 1 whenever 9, ten, or ace appears. This results in one number which is positive, negative or zero. A positive count means that more low cards have been played, and the high cards predominate in the shoe. A neg¬ ative count, on the other hand, means that more high cards have been played, leaving the shoe with an excess of low cards to be played. Note that the count considers the ace a 'high’ card. More about this later. The experienced card counter makes use of the count in basically two ways: he adjusts his bet, and he adjusts his

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play. If the count is positive, he increases his bet. When the count is negative, the bet will be decreased, or the counter will leave the table. If the count is low, he will double down less often, and be more aggressive in drawing cards. When the count is' positive, he will stand more often, and expect to win when the dealer breaks with a stiff hand. He will realize that the deck is in his favor when doubling with 9,10 or 11, and be more confident of the dealer break¬ ing when he doubles with a soft total against the dealers 4,5, or 6. With multiple decks, an experienced card counter will adjust the count via a visual check of the discard pile which indicates how many decks have been played and thus how many decks remain to be played. For example, a count of 8 with four decks remaining, means that the 'true count’ per deck is only 2. The equa¬ tion is True count = Running count / Decks to be played. Note that even though the ace may be used as a 4ow’ card, value 1, it is lumped in with nines and tens. That is because the ace is the strongest card in the deck for the player. (Blackjack pays 3:2, busting is impossible when drawing an ace, and it is ideal when doubling on 9, 10, soft 17 and 18, or drawing to soft 17 and 18). Once an ace has been dealt it’s no longer available and the count should reflect this detriment. Many counters keep a separate count of aces. For instance, if +7 is the count and only four aces have been

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seen after three decks when normally an average of 12 aces should have appeared, then a separate count of aces would be the indicator that excess aces, and not excess tens, remain to be dealt. Play should be adjusted accordingly For example, insurance would not be called for in a situa¬ tion where a high count is due to an excess of aces. In the foregoing example, were the eight aces dealt, the count would have been -1 rather the +7. No method based upon statistics is foolproof. The low count, which indicates a preponderence of low cards remaining in the shoe, can’t warn you that a ten is the next card to be dealt to your stiff total. On the other hand, fate may decree that you get hit with a four when you double down on eleven with the count of +6, or that the dealer shows a four with a seven in the hole, drawing ten when the count is high. Furthermore, it is not impossible to have the nines and tens all bunched behind the plastic end-of-shoe card instead of ready for you when you make a large wager. All card counters have experienced the implausible—and suf¬ fered. However, they all realize that the statistics of the game include such undesirable results. The worst thing they have to deal with is the spouse who will use a streak of bad luck to deprecate card counting in general, and the counter in particular. For a more detailed analysis of the effects of card dis¬ tribution on the card counter, see chapter VIII on Fallacies. You should not try to count in the casino without prac¬ tice beforehand. Practice until you’re satisfied that your count is reasonably error-free.

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39

Take at least two decks. Then turn one card at a time, slowly at first and gradually picking up speed as you adjust the count one card at a time. When the end of the deck(s) is reached, your count should be zero. If not, repeat the procedure. Once mastered, the next step is for you to take groups of cards—two, three, and four at a time. It is actually easier to count groups of cards, since high cards cancel low ones without having to adjust the count. Seeing T, T, 3, 9, 4, 7, A, 2, T, 3, 5, 6, 8, 8, T in front of 7 players and the dealer, you pair off T,T,3,9,4,7 and forget those, then A,2,T,3 and forget those, and then add +1 for the 5,6,T to adjust the running count. It is also useful to take 6 decks and set up distinct piles so you can become proficient in judging the number of complete decks and partial deck in the discard pile. Don’t forget to subtract that number from the total number used, before applying the equation for the true count. Sometimes when you face a really fast dealer, it is not easy to keep up. Utilize the time when players are making their decisions and update your count. If unsure, fall back on basic strategy for the remainder of the shoe.

/

SOME USEFUL FACTS

1. The players long-run expectation with basic strategy versus 8 decks, Atlantic City rules, is -0.508 percent. Expectation may be further subdivided as follows:

0.00508

OVERALL EXPECTATION

-

From Stand or Draw

-0.06206

From Doubling

+0.03202

From Splitting

+0.00233

From Blackjacks

+0.02263

L The player s long-run expectation with basic strategy versus 4 decks, Atlantic City rules, is -0.434 percent. Expectation may be further subdivided as follows:

41

42 / THE ULTIMATE

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0.00434

OVERALL EXPECTATION

-

From Stand or Draw

-0.06210

From Doubling

+0.03263

From Splitting

+0.00240

From Blackjacks

+0.02272

3. If you start with the hard total shown at the left, and draw to 17 or more, the percent chance of busting is shown to the right of the total. The percentage changes whenever the total changes (8 decks). For example, when the dealer shows a 6, his chance of busting is about 42%, without considering the hole card.

Total %Bust

Total %Bust

Total %Bust

Total %Bust

A

17.32

2

34.72

6

41.98

10

21.28

14

55.54

3

38.31

7

25.98

11

20.53

15

58.42

4

39.80

8

23.73

12

48.44

16

61.59

5

41.52

9

22.78

13

52.02

**Blackjack not included These percentages are known as ‘Dealers Chances of Busting’. Note that if the dealer has a 5 or 6 showing, he has a better than 58 percent chance of reaching 17-21 without busting!

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43

Note also that if player stands with a stiff hand against the ace, his chance of losing the hand is better than 82 per¬ cent. But if he draws another card his chance of reaching 19, 20, or 21 is 3/13, or 23 percent, with a fair chance of winning. We are not considering dealer Blackjack, which gives you a loss in any'event. 4. The number of distinct dealer hands, reaching 17 thru Bust, is shown in Appendix B.

5. If you take insurance, out of 13 possible hole-cards, dealer will, on average, have Blackjack 4 times. In that case, your payoff is double. However, there are 9 chances where you lose! You win 8 units, and lose 9. Losing one unit in 13 insurance trials is a loss of 7.7 percent. Remember: casinos want you to take insurance, or they would not offer it. /

6. If you are tempted to deviate from basic, Appendix C’s table indicates the additional negative percent which the deviation is costing you in the long run. Some of these are so small as to be immaterial, but nevertheless are detri¬ mental.

7. The experienced card counter knows that the dealer has more advantage when the shoe contains fewer high cards or more low cards. One may deduce this fact by studying the table in Appendix D, bearing in mind that a normal 8-deck shoe will give the house an advantage of 0.0051 when playing basic strategy. Player expectation when 1, 2, 5, 6, or 7 cards of given value are removed from an 8-deck shoe is given in Appendix

Vi / THE ULTIMATE BLACKJACK BOOK

D. That table clearly indicates that the best card for the dealer is a 5, followed by a 4. Also, when the shoe contains fewer aces and tens, the player is at a distinct disadvantage. Therefore the card counter raises his bet when the cards left in the shoe contain a smaller proportion of fives or a larger proportion of tens and aces.

_VIE_ SOME GOOD ADVICE

D

o not bet more than 2 percent of your bankroll

on any hand (except when splitting or doubling). (See ref¬ erence 15 on optimal betting)

UNLESS you

are a card counter, avoid the temptation to

press the bet.

IF YOU do count cards, do not keep increasing the bet size when your luck is going badly with a good shoe, hoping to recoup your losses quickly. You may be very unpleasantly surprised when your losing streak continues.

DO NOT deviate from basic strategy by playing hunches.

46 / THE ULTIMATE BLACKJACK BOOK

DO NOT permit yourself to be aggravated by the misplays of other players, (see 'Some Common Fallacies’, below)

DO NOT persist in playing when it is obvious that you are having a bad day You are having a bad day when the fol¬ lowing occur with unrelenting regularity: Seven low cards, ranging in value from 2 thru 7, are dealt to seven players, then the dealer gets a ten; Dealers upcard is 4, 5, or 6, and the hole card is 7, 6, 5 respectively, and it usually happens after you have doubled on 10 or 11 and received a stiff total; Dealer’s upcard is 6, and you have soft-doubled on 13, 14, 15, or 16 and received a 10. Dealer’s hole card is ace; You have dutifully split 8’s versus a 10, and deal¬ er’s hole card is 9 or 10; The count is a high plus and you have taken insurance, and the hole card is a nine; You have split 8’s vs dealer’s 5, doubled down on each hand and sit with 20 and 20. The anchor man asks for a hit on 12 and busts with a ten. The deal¬ er’s hole card is 10, and he draws a six; You have lost five hands in a row at $100 per hand. You drop your bet to a minimum $25 and get Blackjack. You let it all ride and break drawing on 12 vs 2. Then dealer breaks; You keep busting with totals of 22; You finally give up your seat in disgust and watch your replacement win six games in a row.

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47

ALWAYS have enough cash on hand for splitting. Its frus¬ trating to have to stand with 8-8 versus four thru eight when splitting does so much to improve ones chances. If your bankroll doesn’t permit a split, then don’t play. Doubling, on the other hand, is not a valid consideration when figuring an adequate bankroll for one hand. If you cannot double after a split, because of lack of funds, just take a hit.

IF YOU

follow the above rules, without deviating, you

should cut your loss to less than one percent of your risk capital in the long run.

, >

✓

_VI1L_ SOME COMMON FALLACIES

1. Always insure a Blackjack. Since your hand has a ten, it is not as likely for the dealer to have a Blackjack than if, for example, you had A-9 or 9-9. Although it is true that you cannot lose, and you will win an amount equal to your original bet (no more) you lower your total win in the long run by taking insurance. CONSIDER:

Of thirteen card values, only four of them are

tens. As I explained earlier, on average, you will win the insur¬ ance bet 4 times out of thirteen. The four times you win, you will be paid 8 betting units. The nine times when deal¬ er does not have the ten your loss is 9 betting units. Out of 13 trials, you win 8 and lose 9. Thus, 13 betting units wagered is a loss of 7.69 percent.

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As a practical matter, there are exceptions to the rule about never taking insurance: a. If you are not a regular player, and you get Blackjack, and the dealer shows an ace, take the insurance, and be happy with a one-unit win instead of trying for one-and-ahalf. It is a disappointment when the dealer turns up a ten. A bird in the hand... b. If you have an unusually big bet down, take insurance and the even money, to avoid a possible push.

2. Always insure a total of twenty. This is especially bad advice if your twenty consists of two tens. Holding two tens reduces the chances for the dealer to have a ten in the hole. See (1). Unless you are counting, do not take insurance.

3. Dealers hole card is probably a ten.

Out of all ten

possible card values, the most likely is a ten. But that does not mean the hole card is probably ten. The ten occurs four times out of 13 possible values—a 30.8% chance. Remember, the casino is willing to give you 2:1 odds that the hole card is not a ten when it offers insurance. Therefore if the dealer has a ten up, relax—until he turns over another ten, or an ace. When the dealer has a six showing, players always assume that the dealer will break. Look at the table in paragraph VI-3. On upcard of six, dealer will only break about 42% of the time. He will break 62% of the time if he has a total of 16. A 10 as hole card occurs only 4 times out of 13. The odds are that he will not break with the six showing.

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4. When other players make mistakes, it always costs me. Not 'always’ although it seems that way. Players remember pain more vividly than gain. The only one who gets hurt, in the long run, is the player making the dumb moves. If he mistakenly draws the extra card and the deal¬ er wins instead of breaks, a 50-50 chance existed for those cards to have been in reverse order. On another occasion, you may be glad that the novice who made the mistake drew a card which would have made the dealer win even though the poor guy busted his own hand. Of course, when you have a big bet down, have split eights and drawn a deuce on each, have doubled and got 20 on each, then logic cannot explain why the guy at the end of the table took a 10 on 14 against a 6 and left the dealer a 5 for a total of 21. Had it worked the other way with third base getting the 5 and dealer busting with the 10, you would have been delirious, but you’d forget it sooner than if you had lost.

5. Hit sixteen against a ten, but not against any other card. It makes more sense to stand with sixteen vs ten than to

stand with

sixteen vs

7.

Refer to

Chapter X,

'Mathematics Behind the Proper Decisions’. In both instances you lose more by standing than drawing a card. However, if you are lucky and don’t break, your chances of beating a seven are better than beating a ten. If you stand, the dealer will break 21 percent of the time if he holds the ten, and 26 percent of the time holding a seven. But if you do not break, your chances of beating the seven are much better than your chances of winning when dealer shows ten.

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6. If things are going poorly, they will continue to go poorly. When things are going poorly, it is likely due to: a. The dealer is drawing tens and aces, and not busting. b. The dealer is not busting when he has a stiff (12-16). If (a) is true, take a break, because not only is luck against you, but the remaining cards will be preponderantly lowvalue, thus making (b) true in subsequent hands. If (b) is true stick around, because a future preponderance of high-value cards will likely make the dealer break more often.

7. The seat you choose has a definite bearing on the out¬ come. The best seat is_(you pick it). There will normally be a best seat, where the player loses the least or wins the most. However, before you sit down there is no way of predicting where that seat will be. After play is done, you will know where that seat was. If you are counting cards it is best to sit at either end. You will know the count when placing your bet, when the first card is dealt, if you sit to the left of the dealer, and your bet can reflect that count. When you sit at dealers right—third base—there will be fewer remaining cards and a more accurate count when you have to make a standor-draw decision. You will also have more time to adjust the count when your turn arrives to make that decision. One drawback: if you must deviate from basic strategy, based upon the count, and you cause the dealer to win, you will have to endure the hostile environment which your

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53

decision has generated among the other players, especial¬ ly if they believe fallacy (4) above.

8. Since dealing stops about 1

1/2 decks from the end

of the shoe, assuming 8 decks, it is futile to count cards. The cards which will not be played because they are behind the plastic indicator can have one of the following effects: a. If the count is negative and there are fewer high cards remaining than with a normal distribution, but most low cards are bunched behind the plastic end-of-shoe card, then the card counter will be betting low, whereas the deck to be played is actually favorable. Result: no pain, probable gain. b. If the shoe has a negative count, and the card counter is betting less, and the high cards are bunched behind the end-of-shoe card, the result will be lower bets but more losses than expected. Result: negligible. c. If the shoe has a positive count and more aces and high cards remain to be played, but those cards are bunched behind the end-of-shoe card, the counter will bet high but lose. Result: much pain. d. If the shoe has a positive count, and fewer high cards are behind the end-of-shoe card, the counter will place higher bets and win more often. Result: much gain.

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e. The cards are normally distributed in front of the endof-shoe card. Result: counting is valid. (a) and (b) will not have any great effect, because the bets are low. (c) represents a nightmare for the card counter. Luckily, this happens only 25% of the time. (d) can be costly, if the dealer keeps pulling tens and aces. Favorable expectations have gone awiy when the dealer keeps drawing that second ten. In the long run, high counts will win.

IX. REASONING BEHIND THE PROPER DECISIONS

11/

1/1/ e

now present the logic

1 1 behind many typical deci¬ sions made in the course of a Blackjack session. For a mathematical analysis of these decisions, see Chapter X 'Mathematics Behind the Proper Decisions’. The two basic decisions made during Blackjack play in a casino are the amount to be wagered on the hand, and the play of the hand. This chapter concerns itself with playing decisions.

STANDING OR DRAWING HARD TOTAL LESS THAN 12. Its obvious that any hard total of less than 12 should be hit, but it is surprising how often players will stand with an ace-five, for example. Their rationale is based upon two

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incorrect assumptions: either that the ace counts only as 11 and drawing might bust the hand, or by drawing a card, they might take the dealers bust card. The ace will only count as 11 if the player has a natur¬ al, or the player chooses to assign 11 as the value of the ace. Making a choice of value 1 or 11 should depend upon ones total with either choice. If choosing 1 as the value of the ace prevents a bust, then choose 1. For instance, hold¬ ing total 12 prior to receiving the ace mandates that the ace be counted as 1 for a total of 13 rather than 23 for a bust. On the other hand, with a total of 10 before receiv¬ ing an ace, you would choose the 11 for a total of 21 rather than opt for 1 for a total of 11. In the ace-five example cited, standing is pure folly. There is no way to worsen the hand by taking a hit, and you might draw a value of 2 thru 5, which makes the hand con¬ siderably better. As for the possibility of giving the dealer a better hand when you draw a card, consider this example: you hold ace-five, and the dealer shows a six. You decide to stand, dealer turns over a five, draws two, then ten and busts. The other players, who had implored you to hit your hard total of 6, heave a sigh of relief that you did not take the 2 which made the dealer bust. Now consider the other side of the coin: the cards in the shoe rather than being in the order 2,10 are in the order 10,2. You did not hit your ace-five. Dealer gets 21 and you get the evil eye from every player at the table. Had you taken a hit, your standing total would have been 16, dealer would have 13 and be drawing with a good chance for a bust.

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As far as the dealer is concerned, you may or may not improve his hand by standing on your hard six. Its a tossup. However, you should never guide your play by guessing what might happen. You should always take a hit if the next card cannot make your hand worse than it is. It just might improve it. ' Remember, always hit a hard total of less than 12.

HARD TOTAL GREATER THAN 16. You must never draw with a hard total of 17 or more. Your chances of bettering your hand are 4 out of 13, if you draw ace, 2, 3, or 4; but drawing 5, 6, 7, 8, 9, or 10 gives you a chance of busting of 9 out of 13. Nonetheless, we see players who do just that. Either they are expert card coun¬ ters who have accurately evaluated their chances with 17, and have concluded that they will lose less in the long run of similar situations and similar card counts by taking a hit; or they are hoping against hope to improve their hand without considering that by standing they will lose less in the long run. No matter what the reason, the expectation is a losing one if you take another card. Remember, never hit a hard total of more than 16.

HOLDING A HARD TOTAL GREATER THAN 11 AGAINST 4, 5 OR 6. Most players are not aware that when they face an upcard of 5 or 6, the dealer’s chance of busting is less than 50 percent. In fact it is normally less than 45 percent. Nevertheless, no total greater than 11 should be hit against the five or six. Never forget that once you bust, you have

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lost your bet, even though the dealer may subsequently bust. That fact, more than any other, accounts for the casi¬ no’s edge in Blackjack. Thus, when your chances of better¬ ing your hand by drawing are slim, such as when you hold a hard total greater than 11, you are better off, i.e. you will lose less in the long run, if you give the dealer a chance to bust before you do. That is true in the cases under discus¬ sion. It is not true when the upcard is greater than six. Then, dealers chance of busting is less than 30 percent and you should attempt to better your hand by taking a hit if you hold a total of less than 17. When facing a four with a total of 12, the decision is not so clear-cut. If you hold 12, dealer’s chance of busting with a 4 showing is about 40 per cent, so standing will cost you. If you draw a card to the 12, your chance of bettering your hand is only 5 out of 13 (drawing 5,6,7,8, or 9) or 38 per¬ cent. So drawing will also cost you. The mathematics of the situation favors standing by a very slight margin. If you feel that the shoe is 10-heavy then stand. If you know that a preponderance of tens have been played, then draw. The other players could be annoyed if you draw and ruin the deal, but you have to play your own game. If you hold more than 11, stand versus 4,5,6.

HOLDING A HARD TOTAL OF 12 VERSUS 2 OR 3. The mathematics demand drawing another card. However, it’s a close call. If you know that the shoe is tenheavy, stand. It will make little difference in the long run. As a general rule, hit 12 against a 2 or 3.

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)9

SPLITTING CARDS SPLITTING ACES Two aces should always be split. The chance of improv¬ ing the hand when starting with eleven is much, much bet¬ ter than when starting with soft twelve (or hard two). In addition, chances of beating the dealer in both of the split hands are excellent because of the possibility of drawing at least one ten, as well as having an ace in the hand with advantage of a soft total. Always split aces.

SPLITTING EIGHTS Two eights should always be split because playing a total of 16 will lose more in the long run than playing two hands, each starting with hard eight. Do not be talked into standing with the two eights, or drawing to the 16 by a so-called authority on casino games, who maintains that risking more money is not warranted when facing nine, ten, or ace with an eight because those are losing games. What he does not consider is that the dealers hole card is probably not a ten, and certainly a ten under the ace loses only one bet in either case. Splitting the eights against a nine or ten will cause you to lose more often than win; but playing the sixteen by not splitting will result in more money lost in the long run. Always split eights.

SPLITTING FOURS If doubling after splitting is permitted, then fours should be split against an upcard of five or six. The math-

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ematics are such that, in the long run, more profit is made with the hands starting with four, than with the single hand with total of 8 because of the possibility of drawing five, six, or seven and doubling down. If doubling down is not per¬ mitted, you’re better off not splitting. Split 4’s against five or six only if you may double later.

SPLITTING FIVES This rates a classification of pure foolishness. A total of ten on two cards should lead to doubling down against 2 thru nine, or drawing against ten or ace with good winning chances. The fool who splits fives is no doubt very happy if the dealer’s upcard is five. Then why play two games with that starting card? Never split fives. Even skilled card counters will not split fives.

SPLITTING NINES AGAINST A SEVEN Since a total of eighteen will usually beat the seven upcard don’t risk a stiff hand by splitting nines. You might draw three thru seven and worsen the hand. Or you may be surprised when the dealer has a three or four in the hole. Stand with eighteen against 7 and win more in the long run.

SPLITTING TENS This is the hallmark of the greedy novice, and should not be practiced. Two tens add up to twenty—a winning total in all but very few instances. And in those instances

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where twenty is not enough to win, certainly starting with ten will not win. Card counters will sometimes split tens against a five or six. Be assured that the shoe is very ten-heavy. Even then, the counter might prefer to be hit with a non-ten, since he would rather save the'tens for the dealers hand. The novice, on the other hand, hopes for the tens, even though leaving low cards for the dealer might be disastrous. Chance of getting 9, T, or A on each of the hands is about 1:5. Never split tens.

OTHER SPLITTING DECISIONS

Where splitting is recommended, analysis of expecta¬ tion with and without splitting proves that splitting either loses less or wins more in the long run than playing the sin¬ gle hand. One of the factors to consider is whether dou¬ bling is permitted on the split hand. Check the casino rules where you are playing. If you cannot double after the split, it may be better to hit or to stand.

DOUBLING DOWN You will normally win more often if you just draw a card and preserve your right to draw another card or cards. For example if you double on a total of nine and draw a two, or if you are doubling eleven versus ten and draw 3, 4, or 5. However, there are distinct advantages to doubling, even if this results in fewer actual wins. Consider the fol¬ lowing example: if you draw your probability of winning is 60 percent, but if you double, your probability is only 56

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percent. In the former case, you win a net 20 units (60 minus 40), but in the doubling case you win a net 24 units (2 x (56 - 44)). In general, double if the probability of winning exceeds fifty percent with a single additional card. However, in those cases which have an overwhelming probability of winning, for example holding ace-nine versus six, do not risk a clear win by taking an additional card. A bird in the hand... Strategy tables have analyzed the doubling cases. Use them.

✓

_I_ MATHEMATICS BEHIND THE PROPER DECISIONS

T

he logic behind all strategy

A decisions made in Blackjack is a table of dealer probability for reaching given totals starting with a given upcard. One could generate hundreds of millions of such tables, with different values dependent upon the players drawn cards, and the remaining cards in the shoe. Fortunately, the disparities between tables are minor, and I am confident in selecting a representative table on which to base decisions for each of the situations encountered in Blackjack play. Let’s make the following assumptions: a. There are five decks remaining in the shoe when dealer draws his remaining cards. In an 8deck shoe, with two decks cut away, that is a good average number. Tables for five and six decks are presented for comparison (see Appendix E).

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b. Ignore the players cards. To get an idea of the effect of that assumption, Appendix E shows tables wherein player holds 1) a ten and a two and 2) 5 and 6. The computer analyses, using given assumptions, are exact and are based upon dealer drawing a maximum of eight cards. After presenting four tables for comparison purposes, we then shall make simplifying assumptions and present a final table for the purpose of making our strate¬ gy decisions. In the tables of Appendix E, dealer naturals are excluded. Since differences between tables are of the order of 1-2 percent, we adopt table 1 for further discussion, with values rounded as shown below:

TABLE A: DEALER PROBABILITIES WITH UPCARD AS SHOWN (8 DECKS). 17

18

19

20

21

BUST

1

1878

1892

1896

1902

0774

1659

2

1398

1344

1307

1245

1189

3518

3

1344

1307

1257

1214

1154

3725

4

1304

1239

1218

1170

1125

3944

5

1201

1224

1183

1122

1085

4186

6

1664

1043

1069

1021

0978

4226

7

3678

1385

0779

0791

0742

2626

8

1295

3585

1297

0686

0698

2441

9

1208

1171

3508

1212

0602

2299

10

1218

1215

1223

3667

0379

2298

REACH: UPCARD

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Using table A, each of our decisions will be optimum under the assumptions given.

A SUMMARY OF THE PROOFS IS GIVEN HERE: (expectations in parentheses) ✓

1. Stand or draw holding 16 vs dealers 10: Draw (-0.5389 vs -0.5404) 2. Stand or draw holding 16 vs dealers 9: Draw (-0.5090 vs -0.5402) 3. Stand or draw holding 16 vs dealers 8: Draw (-0.4586 vs -0.5120) 4. Stand or draw holding 16 vs dealers 7: Draw (-0.4150 vs -0.4749) 5. Stand or draw holding 12 vs dealers 4: Stand (-0.2112 vs -0.2145) Note here that deviating from the correct strategy is less costly with 16 vs 10 than with holding 16 vs 7, 8, or 9. Also, standing with 12 vs 4 is an extremely close call.

DETAIL ANALYSES ARE SHOWN RELOW: (refer to Appendix E) DECISION 1: Stand or draw holding 16 vs dealers 10: The results are: Stand, expectation is -0.5404 Draw, expectation is -0.5389

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Extremely close! (slightly better to draw) The mathematics are as follows: From the above table, dealer’s chances of reaching 17,18,19,20,21,Bust are, respectively, .1218, .1215, .1223, .3667, .0379, .2298. a) Player stands If player stands, his chances of reaching 17-bust are all zero. His expectation is then computed as follows: E = -(.1218 + .1215 + .1223 + .3667 + .0379) + 0.2298

0.5404

- -

The positive value of 0.2298 represents dealer busting. All others represent player losing. b) Player draws one card (any further draws would be worse). Players chance of getting any one of values 1 thru 9 is approximately 1 out of 13 (0.0769), and his chance of drawing a 10 is approximately 4 out of 13 (0.3077). Then, player’s probabilities of reaching 17-bust are, respectively, 0.0769, 0.0769, 0.0769, 0.0769, 0.0769, 0.6154. His expectation is then computed as follows: PLAYER DEALER HAS HAS 17-21

Bust

+0.0769(.1218+.1215+.1223+.3667)

21

17,18,19,20

+0.0769(.1218+.1215+.1223-.0379)

20

17,18,19,21

+0.0769(.1218+.1215-.3667-.0379)

19

17,18,20,21

+0.0769(.1218-.1223-.3667-.0379)

18

17,19,20,21

-0.0769(. 1215+. 1223+.3667+.0379)

17

18,19,20,21

-0.6154(1)

Bust

17-Bust

0.0769(.2298)(5)

-

0.5389

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DECISION 2: Stand or draw holding 16 vs dealers 9: The results are: Stand, expectation is -0.5402 Draw, expectation is -0.5090 Note that this call to draw is a bit more clearcut than stand or draw vs dealer 10. The mathematics are as follows: From the above table, dealers chances of reaching 17,18,19,20,21,Bust are, respectively, .1208, .1171, .3508, .1212, .0602, .2299. a) Player stands If player stands, his chances of reaching 17-bust are all zero. His expectation is then computed as follows: E = -(.1208 + .1171 + .3508 + .1212 + .0602) + 0.2299 =

0.5402

-

The positive value of 0.2299 represents dealer busting. All others represent player losing. b) Player draws one card (any further draws would be worse). The analysis is similar to 16 vs 10. His expectation is then computed as follows: PLAYER DEALER HAS HAS 0.0769(.2299X5)

17-21

Bust

+0.0769(. 1208+. 1171+ .3508+. 1212)

21

17,18,19,20

+0.0769(. 1208+. 1171 + .3508-.0602)

20

17,18,19,21

+0.0769(.1208+.1171-.1212-.0602)

19

17,18,20,21

+0.0769(.1208-.3508-.1212-.0602)

18

17,19,20,21

-0.0769(.l 171+ .3508+.1212+.0602)

17

18,19,20,21

Bust

17-Bust

-0.6154(1) 0.5090

= -

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DECISION 3: Stand or draw holding 16 vs dealers 8: The results are: Stand, expectation is -0.5120 Draw, expectation is -0.4586 Note that this call to draw is more clearcut than stand or draw vs dealer 10 or dealer 9. The mathematics are as follows: From the above table, dealers chances of reaching 17,18,19,20,21,Bust are, respectively, .1295, .3585, .1297, .0686, .0698, .2441. a) Player stands If player stands, his chances of reaching 17-bust are all zero. His expectation is then computed as follows: E = -(.1295 + .3585 + .1297 + .0686 + .0698) + 0.2441

0.5120

= -

The positive value of 0.2441 represents dealer busting. All others represent player losing. b) Player draws one card (any further draws would be worse). The analysis is similar to 16 vs 10. His expectation is then computed as follows: PLAYER DEALER HAS HAS 0.0769(.2441)(5)

17-21

Bust

+0.0769(. 1295+.3585+. 1297+.0686)

21

17,18,19,20

+0.0769(. 1295+.3585+. 1297-.0698)

20

17,18,19,21

+0.0769(. 1295+.3585- .0686-.0698)

19

17,18,20,21

+0.0769(. 1295-. 1297-.0686-.0698)

18

17,19,20,21

-0.0769(.3585+. 1297+.0686+.0698)

17

18,19,20,21

Bust

17-Bust

-0.6154(1) -

0.4586

-

THE

ULTIMATE

BLACKJACK

BOOK

/

69

DECISION 4: Stand or draw holding 16 vs dealer’s 7: The results are: Stand, expectation is -0.4749 Draw , expectation is -0.4150 Note that this call to draw is much more clearcut than stand or draw vs dealer" 10 or 9 or 8! The mathematics are as follows: From the above table, dealers chances of reaching 17,18,19,20,21,Bust are, respectively, .3678, .1385, .0779, .0791, .0742, .2626. a) Player stands If player stands, his chances of reaching 17-bust are all zero. His expectation is then computed as follows: E = -(.3678 + .1385 + .0779 + .0791 + .0742) + 0.2626 =

0.4749

-

The positive value of 0.2626 represents dealer busting. All others represent player losing. b) Player draws one card (any further draws would be worse). The analysis is similar to 16 vs 10. His expectation is then computed as follows: PLAYER DEALER HAS HAS 17-21

Bust

+0.0769(.3678+. 1385+.0779+.0791)

21

17,18,19,20

+0.0769(.3678+. 1385+.0779-.0742)

20

17,18,19,21

+0.0769(.3678+. 1385-.0791-.0742)

19

17,18,20,21

+0.0769(.3678-.0779-.0791-.0742)

18

17,19,20,21

-0.0769(. 1385+.0779+.0791 + .0742)

17

18,19,20,21

Bust

17-Bust

0.0769(.2626)(5)

-0.6154(1)

0.4150

-

70

/ THE

ULTIMATE

BLACKJACK BOOK

DECISION 5: Stand or draw holding 12 vs dealers 4: The results are: Stand, expectation is -0.2112 Draw, expectation is -0.2145 Note that this call is closer than commonly believed! The mathematics are as follows: From the above table, dealers chances of reaching 17,18,19,20,21,Bust are, respectively, .1304, .1239, .1218, .1170, .1125, .3944. a) Player stands If player stands, his chances of reaching 17-bust are all zero. His expectation is then computed as follows: E = -(.1304 + .1239 + .1218 + .1170 + .1125) + 0.3944

0.2112

- -

The positive value of 0.3944 represents dealer busting. All others represent player losing. b) Player draws one card (any further draws would be worse). The analysis is similar to 16 vs 10, except we must consider player’s reaching 13 thru 16. His expectation is then computed as follows: PLAYER DEALER HAS HAS 0.0769(.3944)(9)

-

13-21

Bust

+0.0769(.1304+.1239+.1218+.1170)

21

17,18,19,20

+0.0769(. 1304+. 1239+.1218-. 1125)

20

17,18,19,21

+0.0769(. 1304+. 1239-.1170-. 1125)

19

17,18,20,21

+0.0769(.1304-.1218-.1170-.1125)

18

17,19,20,21

-0.0769(.1239+.1218+.1170+.1125)

17

18,19,20,21

-0.3077(1.0-.3944)

13-16

17-21

-0.3077(1)

Bust

17-Bust

0.2145

-

✓

IX. ANALYSIS OF THE SUPER SEVENS SIDE BET

I ertain Blackjack tables have V a curved strip on the layout, indicating that Super Sevens is available as a side bet. Prior to any regular game, the player may place a $1 side bet in a circle so designated in front of his regular bet. If the first card dealt to his hand is 7, player receives $3 in addition to the return of his $1 bet. Any other card loses the $1 bet. If the second card is another 7 of a different suit, play¬ er has won $50 and has the option of drawing for the third 7, which could net him $500. If the second card is another 7, of the same suit, player has won $100, and may opt for another 7 of the same suit, which would net the grand prize of $5,000. If the 3rd card is a non-matching 7, the win is a net $500.

71

11

/ THE ULTIMATE BLACKJACK BOOK

If player, because of strategy considerations, opts a split of the two sevens, he keeps the $50 (or $100), but forfeits the right to try for a third seven. The author was a witness to the following play at Foxwoods in the summer of 1994. A lady had received two sevens of spades after hav¬ ing placed a $1 bet on the Super Sevens. The deal¬ er’s upcard was a ten which demanded she draw a card. Prior to her play, the player to her right had asked for a hit and received—you guessed it—a seven of spades! The lady groaned and turned away from the table and took a few steps from the action and just stood there staring at the cards. When she returned, dealer dealt her another seven of spades! All the action was stopped, as everyone waited for the pit boss to bring out the necessary papers to complete IRS requirements and, finally, to award her the $5,000. The winner was too dumbfounded to speak. She threw the dealer a $25 chip as tip, cra¬ dled the winning chips and left. As might be expected, the Sevens side bet is a sucker bet for the unknowing, with a negative expectation of over 14% at the start of an 8-deck shoe. However, for those with patience and a desire to beat this game, the table given below indicates the point at which it might be wise to play. The only count required is the number of sevens which have been dealt to any point of play. In addition, one must be able to determine, to a reasonable degree of accuracy, the number of decks in the discard pile, preferably to the nearest tenth. Some practice at home helps.

THE ULTIMATE

BLACKJACK BOOK

/

73

EXPLANATION OF TABLE AND PLAYER GUIDELINES FOR WINNING: First line of the table displays expectations prior to first deal. Column 4 is the player’s expectation in percent. Columns 5 thru 9 are the fractional expectation values for each of the above five wins (7, 2 non-match, 3 non-match, 2 matching, 3 matching). When they are added together and subtracted from 1, the result is the overall expectation. If the value in columns 5 thru 9 is divided by the pay¬ off, result is the probability of getting that payoff. For example, at the outset, the probability of getting 3 samesuit sevens may be calculated as 0.099/5000, or about 20 chances in a million. The lines following the first line give, in columns 1 to 3, player guidelines for placing a bet when the expectation is positive. The value in column 2 is the number of sevens which have been seen at any stage. In order to have a minimum positive expectation (shown in column 4 on that line), the number of discarded decks must not be less than the number shown in column 1. Column 3 gives the ‘shortage’ of sevens in the discard pile (normal number less number actually seen). For example, assume you have seen six sevens at any stage of play. Refer to the 6 in column 2, line 8. Now check the discard pile and refer to column 1. The discard indica¬ tor is 2.009. Thus, if the discard pile is less than 2 decks, do not place a bet. If it were greater than 2 decks, you would have about an even chance (given in column 4). The more cards in the discard pile exceeding 2 decks, the higher your expectation.

74 / THE ULTIMATE

BLACKJACK BOOK

NUMBER OF DECKS 8 COLUMNS: 5

6

7

8

9

0.00 14.386

0. 231

0.226

0.170

0.132

0.099

2,30

0.000

0. 249

0.263

0.213

0.153

0.124

1

2.25

0.001

0. 249

0.264

0.214

0.153

0.123

1.052

2

2.21

0.000

0. 249

0.264

0.214

0.153

0.122

1.291

3

2.17

0.001

0. 249

0.265

0.215

0.152

0.120

1.530

4

2.12

0.000

0. 250

0.266

0.216

0.152

0.119

1.770

5

2.08

0.001

0. 250

0.267

0.217

0.152

0.117

2.009

6

2.03

0.000

0. 250

0.268

0.218

0.151

0.115

2.248

7

1.99

0.003

0. 251

0.269

0.219

0.151

0.113

2.487

8

1.95

0.003

0. 251

0.270

0.220

0.150

0.111

2.726

9

1.90

0.003

0. 252

0.272

0.221

0.150

0.109

2.965 10

1.86

0.000

0. 252

0.273

0.222

0.149

0.106

3.204 11

1.82

0.001

0. 253

0.275

0.224

0.148

0.103

3.443 12

1.77

0.000

0. 253

0.276

0.225

0.147

0.100

3.682 13

1.73

0.000

0. 254

0.278

0.227

0.147

0.097

3.921 14

1.68

0.002

0. 255

0.281

0.229

0.146

0.093

4.160 15

1.64

0.001

0. 255

0.283

0.231

0.144

0.088

4.399 16

1.60

0.002

0. 256

0.286

0.234

0.143

0.084

4.638 17

1.55

0.001

0. 257

0.289

0.237

0.141

0.078

4.876 18

1,51

0.004

0. 259

0.293

0.240

0.140

0.071

5.115 19

1.46

0.000

0. 260

0.297

0.244

0.137

0.064

5.353 20

1.41

0.004

0. 262

0.303

0.249

0.135

0.055

5.591 21

1.36

0.003

0. 263

0.309

0.254

0.131

0.045

5.829 22

1.32

0.007

0. 266

0.317

0.261

0.127

0.033

6.066 23

1.26

0.000

0. 268

0.326

0.270

0.121

0.018

6.301 24

1.21

0.004

0. 272

0.338

0.281

0.113

0.000

1

2

0.000

0

0.574

0

0.813

4

3 -

V

/ t

APPENDIX A:

THE UNDER-OVER SIDE BET

I

n February, 1992, the Mashantucket Indian tribe, located

in the southeast comer of Connecticut, established a casino in the area of Ledyard, Connecticut. Permission to offer table games was granted after long drawnout legal battles in the courts. The application was approved because many forms of gambling were available at charity-sponsored events throughout the state. The casino, known as Foxwoods, has since been crowd¬ ed seven days per week, 24 hours per day. It offers Bingo, Poker, Slot machines, Blackjack, Craps, Baccarat, and other games. At some Blackjack tables, the player may place a side bet up to the original bet, that his first two cards will add up to a total of under or over 13, with ace counted as one. A total of exactly 13 loses.

75

76

/ THE

ULTIMATE

BLACKJACK BOOK

As shown below, the bet is very disadvantageous to the player with the house percentage for five decks remaining ranging from 10 percent for ‘Under’ to 6.5 percent for ‘Over’. A proficient card counter may watch for player advan¬ tage by tracking card values A,2,3 and 9,T. Even when the shoe is heavy with aces and twos, if you want to bet ‘Under’ with 4 decks remaining, you would need 11 extra cards of either value to result in a 0.733% edge. Conversely, if you wish to bet ‘Over’ with 4 decks remaining, 11 extra tens would yield a 6% edge to the player. For the unsuspecting novice who places an Under-Over bet with the first hand dealt from 8 decks, adding twenty extra tens, or 20 extra aces, will not yield an advantage. The tables shown on the following pages indicate a very few favorable situations which rarely occur. For example, adding 3 aces or 3 deuces to a single deck will result in a positive expectation for the player when playing ‘Under’. Also, adding 4 tens will yield a positive advantage with one deck when playing ‘Over’. However, with multiple decks, favorable situations appear so rarely so as to warrant this game being called a ‘sucker bet’. For example, using five decks would require 13 extra aces or deuces to obtain an advantage with ‘Under’. Some suggested counting methods are shown following the tables.

THE ULTIMATE

BLACKJACK BOOK

/

77

In the tables below, Columns 1 thru 10 give the shoe composition from ace thru 10. Column 11 is total cards per shoe. Columns 12, 13 give probability for over or under, respectively. Columns 14, 16 give the respective expectations. Column 15 gives the increment in expectation due to a change in shoe composition from normal (normal is 1st line), for 'Over’. For example, adding one 7 to a one-deck shoe changes expectation for 'Over' from -6.787 to -6.241; adding 1 ace changes ‘Under’ expectation from -10.106 to -5.951.

A

234

5

6789T

1 deck 4

Diff Under%

Effect of adding one card of each value

4

4

16

52 0.466 0.449

5

44444444

16

53 0.448 0.470 -10.305 -3.52

-5.951

4

5

4

16

53 0.448 0.470 -10.305 -3.52

-5.951

4

45444444

16

53 0.448 0.459 -10.305 -3.52

-8.273

4

44544444

16

53 0.460 0.456

-7.983 -1.20

-8.853

4

44454444

16

53 0.463 0.453

-7.402 -0.61

-9.434

4

44445444

16

53 0.466 0.450

-6.821 -0.03 -10.015

4

4

4

16

53 0.469 0.447

-6.241

0.55 -10.595

4

44444454

16

53 0.472 0.444

-5.660

1.13 -11.176

4

44444445

16

53 0.475 0.441

-5.080

1.71 -11.756

4

4

17

53 0.478 0.438

-4.499

2.29 -12,337

4

4

4

4

4

4

Under Over%

4

4

4

DK Over

4

4

4

4

4

4

4

4

4

5

4

4

4

4

4

4

-6.787

-10.106

A

78

/ THE ULTIMATE

2

3

4

5

6

1 deck

7

8

9

T

BLACKJACK BOOK

DK Over

Under Over%

Diff Under%

Effect of adding two cards of each value

4

4

4

4

4

4

4

4

4

16

52 0.466 0.449 . -6.787

-10.106

6

4

4

4

4

4

4

4

4

16

54 0.432 0.490 -13.627 -6.84

-2.027

4

6

4

4

4

4

4

4

4

16

54 0.432 0.490 -13.627 -6.84

-2.027

4

4

6

4

4

4

4

4

4

16

54 0.432 0.468 -13.627 -6.84

-6.499

4

4

4

6

4

4

4

4

4

16

54 0.454 0.462

-9.154 -2.37

-7.617

4

4

4

4

6

4

4

4

4

16

54 0.460 0.456

-8.036 -1.25

-8.735

4

4

4

4

4

6

4

4

4

16

54 0.465 0.451

-6.918 -0.13

-9.853

4

4

4

4

4

4

6

4

4

16

54 0.472 0.444

-5.660

1.13 11.111

4

4

4

4

4

4

4

6

4

16

54 0.477 0.439

-4.542

2.25 -12.229

4

4

4

4

4

4

4

4

6

16

54 0.483 0.433

-3.424

3.36 -13.347

4

4

4

4

4

4

4

4

4

18

54 0.488 0.428

-2.306

4.48 -14.465

-

1 deck

Effect of adding three cards of each value

4

4

4

4

4

4

4

4

4

16

52 0.466 0.449

7

4

4

4

4

4

4

4

4

16

55 0.416 0,508 -16.768 -9.98

1.684

4

7

4

4

4

4

4

4

4

16

55 0.416 0.508 -16.768 -9.98

1.684

4

4

7

4

4

4

4

4

4

16

55 0.416 0.476 -16.768 -9.98

-4.781

4

4

4

7

4

4

4

4

4

16

55 0.448 0.468 -10.303 -3.52

-6.397

4

4

4

4

7

4

4

4

4

16

55 0.457 0.460

-8.687 -1.90

-8.013

4

4

4

4

4

7

4

4

4

16

55 0.465 0.452

-7.071 -0.28

-9.630

4

4

4

4

4

4

7

4

4

16

55 0.475 0.442

-5.051

1.74 -11.650

4

4

4

4

4

4

4

7

4

16

55 0.483 0.434

-3.434

3.35 -13.266

4

4

4

4

4

4

4

4

7

16

55 0.491 0.426

-1.818

4.97 -14.882

4

4

4

4

4

4

4

4

4

19

55 0.499 0.418

-0.202

6.59 -16.498

-6.787

-10.106

THE

2

3

4

5

6

1 deck

ULTIMATE

7

8

9

T

BLACKJACK BOOK

DK Over

Under Over%

/

79

Diff Under%

Effect of adding four card of each value

4

4

4

4

4

4

4

4

16

52 0.466 0.449

-10.106

4

4

4

4

4

4

4

4

16

56 0.401 0,526 -19.740 -12.95

5.195

8

4

4

4

4

4

4

4

16

56 0.401 0.526 -19.740 -12.95

5.195

4

8

4

4

4

4

4

4

16

56 0.401 0.484 -19.740 -12.95

-3.117

4

4

8

4

4

4

4

4

16

56 0.443 0.474 -11.429

-4.64

-5.195

4

4

4

8

4

4

4

4

16

56 0.453 0.464

-9,351

-2.56

-7.273

4

4

4

4

8

4

4

4

16

56 0.464 0.453

-7.273

-0.49

-9.351

4

4

4

4

4

8

4

4

16

56 0.478 0.439

-4.416

2.37 -12.208

4

4

4

4

4

4

8

4

16

56 0.488 0.429

-2.338

4.45 -14.286

4

4

4

4

4

4

4

8

16

56 0.499 0.418

-0.260

6,53 -16.364

4

4

4

4

4

4

4

4

20

56 0.509 0.408

1.818

8.61 -18.442

-6.787

2 decks Effect of adding one card of each value

-10.082

8

8

8

8

8

8

8

8

32 104 0.467 0.450

-6.647

8

8

8

8

8

8

8

8

32 105 0.458 0.460

-8.425 -1.78

-7.985

9

8

8

8

8

8

8

8

32 105 0.458 0.460

-8.425 -1.78

-7.985

8

9

8

8

8

8

8

8

32 105 0.458 0.454

-8.425 -1.78

-9.158

8

8

9

8

8

8

8

8

32 105 0.464 0.453

-7.253 -0.61

-9.451

8

8

8

9

8

8

8

8

32 105 0.465 0.451

-6.960 -0.31

-9.744

8

8

8

8

9

8

8

8

32 105 0.467 0.450

-6.667 -0.02 -10.037

8

8

8

8

8

9

8

8

32 105 0.468 0.448

-6.374

0.27 -10.330

8

8

8

8

8

8

9

8

32 105 0.470 0.447

-6.081

0.57 -10.623

8

8

8

8

8

8

8

9

32 105 0.471 0.445

-5.788

0.86 -10.916

8

8

8

8

8

8

8

8

33 105 0.473 0.444

-5.495

1.15 -11.209

80 A

2

/ THE ULTIMATE

3

4

5

6

2 decks

7

8

9

T

BLACKJACK BOOK

DK Over

Under Over%

Diff Under%

Effect of adding three card of each value

8

8

8

8

8

8

8

8

32 104 0.467 0.450

11 8

8

8

8

8

8

8

8

32 107 0.441 0.480 -11.832 -5.19

-3.968

8

8 11 8

8

8

8

8

8

32 107 0.441 0.463 -11.832 -5.19

-7.353

8

8

8 11 8

8

8

8

8

32 107 0.458 0.459

-8.446 -1.80

-8.200

8

8

8

8 11 8

8

8

8

32 107 0.462 0.455

-7.600 -0.95

-9.046

8

8

8

8

8 11 8

8

8

32 107 0.466 0.451

-6.754 -0.11

-9.892

8

8

8

8

8

8 11 8

8

32 107 0.471 0.446

-5.801

0.85 -10.845

8

8

8

8

8

8

8 11 8

32 107 0.475 0.442

-4.955

1.69 -11.691

8

8

8

8

8

8

8

8 11 32 107 0.479 0.437

-4.109

2.54 -12.537

8

8

8

8

8

8

8

8

-3.262

3.38 -13.384

8

8

35 107 0.484 0.433

-6.647

-10.082

2 decks Effect of adding six cards of each value

8

8

8

8

8

8

8

8

8

32 104 0.467 0.450

14 8

8

8

8

8

8

8

8

32 110 0.417 0.508 -16.597

0.00

1.651

8 14 8

8

8

8

8

8

8

32 110 0.417 0,508 -16.597 -9.95

1.651

8

8 14 8

8

8

8

8

8

32 110 0.417 0.476 -16.597 -9.95

-4.754

8

8

8 14 8

8

8

8

8

32 110 0.449 0.468 -10.192 -3.55

-6.355

8

8

8

8 14 8

8

8

8

32 110 0.457 0.460

-8.590 -1.94

-7.957

8

8

8

8

8 14 8

8

8

32 110 0.465 0.452

-6.989 -0.34

-9.558

8

8

8

8

8

8 14 8

8

32 110 0.476 0.442

-4.887

1.76 -11.660

8

8

8

8

8

8

8 14 8

32 110 0.484 0.434

-3.286

3.36 -13.261

8

8

8

8

8

8

8

8 14 32 110 0.492 0.426

-1.685

4.96 -14.862

8

8

8

8

8

8

8

8

-0.083

6.56 -16.464

8

38 110 0.500 0.418

-6.647

-10.082

THE ULTIMATE

A

2

3

4

5

3' decks

6

7

8

9

T

BLACKJACK BOOK DK

Over

Under Over%

/

81

Diff Under%

Effect: of adding one card of each value

12 12 12 12 12 12 12 12 12 48 156 0.467 0.450

-6.600

13 12 12 12 12 12 12 12 12 48 157 0.461 0.457

-7.790 -1.19

-8.672

12 13 12 12 12 12 12 12 12 "48 157 0.461 0.457

-7.790 -1.19

-8.672

12 12 13 12 12 12 12 12 12 48 157 0.461 0.453

-7.790 -1.19

-9.456

12 12 12 13 12 12 12 12 12 48 157 0.465 0.452

-7.006 -0.41

-9.652

12 12 12 12 13 12 12 12 12 48 157 0.466 0.451

-6.810 -0.21

-9.848

12 12 12 12 12 13 12 12 12 48 157 0.467 0.450

-6.614 -0.01 -10.044

12 12 12 12 12 12 13 12 12 48 157 0.468 0.449

-6.418

0.18 -10.240

12 12 12 12 12 12 12 13 12 48 157 0.469 0.448

-6.222

0.38 -10.436

12 12 12 12 12 12 12 12 13 48 157 0.470 0.447

-6.026

0.57 -10.632

12 12 12 12 12 12 12 12 12 49 157 0.471 0.446

-5.830

0.77 -10.828

-10.074

3> decks Effect of adding two cards of each value

10.074

12 12 12 12 12 12 12 12 12 48 156 0.467 0.450

6.600

14 12 12 12 12 12 12 12 12 48 158 0.455 0.464

8.958

2.36

7.297

12 14 12 12 12 12 12 12 12 48 158 0.455 0.464

8.958

2.36

7.297

12 12 14 12 12 12 12 12 12 48 158 0.455 0.456

-8.958 -2.36

-8.845

12 12 12 14 12 12 12 12 12 48 158 0.463 0.454

-7.409 -0.81

-9.232

12 12 12 12 14 12 12 12 12 48 158 0.465 0.452

-7.022 -0.42

-9.619

12 12 12 12 12 14 12 12 12 48 158 0.467 0.450

-6.635 -0.03 -10.006

12 12 12 12 12 12 14 12 12 48 158 0.469 0.448

-6.232

0.37 -10.409

12 12 12 12 12 12 12 14 12 48 158 0.471 0.446

-5.845

0.76 -10.796

12 12 12 12 12 12 12 12 14 48 158 0.473 0.444

-5.458

1.14 -11.183

12 12 12 12 12 12 12 12 12 50 158 0.475 0.442

-5.071

1,53 -11.570

82 / THE U LT IMATE BLACKJ,ACK BOOK A

2

3

4

5

3 .decks

6

7

8

9

T

DK Over

Under Over%

Diff Under %

Effect of adding eight cards of each value

12 12 12 12 12 12 12 12 12 48 156 0.467 0.450

-10.074

-6.600 \

20 12 12 12 12 12 12 12 12 48 164 0.422 0,502- 15.517

-8.92

0.434

12 20 12 12 12 12 12 12 12 48 164 0.422 0.502--15.517

-8.92

0.434

12 12 20 12 12 12 12 12 12 48 164 0.422 0.473--15.517

-8.92

-5.312

12 12 12 20 12 12 12 12 12 48 164 0.451 0.466

-9.771 -3.17

-6.748

12 12 12 12 20 12 12 12 12 48 164 0.458 0.459

-8.335 -1.73

-8.185

12 12 12 12 12 20 12 12 12 48 164 0.466 0.452

-6.898 -0.30

-9.621

12 12 12 12 12 12 20 12 12 48 164 0.475 0.443

-5.043

1.56 -11.477

12 12 12 12 12 12 12 20 12 48 164 0.482 0.435

-3.606

2.99 -12.913

12 12 12 12 12 12 12 12 20 48 164 0.489 0.428

-2.170

4.43 -14.350

12 12 12 12 12 12 12 12 12 56 164 0.496 0.421

-0.733

5.87 -15.786

4 decks Effect of adding one card of each value 16 16 16 16 16 16 16 16 16 64 208 0.467 0.450

-6.577

10.071

17 16 16 16 16 16 16 16 16 64 209 0.463 0.455

-7.471 -0.89

-9.017

16 17 16 16 16 16 16 16 16 64 209 0.463 0.455

-7.471 -0.89

-9.017

16 16 17 16 16 16 16 16 16 64 209 0.463 0.452

-7.471 -0.89

-9.606

16 16 16 17 16 16 16 16 16 64 209 0.466 0.451

-6.883 -0.31

-9.753

16 16 16 16 17 16 16 16 16 64 209 0.466 0.450

-6.735 -0.16

-9.901

16 16 16 16 16 17 16 16 16 64 209 0.467 0.450

-6.588 -0.01 -10.048

16 16 16 16 16 16 17 16 16 64 209 0.468 0.449

-6.441

0.14 -10.195

16 16 16 16 16 16 16 17 16 64 209 0.469 0.448

-6.294

0.28 -10.342

16 16 16 16 16 16 16 16 17 64 209 0.469 0.448

-6.146

0.43 -10.490

16 16 16 16 16 16 16 16 16 65 209 0.470 0.447

-5.999

0.58 -10.637

THE ULTIMATE

A

2

3

4

5

4 decks

6

7

8

9

T

BLACKJACK BOOK

DK Over

Under Over%

83

/

Diff Under%

Effect of adding eleven cards of each value

16 16 16 16 16 16 16 16 16 64 208 0.467 0.450

-6.577

-10.071

27 16 16 16 16 16 16 16 16 64 219 0.421 0.504 -15.747 -9.17

0.733

16 27 16 16 16 16 16 16 16 64 219 0.421 0.504 -15.747 -9.17

0.733

16 16 27 16 16 16 16 16 16 64 219 0.421 0.474- 15.747

-9.17

-5.165

16 16 16 27 16 16 16 16 16 64 219 0.451 0.467

-9.849 -3.27

-6.640

16 16 16 16 27 16 16 16 16 64 219 0.458 0.459

-8.374 -1.80

-8.114

16 16 16 16 16 27 16 16 16 64 219 0.466 0.452

-6.900 -0.32

-9.589

16 16 16 16 16 16 27 16 16 64 219 0.475 0.442

-4.964

1.61 -11,524

16 16 16 16 16 16 16 27 16 64 219 0.483 0.435

-3.490

3.09 -12.999

16 16 16 16 16 16 16 16 27 64 219 0.490 0.428

-2.015

4,56 -14.474

16 16 16 16 16 16 16 16 16 75 219 0.497 0.420

-0.540

6.04 -15.948

5 decks ]Effect of adding one card of each value

20 20 20 20 20 20 20 20 20 80 260 0.467 0.450

-6.564

21 20 20 20 20 20 20 20 20 80 261 0.464 0.454

-7.280

0.72

-9.225

20 21 20 20 20 20 20 20 20 80 261 0.464 0.454

-7.280 -0.72

-9.225

20 20 21 20 20 20 20 20 20 80 261 0.464 0.452

-7.280 -0.72

-9.696

20 20 20 21 20 20 20 20 20 80 261 0.466 0.451

-6.808 -0.24

-9.814

20 20 20 20 21 20 20 20 20 80 261 0.467 0.450

-6.690 -0.13

-9.932

20 20 20 20 20 21 20 20 20 80 261 0.467 0.450

-6.572 -0.01 -10.050

20 20 20 20 20 20 21 20 20 80 261 0.468 0.449

-6.454

0.11 -10.168

20 20 20 20 20 20 20 21 20 80 261 0.468 0.449

-6.337

0.23 -10.286

20 20 20 20 20 20 20 20 21 80 261 0.469 0.448

-6.219

0.35 -10.404

20 20 20 20 20 20 20 20 20 81 261 0.469 0.447

-6.101

0.46 -10,522

-10.068

84 / THE ULTIMATE BLACKJACK BOOK A

2

3

4

5

6

7

8

9

T

DK Over

Under Over%

Diff Under%

5 decks Effect of adding thirteen cards of each value

20 20 20 20 20 20 20 20 20 80 260 0.467 0.450

a

-6,564

-10.068

33 20 20 20 20 20 20 20 20 80 273 0.424 0,501 -15.266 -8.70

0.183

20 33 20 20 20 20 20 20 20 80 273 0.424 0.501 -15.266 -8.70

0.183

20 20 33 20 20 20 20 20 20 80 273 0.424 0.473 -15.266 -8.70

-5.419

20 20 20 33 20 20 20 20 20 80 273 0.452 0.466

-9.664 -3.10

-6.820

20 20 20 20 33 20 20 20 20 80 273 0.459 0.459

-8.263 -1.70

-8.220

20 20 20 20 20 33 20 20 20 80 273 0.466 0.452

-6.863 -0.30

-9.621

20 20 20 20 20 20 33 20 20 80 273 0.475 0.443

-5.042

1.52 -11.441

20 20 20 20 20 20 20 33 20 80 273 0.482 0.436

-3.641

2.92 -12.842

20 20 20 20 20 20 20 20 33 80 273 0.489 0.429

-2.241

4.32 -14.243

20 20 20 20 20 20 20 20 20 93 273 0.496 0.422

-0.840

5.72 -15.643

6 decks :Effect of adding one card of each value 24 24 24 24 24 24 24 24 24 96 312 0.467 0.450

-6.555

25 24 24 24 24 24 24 24 24 96 313 0.464 0.453

-7.152 -0.60

-9.363

24 25 24 24 24 24 24 24 24 96 313 0.464 0.453

-7.152 -0.60

-9.363

24 24 25 24 24 24 24 24 24 96 313 0.464 0.451

-7.152 -0.60

-9.757

24 24 24 25 24 24 24 24 24 96 313 0.466 0.451

-6.758 -0.20

-9.855

24 24 24 24 25 24 24 24 24 96 313 0.467 0.450

-6.660 -0.11

-9.953

24 24 24 24 24 25 24 24 24 96 313 0.467 0.450

-6.562 -0.01 -10.052

24 24 24 24 24 24 25 24 24 96 313 0.468 0.449

-6.464

0.09 -10.150

24 24 24 24 24 24 24 25 24 96 313 0.468 0.449

-6.365

0.19 -10.248

24 24 24 24 24 24 24 24 25 96 313 0.469 0.448

-6.267

0.29 -10,347

24 24 24 24 24 24 24 24 24 97 313 0.469 0.448

-6.169

0.39 -10.445

-10.067

THE ULTIMATE

A

2

3

4

5

6

7

8

9

T

BLACKJACK BOOK DK

Over

Under Over%

85

/

Diff Under%

6 decks Effect of adding sixteen cards of each value 24 24 24 24 24 24 24 24 24 96 312 0.467 0.450

-10.067

-6.555

40 24 24 24 24 24 24 24 24 96 328 0.423 0,502 -15.462 -8.91

0.425

24 40 24 24 24 24 24 24 24 96 328 0.423 0,502 -15.462 -8.91

0.425

24 24 40 24 24 24 24 24 24 96 328 0.423 0.473 -15.462 -8.91

-5.303

24 24 24 40 24 24 24 24 24 96 328 0.451 0.466

-9.734 -3.18

-6.735

24 24 24 24 40 24 24 24 24 96 328 0.458 0.459

-8.302 -1.75

-8.167

24 24 24 24 24 40 24 24 24 96 328 0.466 0.452

-6.870 -0.32

-9.599

24 24 24 24 24 24 40 24 24 96 328 0.475 0.443

-4.990

1.56 -11.479

24 24 24 24 24 24 24 40 24 96 328 0.482 0.435

-3.558

3.00 -12.911

24 24 24 24 24 24 24 24 40 96 328 0.489 0.428

-2.126

4.43 -14.343

24 24 24 24 24 24 24 24 24 136 352 0.535 0.384

6.993 13.55 -23.155

7 decks Effect of removing one card of each value 28 28 28 28 28 28 28 28 28 112 364 0.467 0.450

-6.548

-10.066

27 28 28 28 28 28 28 28 28 112 363 0.470 0.447

-6.032

0,52 -10.674

28 27 28 28 28 28 28 28 28 112 363 0.470 0.447

-6.032

0.52 -10.674

28 28 27 28 28 28 28 28 28 112 363 0.470 0.448

-6.032

0,52 -10.333

28 28 28 27 28 28 28 28 28 112 363 0.468 0.449

-6.373

0.18 -10.248

28 28 28 28 27 28 28 28 28 112 363 0.468 0.449

-6.458

0.09 -10.162

28 28 28 28 28 27 28 28 28 112 363 0.467 0.450

-6.543

0.00 -10.077

28 28 28 28 28 28 27 28 28 112 363 0.467 0.450

-6.625 -0.08

-9.995

28 28 28 28 28 28 28 27 28 112 363 0.466 0.450

-6.711 -0.16

-9.910

28 28 28 28 28 28 28 28 .27 112 363 0.466 0.451

-6.796 -0.25

-9.825

28 28 28 28 28 28 28 28 28 111 363 0.466 0.451

-6.881 -0.33

-9.739

86 / THE U LT IMATE B L AC KJi\CK BOOK A

2

3

4

5

6

7

8

9

T

DK

Over

Under Over%

Diff Under%

7 decks Effect of adding one card of‘ each value 28 28 28 28 28 28 28 28 28 112 364 0.467 0.450 , -6.548

-10.066

29 28 28 28 28 28 28 28 28 112 365 0.465 0.453

-7.060 -0.51

-9.463

28 29 28 28 28 28 28 28 28 112 365 0.465 0.453

-7.060 -0.51

-9.463

28 28 29 28 28 28 28 28 28 112 365 0.465 0.451

-7.060 -0.51

-9.800

28 28 28 29 28 28 28 28 28 112 365 0.466 0.451

-6.723 -0.17

-9.884

28 28 28 28 29 28 28 28 28 112 365 0.467 0.450

-6.639 -0.09

-9.968

28 28 28 28 28 29 28 28 28 112 365 0.467 0.450

-6.554 -0.01 -10.053

28 28 28 28 28 28 29 28 28 112 365 0.468 0.449

-6.470

0.08 -10.137

28 28 28 28 28 28 28 29 28 112 365 0.468 0.449

-6.386

0.16 -10.221

28 28 28 28 28 28 28 28 29 112 365 0.468 0.448

-6.301

0.25 -10.306

28 28 28 28 28 28 28 28 28 113 365 0.469 0.448

-6.217

0.33 -10.390

7 decks Effect of adding eighteen cards of each value 28 28 28 28 28 28 28 28 28 112 364 0.467 0.450

-6,548

-10.066

46 28 28 28 28 28 28 28 28 112 382 0.424 0.500 -15.159 -8.61

0.076

28 46 28 28 28 28 28 28 28 112 382 0.424 0.500 -15.159 -8.61

0.076

28 28 46 28 28 28 28 28 28 112 382 0.424 0.473 -15.159 -8.61

-5.465

28 28 28 46 28 28 28 28 28 112 382 0.452 0.466

-9.618 -3.07

-6.850

28 28 28 28 46 28 28 28 28 112 382 0.459 0.459

-8.233 -1.68

-8.235

28 28 28 28 28 46 28 28 28 112 382 0.466 0.452

-6.848 -0.30

-9.621

28 28 28 28 28 28 46 28 28 112 382 0.475 0.443

-5.042

1,51 -11.426

28 28 28 28 28 28 28 46 28 112 382 0.482 0.436

-3.657

2.89 -12.811

28 28 28 28 28 28 28 28 46 112 382 0.489 0.429

-2.272

4.28 -14.197

28 28 28 28 28 28 28 28 28 130 382 0.496 0.422

-0.886

5.66 -15,582

THE ULTIMATE

A

2

3

4

5

6

7

8

9

T

BLACKJACK BOOK

DK Over

Under Over%

87

/

Diff Under%

8 decks Effect of adding one card of each value 32 32 32 32 32 32 32 32 32 128 416 0.467 0.450

-6.543

-10.065

33 32 32 32 32 32 32 32 32 128 417 0.465 0.452

-6.991 -0.45

-9.537

32 33 32 32 32 32 32 32 32 128 417 0.465 0.452

-6.991 -0.45

-9.537

32 32 33 32 32 32 32 32 32 128 417 0.465 0.451

-6.991 -0.45

-9.832

32 32 32 33 32 32 32 32 32 128 417 0.467 0.450

-6.696 -0.15

-9.906

32 32 32 32 33 32 32 32 32 128 417 0.467 0.450

-6.622 -0.08

-9.980

32 32 32 32 32 33 32 32 32 128 417 0.467 0.450

-6.549 -0.01 -10.053

32 32 32 32 32 32 33 32 32 128 417 0.468 0.449

-6.475

0.07 -10.127

32 32 32 32 32 32 32 33 32 128 417 0.468 0.449

-6.401

0.14 -10.201

32 32 32 32 32 32 32 32 33 128 417 0.468 0.449

-6.327

0.22 -10.275

32 32 32 32 32 32 32 32 32 129 417 0.469 0.448

-6.253

0.29 -10.349

8 decks Effect of adding twenty 5 = +1; 9,10 = -1 ignore all others

A

23456789TDK Over 1 deck

Under Over%

Diff Under%

Effect of typical high count of +2

4

4

4

4

4

4

4

4

4

16

52 0.466 0.449

-6.787

-10.106

3

3

3

2

2

3

4

4

3

12

39 0.491 0.425

-1.754

5.03 -14.980

2

2

2

2

4

4

4

4

2

12

38 0.545 0.370

8.962 15.75 -26.031

2

4

4

4

4

4

4

4

4

16

50 0.504 0.404

0.898

7.69 -19.184

3

3

3

3

4

3

4

4

3

15

45 0.513 0.404

2.626

9.41 -19.192

4

2

2

4

4

0

4

4

4

14

42 0,526 0.404

5.226 12.01 -19.164

4

4

4

2

2

4

4

4

4

14

46 0.461 0.454

-7.826 -1.04

2

4

4

4

2

4

4

4

4

14

46 0.488 0.419

-2.415

2

4

2

4

4

4

4

4

4

14

46 0.523 0.404

4,541 11.33 -19.227

0

0

4

4

4

4

4

4

4

10

38 0,568 0.307

13,514 20.30 -38.549

3

1

4

4

4

4

4

4

3

15

46 0.527 0.373

5.314 12.10 -25.411

1 deck

Effect of typical low count of -2

-9.179

4.37 -16.135

4

4

4

4

4

4

4

4

4

16

52 0.466 0.449

3

4

4

2

2

3

4

4

3

10

39 0.414 0.497 -17.139-10.35

-0.675

4

4

4

4

4

4

4

4

4

14

50 0.442 0.473 -11.673 -4.89

-5.306

3

3

3

3

3

3

3

3

3

10

37 0.432 0.482 -13,514 -6.73

-3.604

3

3

3

3

3

3

4

4

3

15

44 0.518 0.400

4

4

4

2

2

4

4

4

3

11

42 0.410 0.504 -18.002-11.21

0

0

0

0

4

4

4

4

0

2

18 0.503 0.288

0.654

7.44 -42.484

1

1

1

1

4

4

4

4

4

2

26 0.483 0.400

-3.385

3.40 -20.000

2

2

2

2

2

2

2

2

2

6

24 0.413 0,500 -17.391-10.60

3

3

2

2

4

4

4

4

4

8

38 0.452 0.468

-9.531 -2.74

-6.401

4

4

0

0

0

4

4

4

4

2

26 0.403 0,548 -19.385-12.60

9,538

-6.787

-10.106

3,594 10.38 -20.085 0.813

0.000

THE

ULTIMATE

BLACKJACK BOOK

/

89

2. Assume a new count: ace,2 = +2; 9,10 = -2; ignore all others

A234

5

6789T

1 deck

DK Over

Under Over%

Diff Under%

Effectxof typical high count of +2

4

4

4

4

4

4

4

4

4

16

52 0.466 0.449

2

4

4

2

2

3

4

4

3

16

44 0.518 0.387

3,594 10,38 -22.622

3

3

4

4

4

4

4

4

4

16

50 0.504 0.404

0.898

0

0

4

4

4

4

4

4

2

11

37 0.562 0,312

12.312 19.10 -37,538

0

0

3

3

3

3

3

3

2

11

31

25.161 31.95 -49.677

0.626 0.252

-6.787

-10.106

7.69 -19.184

3. Assume a new count: ace = +3; Ten = -1; ignore all others

A

2

3

4

5

6

7

1 deck

8

9

T

DK Over

Under Over%

Diff Under %

Effect of typical high count of + 2

4

4

4

4

4

4

4

4

4

16

52 0.466 0.449

-6.787

-10.106

2

4

4

2

2

3

4

4

3

14

42 0.494 0.411

-1.278

5,51 -17.770

1

4

4

4

4

4

4

4

4

13

46 0.487 0.416

-2.609

4.18 -16.715

0

3

3

3

3

3

3

3

3

13

37 0,563 0.338

12.613 19.40 -32.432

/

APPENDIX B:

Number of Distinct Dealer Hands, reaching 17 thru bust

NUMBER OF

TOTAL

POSSIBLE HANDS

REACHED 17

5447

18

5442

19

5434

20

5421

21

5400 21497

Bust

** considers card order; e.g. T61, 6T1 are each counted separately

91

92

/ THE

ULTIMATE

BLACKJACK BOOK

BROKEN DOWN BY UPCARD: (Blackjack not considered) Reach

17

18

19

20

21

Bust

Row Total

\

Up

A

768

768

768

768

768

2953

6793

2

1811

1809

1606

1801

1793

7101

16121

3

1134

1133

1131

1128

1123

4493

10142

4

696

695

694

692

689

2777

6243

5

434

434

433

432

430

1742

3905

6

253

252

252

251

250

1017

2275

7

159

159

158

158

157

640

1431

8

96

96

96

95

95

387

865

9

64

64

64

64

63

258

577

T

32

32

32

32

32

129

289

5447

5442

5434

5421

5400

21497

48641

Total

APPENDIX C:

(ost of Deviating from Basic Strategy

Please note that the following plays are NOT recommend¬ ed. Also, note that card counting is not considered. The numbers are based on 8 decks.

WHAT IT COSTS IN PERCENT

DEVIATION FROM BASIC STRATEGY

a. Stand with soft 18 vs ace,

.000001

holding 2-2 or 3-3 b. Stand with soft 18 vs ace, any cards

.000009

c. Stand with soft 18 vs 9,

.000040

holding 2 cards under 5 d. Double ace-2 vs 4

.000045

e. Do not double ace-4 vs 4

.000160

f. Draw on 12 vs 4

.000174

g. Draw on 3-card 12 vs 3

.000175

h. Do not split 3-3 vs 2

.000231

93

% / THE ULTIMATE

BLACKJACK BOOK

i. Double soft 18 vs 2

.000238

j. Double soft 17 vs 2

.000307

k. Split 3-3 vs 8

.000430

1. Split 2-2 vs 8

.000495

m. Stand with 7-9 vs T n. Double 9 vs 2

.000498 .000710

Note that novices regularly do some of the following: aa. Double soft 19 vs 6

.00139

bb. Do not double T vs 9

.00141

cc. Stand with 9-3 vs 3

.00148

dd. Double 11 vs ace

.00149

ee. Double ace-3 vs 4

.00155

ff. Draw on 12 vs 6

.00179

gg. Double 8 vs 6

.00229

hh. Draw on 12 vs 5

.00275

ii. Stand with ace-6 vs 7

.00545

jj. Double 9 vs 7

.00561

kk. Do not split 8 vs T 11. Double T vs T mm. Do not double 11 vs.T

.00811 .01080 .01984

nn. Hit 13 vs 2 or 3

.02000

oo. Stand with 16 vs 7

.02398

pp. Split tens vs 6

.09607

APPENDIX D:

Player Expectation with Certain Cards Removed

Player expectation when 1, 2, 5, 6, or 7 cards of given value are removed from an 8-deck shoe is shown in the table below. Card Value

Expectation

Removed

Remove 1

Remove 2

Remove 5

Remove 6

Remove 7

A

-0.0058

-0.0064

-0.0085

-0.0093

-0.0099

2

-0.0046

-0.0041

-0.0027

-0.0023

-0.0017

3

-0.0045

-0.0040

-0.0023

-0.0018

-0.0012

4

-0.0044

-0.0037

-0.0015

-0.0009

-0.0001

5

-0.0042

-0.0033

-0.0006

-0.0003

-0.0012

6

-0.0045

-0.0041

-0.0024

-0.0018

-0.0013

7

-0.0047

-0.0045

-0.0033

-0.0030

-0.0026

8

-0.0053

-0.0051

-0.0053

-0.0055

-0.0053

9

-0.0057

-0.0054

-0.0064

-0.0070

-0.0069

T

-0.0057

-0.0063

-0.0081

-0.0087

-0.0093

APPENDIX E:

Dealer Probability Tables

Dealer naturals are excluded. TABLE 1: 5 DECKS, PLAYER HOLDS 10 AND 2. Example: probability of busting, showing 5, is 0.4185523. Reach

17

18

19

20

21

Bust

Up 1

1877948 1891674 1896148 1901665 0773934 1658628

2

1398043 1343725 1306504 1244954 1889280 3517843

3

1343752 1306643 1256993 1213913 1537050 3724992

4

1304150 1238946 1218300 1169721 1245253 3944354

5

1200701

6

1663570 1043248 1069336 1020709 0977628 4225507

7

3677852 1384631 0778695 0790732 0742327 2625764

8

1294879 3584537 1296568 0685649 0697652 2440708

9

1207566 1171413 3507725 1212445 0601518 2299330

10

1224002 1183052 1121878 0848410 4185523

1218139 1214611

1222619 3666898 0379308 2298422

97

98

/ THE ULTIMATE BLACKJACK BOOK

TABLE 2: 5 DECKS, PLAYER HOLDS 5 AND 6. Example: probability of busting, showing 5, is 0.4229755. Reach

17

21

20

19

18

Bust

Up 1

1838384 1901242 1896286 1901291 0767055 1695740

2

1379615 1322579 1300854 1243053 1186575 3567321

3

1343394 1287430 1230468 1208100 1151779 3778826

4

1308209 1237175 1193557 1142453 1118064 4000538

5

1219599 1225500 1174861 1094700 1055581 4229755

6

1661739 1061963 1063636 1012861 0951139 4248658

7

3712167 1379634 0780427 0782632 0731767 2613371

8

1290576 3618695 1286932 0687390 0689843 2426561

9

1203797 1167521 3538113 1203246 0603879 2283442

10

1217388 1210574 1213710 3699914 0369559 2288852

TABLE 3: 6 DECKS, PLAYER HOLDS 10 AND 2. Example: probability of busting, showing 5, is 0.4181889. Reach

17

18

19

20

21

Bust

Up 1

1879832 1891255 1894991 1899569 0774644 1659706

2

1398042 1344628 1304842 1244174 1187424 3520886

3

1344850 1306304 1256779 1212134 1152588 3727343

4

1304291 1242359 1217567 1168881 1122477 3944422

5

1204384 1223758 1182028 1123487 1084451 4181889

6

1662038 1046478 1068225 1020106 0976623 4226528

7

3679180 1383515 0779974 0789986 0742016 2625326

8

1293334 3585996 1294744 0687052 0697041 2441830

9

1206289 1176216 3507714 1210326 0602655 2296796

10

1216283 1213352 1220002 3673672 0378372 2298317

THE ULTIMATE BLACKJACK BOOK

/

99

TABLE 4: 6 DECKS, PLAYER HOLDS 5 AND 6. Example: probability of busting, showing 5, is 0.4218612. Reach Up

17

18

19

20

21

Bust

s

1

1846932 1899195 1895134 1899272 0768886 1690577

2

1382754 1327085 1300114 1242564 1185451 3562029

3

1344580 1290384 1234758 1207286 1150954 3772035

4

1307641 1240951 1197010 1146261 1117096 3991038

5

1220062 1224983 1175220 1100956 1060163 4218612

6

1660451 1061975 1063483 1013590 0954628 4245869

7

3707706 1379344 0781354 0783219 0733260 2615114

8

1289771 3614428 1286709 0688479 0690513 2430098

9

1203174 1172992 3532970 1202675 0604591 2283594

10

1215666 1210016 1212613 3701106 0370241 2290355

••

'

APPENDIX F:

Multi-Action Blackjack

A

version of Blackjack which is being offered at more

and more casinos around the country is Multi-Action. In this version of the game, there are normally six players. The table layout is different. Each players spot has room for three bets and each of these spaces is subdivided and labeled for insurance, split, and double. DEALER UPCARD GAME NUMBER

3 I N S u R A N C E

1

2 • • •

PLAYER

D O U B L E

S P L I T

GAME 3 GAME 2 GAME 1

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/ THE ULTIMATE BLACKJACK BOOK

Dealer deals two cards to each player, and draws his upcard. Players play out their hands, as in regular black¬ jack. If the player breaks, he loses all his bets. Then dealer draws to his upcard, which is located in his leftmost square. Payoffs are made as usual, affecting player bets for Game 1. Then all dealers cards are collected except the upcard, which shifts to his middle square for Game 2 and dealer again completes a hand and collects bets or makes a payoff for Game 2. He then repeats the procedure one more time for Game 3. Thus, dealer plays three games starting with the same upcard. A player must play a minimum of the first two games. Bets may be different for each game played but once play¬ er has reached a decision on his hand, that hand applies to each of dealers three hands. Note that when dealer draws a blackjack to an ace or ten that blackjack applies only to one dealer hand and the other hands may turn out differently. Most players assume that they lose less standing with a stiff against a ten, believing that their loss will be less than if they bust the hand providing that dealer cooperates and busts at least once. In that event they lose a net of one game, and not all three were they to break at the outset. This fallacy is what the casino is relying on by offering Multi-Action. What players do not bother to consider is that they just might draw a good card to their stiff and possibly win three hands. The analysis appears below.

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ANALYSIS OF STAND VS DRAW WITH 16 VS 10 (MULTIPLE-ACTION) (Refer to the analysis shown in Decision 1, Chapter X) The expectation for one game, when standing, is -0.5404, and when drawing, it is -0.5389 (8 decks). Also, dealer’s chance of breaking, with 10 up, is about 0.23 (see Appendix E). Assume the player stands. Then the dealer draws to his 10 in each of three games. In game 1, players expectation is -0.5404. In the second and third games this expectation is affected very

slightly by card depletion for the prior

game(s). However,we are dealing from a multiple-deck shoe and the change will be negligible. The player hopes that dealer will break at least least once. The probability for dealer to reach 17 thru 21 in three games with 10 up is (1-0.23) cubed, or 0.46. Thus the play¬ er is correct in presuming that, by standing, he will proba¬ bly not lose all three games (chances are less than half). However, that does not mean he improves his expectation by standing vs drawing. Consider the following: CONDITION

PROBABILITY

Dealer wins all three games

0.46

Dealer wins two, loses one

0.41

Dealer wins one, loses two

0.12

Dealer loses all three games

0.01

Then the expectation is 0.46(-3) + 0.41(-1) + 0.12( + 1) + 0.01 (+3) or-1.64 with 3 units bet, which is -0.54 per unit.

The result is identical to that of the single game!

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/ THE ULTIMATE BLACKJACK BOOK

The same logic applies to other situations wherein player has a stiff hand. He should play basic strategy, unless he is counting. Insurance is another matter for analysis. In basic strat¬ egy its best not to take insurance, for reasons discussed before. With Multiple-Action, if dealer has drawn one ten the odds change for the next hand(s). For example, assume 5 decks remain, with a normal dis¬ tribution of tens, and dealer has ace up. He is going to draw three times to that ace. What are the blackjack expectations? For the first draw, chance of a natural is 80/260 or 0.3077. If a ten is drawn, chance of a second natural is 79/259, 0.3050. If a second ten is drawn, chance of a third natural is 78/258, or 0.3023. Summarizing, Chance of drawing one ten in three draws is 0.4451. (There are three possibilities: game 1, 2, or 3: 3 x 80 x 180 x 179 / 260/259/258 = 0.4451) Chance of drawing two tens in three draws is 0.1964. Chance of drawing three tens in three draws is 0.0284. Chance of not drawing any tens is 0.3301. If player buys insurance for each game, the insurance pay¬ off will be

-3(.3301) + 0(.4451) + 3(.1964) + 6(.0284) -

-0.2307. That is equivalent to -0.2307/3 per unit, or -0.0769 per unit. In the discussion on insurance in a previous chapter, the loss to be expected is one unit in 13, or -0.0769!

In summary, when playing at a Multiple-Action table, play Basic Strategy and do not hesitate to hit a stiff when warranted.

APPENDIX G:

SOME PERSONAL EXPERIENCES

I

n 1966, I decided to get out of the rat race of overtime

hours and commuting. Having taught Civil Engineering in the evenings at a local University, I grew to love teaching. When an offer to teach full-time was received, I gladly accepted. Being an Assistant Professor and having summers off gave me and my family the opportunity for long vacations. In 1968, we headed west on an extended trip which includ¬ ed one night in Las Vegas at the Riviera Hotel and Casino. While my wife and children enjoyed the pool, I played roulette and had what I now know was extraordinary luck. I enjoyed it so much that we extended our stay one night, then two. For the third night, we bought tickets to Fiddler

on the Roof. These cost $55 for the five of us. As the oth¬ ers were getting dressed, I decided to go downstairs to the

105

106

/ THE ULTIMATE BLACKJACK BOOK

casino to win the money for the show. At that point, I was $200 ahead at roulette. I placed $55 on black to win, and red came up. I then placed $110 on black; red came up again.,I then bet $220 on black and red came up. Goodbye my winnings and then some. I was too afraid to double up again and I placed $50 on black. Red again. Too afraid again to double up, again I placed $50 on black. The ball settled again in the red. To make a long story short, the red slot attracted the ball eleven straight times, with my money on black eveiy time. I went upstairs and announced, “I just lost $540. We leave in the morning,” I added, “The pit boss felt so sorry for me that he’s treating us to the show.” Big deal. A lot of water has gone over the dam since then. I developed an interest in gambling and read that you can¬ not beat the 5.26 per cent house advantage in roulette, no matter how you bet. I heard about Beat the Dealer by Edward Thorp, a professor who claimed that in Blackjack the player could gain an edge by following a strategy out¬ lined in the book. Being a cynic by nature, the proof was in the pudding and I memorized Thorp s Basic Strategy. Memorizing was a cinch in those days. On our next vacation, I decided to put my new knowledge to the test. Our destination? Again Las Vegas. Again the Riviera Hotel. We stayed three nights and—lo and behold—I lost only $350. I was convinced Thorp s method was a flop. My next opportunity to play Blackjack was on a trip to Paradise Island in the Bahamas. I lost again, but not as much as before. One experience lingers in my mind. At that time the dealer in the Paradise Island Casino had to check his hole

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107

card if he showed an ace or a ten, in order to determine whether he had a natural. This required lifting the corner of the card and taking a peek. I sat to dealers left and, much to my surprise, whenever he peeked I could see what the hole card was! What an opportunity to make a fortune! On the first hand with an ace up, I had 17 and I saw a seven in the hole. I did not think I would attract too much attention by asking for a hit with 17 against the ace. I drew a 6 and busted. The next time, the dealer had a ten up, and I saw that he had a ten in the hole. I had hard 19, but decided not to draw a card and lost the hand, figuring that the night was young, and I did not want to flag the dealer about his slop¬ piness in peeking by making such an improbable move as hitting hard 19. The next time the dealer showed a ten, his hole card was a four and I had fifteen showing. I stayed, according to strategy. Instead, he drew a five to his fourteen and I lost. Normally, Td have hit the fifteen against the ten. As it was, not one of the players hit their hands and the first card on the next deal was a ten. I was annoyed that my surrepti¬ tious knowledge cost me. And so it went until the relief dealer appeared. I had lost six hands deciding to stand when the dealer total was a stiff based upon his hole card. In each instance, he made the hand good. That trip also turned out to be a losing one for me. On my next trip to Paradise Island, along with my sis¬ ter-in-law and her husband, I tried playing craps. I had read that the bet with the least damage was Don’t Pass and Don’t Come (a 1.4 pet house advantage if I limited myself to those bets.) My brother-in-law played Pass, the num-

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bers, and the Hard Way bets. He would take full odds and often pressed his bet. I remarked that those bets were not the ones with the least house advantage, but he did not want to bet wrong’. With money on Don’t Come for each roll, one woman held the dice for 15 minutes. I lost almost every roll, win¬ ning only when the dice showed 2 or 3. Then the dice were passed to me, and I dutifully put my money on Don’t Pass. When I rolled seven, the players yelled with approval. I then rolled an eleven, with the same result. Then came an 8, and I put my chip on Don’t Come. I rolled a five. Each time I rolled, I put another chip on Don’t Come. In quick succession, I rolled 8 and 5, losing both chips. Then, with my money on Don’t Pass, I finally rolled the seven. The players were screaming with delight! After 20 minutes of almost constant losses for me (except for an occasional two or three), and continuously holding the dice, I finally dost’ by rolling 7 instead of the point, collected my bet and passed the dice. The damage had been done. Not only had I exceeded my self-imposed limit but now I got a dressing-down for betting wrong’ when eveiyone else was betting Tight’. I decided that I was through with craps, and I turned my attention to Blackjack. For the next few years I did lots of consulting work, developing computer programs. I test¬ ed them in the evening hours so as not to interfere with the staff’s use of the computer. This gave me the chance to use a high-powered machine to develop Blackjack programs at a time when it would normally be idle. Since Paradise Island Casino used four decks and the available literature was mostly confined to a 1-deck strate¬ gy, I analyzed the effect of the multiple-deck shoe on the

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109

player expectation. With Basic Strategy as published by Thorp, the player was at a disadvantage with four decks. I did not want to play against a game which would beat me in the long run. Therefore, the next step was to become a card counter. I selected a counting method, using my own judgment. It was the method outlined in Chapter V—and it has served me well. Later, I was very pleased to see the method listed in Reference 7 of the Bibliography where it was referred to as the Green Fountain method (with an asterisk explaining that it is already a well-known counting method). The year was 1977. The Atlantic City casinos were scheduled to open in 1978. When Resorts International opened a casino on the Boardwalk in Atlantic City, I was full of anticipation. I had done much of the analysis I set out to do with the ‘new’ counting method, practicing it until there was no problem keeping track of the count no matter how fast I dealt the cards to myself. I’d made two more trips to Paradise Island where one dealer in particular offered a tremendous chal¬ lenge. British, with slick black hair and a haughty demeanor, his fellow dealers referred to him as Speedy Gonzales. His hands moved like lightning over the cloth. Whenever a player lost the cards were scooped up in less than a second. However, I had trained myself well and was able to match his speed. Nevertheless, I lost. But, I thought, the odds were with me in the long run, and I never lost faith. My wife and I opened a credit line at Resorts. The consulting business was good, and we played $25 a hand. The first visit resulted in a win of $400, and we were hooked.

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We had been playing at Resorts for about four months, when my brother-in-law asked me whether, ‘with your action’, we were being ‘comped’. I had never thought about asking for comps. Up to that point, we had eaten in the casino cafeteria, paying our own way, stayed overnight paying for our rooms. When we saw shows it was at our own expense. Those days ended. Since then we have never paid for a meal, a room, or a show. Brothers-in-law come in handy on occasion! That year I had an amazing streak of good fortune, which came to a screeching halt when I pushed my luck and started betting over my head. The trouble was bore¬ dom after many hours of play filling my head with ‘the count’, concentrating all the time, while others enjoyed themselves animatedly. A losing streak at $100 per hand can dig deeply into one’s winnings. When Golden Nugget (now Bally’s Grand) and Caesars opened, we applied and received credit. Later it was Bally’s, followed by the Trump constructions. We drove down—a 3-hour trip—about 3 times per month. Paradise Island was no longer attractive to us. Not only did the Atlantic City casinos offer early surrender then (player could, at his option, give up half his bet, and sur¬ render the hand), but Paradise Island did not permit soft doubling. Doubling was limited to a 2-card total of 9, 10, or 11. The early surrender feature was a boon. No wonder they discontinued it. During 1984, I requested, and received, time on the University mainframe to determine the player expectation in Blackjack, if multiple decks were used. I soon discov¬ ered that even with the speed of a computer, the billions of

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111

calculations required for exact analysis chewed up a great deal of time. This led me to develop a method of analysis (Ref. 19) which was presented at Bally’s in Atlantic City in 1985. Even though the method was an approximation, it was close to exact for all practical purposes, and cut com¬ puter time by 90 percent. The presentation went well, the paper was published, but more important for me was meeting some of the shin¬ ing lights in the Blackjack analysis field, such as Professor Peter Griffin. Little did I think that a scholarly paper would get me barred from playing Blackjack at Paradise Island a few years later! In 1987, my son and I took a trip to Paradise Island, memorable in that we won handily (he played the game for the first time) and because of one particular hand which I will never forget. We had flown down after receiving apromotional price from Resorts. It said that $687.50 for two people would include the flight and room for three nights. Food comps were determined by the action. I played big for me—$25 to $100 per hand—and luck was with me for two nights. On the third evening, I approached the shift boss and said, “Mario, I think we’re giving you enough action to deserve a full comp”.

He

replied, “Ed, the evening is young. Let’s wait and see.” Two hours later, I got the same response. We played for another 4 hours. It was now after mid¬ night, and Mario approached my table. He asked, “How much do we owe you?” and I replied “$687.50.” Mario looked at the dealer and said, “Pay him $687.50 in chips.”

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I had been counting and the shoe was almost down to the plastic. The running count was 18, and the ace count indi¬ cated there were ten extra aces left in the shoe. The dealer passed me $687.50 in chips, and signed and dropped a neatly folded paper into the cash slot. Mario said, “OK, Ed?” And I said, “Thanks. Play it!” Suddenly, 5 black chips, 7 green chips, two reds, and a pink were sitting on the circle in front of me. The dealer gave me a ten, and came around with an ace on top of it! He announced “Blackjack” and paid me $1,032. Mario made a wry smile and I said, “Gotta call it a night. Got a flight tomorrow” and walked away. The dealer uttered a sarcastic “Thank you, sir.” I make it a practice never to tip, unless I have won with the dealer’s enthusiastic best wishes making me feel better along the way. I never tip a matter-of-fact dealer, who appears to enjoy collecting losing bets, and who exudes an air of “what suckers.” Those British dealers had just such mannerisms, and it made that encounter doubly sweet. Later that year, I received an invitation for a junket to come to Paradise Island. I would pay the airfare, and the casino would supply the room. Food and a possible refund of the fare would depend on my ‘action’. I took my publi¬ cation along. On my previous trips I had become friendly with a pit boss by the name of Phil, and we had often discussed Basic Strategy. Now I showed him my paper, and loaned him the Proceedings for reading at his leisure, while I sat down to play. Luck was with me from the word ‘go’. I drew low cards to the stiff hands, and drew high cards on my double¬ downs. The dealer busted when he wasn’t supposed to

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bust, and the players got all the naturals. It was a players dream shoe. I liked to play with marker money. It gave me a psy¬ chological boost to think I was playing with ‘their’ money. I still feel that way, even though I realize its not so. On this occasion, I would take $500 in marker chips, win it back, walk to the cashier and buy back my marker, then go back to the table and take out another marker for $500. After four such occasions, at my next request for a marker. Abel, a floor manager, came to my table and the conversation went something like this: “Why don’t you use your winnings instead of asking for a marker?” “I like playing with borrowed money and paying it back.” “Well, you’re too good a player and these chandeliers are very expensive.” “Okay, I’ll use my winnings.” “Not at Blackjack, unless you agree to play only $25 bets, no more, no less. If you want to play craps, roulette, or any other game—no restrictions.” “By the way,” he added, “ your credit line is cancelled.” “Then I’ll fly back today, and I won’t pay for the trip.” Abel shrugged and walked away. I walked over to Phil, and told him what had happened. Phil said, “It’s probably my fault. He came over after you handed me your article and asked what it was about. I showed it to him and he must have felt you were a card counter.” I played a few more hands at $25, being frustrated at not being able to vary my bet. True to my promise to Abel, I flew home the next morning after one night in Paradise.

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I notified the Bank of New York, my charge card carri¬ er, that the invoice for the trip would not be paid, because I was invited by the merchant to “come and play in Paradise...,7’ and he had reneged on the offer by not allow¬ ing me play my game freely in the normal manner. After several attempts to collect via letters, the matter was dropped. Our favorite casino in Atlantic City is Ballys Grand. Their entertainers, such as Frank Sinatra, are top-notch. Vanessas is tops in dining. So is Lily Langtry’s and Charlie’s. Also, we were winning on two out of three visits. Every now and then, the old boredom returned, and I sought the excitement of big bets. Usually, that cost me big chunks of the winnings I had slowly accumulated. After ten of fifteen visits, winning $400 or $500 on average, my betting recklessness would cost us $4,000 or $5,000. On one occasion, our car developed radiator problems and we stopped in a local gas station about 5 miles from Bally’s Grand. I called the host and asked how we could get there. Within 15 minutes, a stretch limo picked us up! One of the most frustrating things in my experience with this game of Blackjack is my inability to convince my wife not to play hunches. This is especially true with insur¬ ance when she has Blackjack. She always takes insurance and even money. I talk myself blue in the face trying to explain that she gives up money in the long run with this technique, but to no avail. I have to admit, however, her money management is superior to mine. She has rarely lost more than a few hundred dollars. At Bally’s Grand one day I requested a comp for an up¬ coming show, and I was told that the shift manager want-

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ed to see me. I waited for him in the lobby after he intro¬ duced himself, asked what I wanted. I mentioned the show. To my surprise he responded that I was “too skilled" a player, and I would no longer be eligible for such comps. I was flattered by the compliment, but disappointed. I had made out a check to Ballys for $2000 one week earlier and believed they would ignore my lucky string of prior wins. I was wrong. I had never concealed my knowledge of the game. At the tables, I would cite percentage odds to other players, especially the dealers chances of not breaking when he had a five or six as his upcard. I suppose that wasn’t too bright. My consolation lay in the fact that the shift manag¬ er felt I was that skilled. Both our credit lines were can¬ celled. My wife and I could no longer enjoy the Grand’s hospitality. When Trump’s Taj Mahal opened, we opened a line of credit. We used it whenever we had a losing day at another casino. Strangely, the Taj would always give us a winning ses¬ sion. On one particular occasion, we were behind $3,500 after playing at Harrah’s when we arrived at the Taj. I told my wife the only way we had a chance to win back our money was to play high. Accordingly, we seated ourselves in the $100 pit. We did not begin play until a new shoe started. Whenever the count was favorable, we both raised our bets, and I played two hands at a time. We’d won $3,700 when the pit boss walked over and remarked, “The next time you raise your bet, the dealer will shuffle.’’ I asked why, as did the woman seated to my left. He responded, “You know why.’’ We quit and cashed in our chips.

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The next time I asked for credit, my line was cancelled. So we used my wife’s credit. The next time she asked for a comp, we were turned down and her credit cancelled. Atlantic City is not permitted to bar counters, but they can make them uncomfortable. Anyhow, there was still Bally’s, Harrahs, Sands, Trump Castle and Trump Plaza, Showboat, Caesar’s, Claridge, and Resorts. We have taken many cruises, and enjoyed Blackjack aboard ship. Some years ago the rules on board were so bad for the player that we refused to play. For example, on one ship, a push was a dealer win! One day on the Rotterdam, cruising to Alaska, someone remarked that whales had been sighted on the starboard side. All the dealers went over to the windows, leaving their chips and the players! Imagine any dealer in a Vegas, Reno, or Atlantic City casino ever abandoning his tray! These

reminiscences would be

incomplete without

including the following experience. In the early 1980’s we vacationed in Aruba and played at one of its hotels. The dealer used a shoe, and one customer was a high roller, betting up to $500 a hand. I sat to dealer’s left and I noticed that he kept his hand on top of the shoe, with his index finger on the next card to be extracted. As I watched, he dealt SECONDS, pulling back the top card for the player and waiting until he need¬ ed the pulled-back card, either to give himself a winning hand, or to beat the high roller with it. WATCH OUT for dealers who keep their hand on the shoe. I did not dare say anything. I might not have gotten home.

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At the present time, I am retired and writing articles. We spend at least five days monthly at the Foxwoods Casino in Connecticut. You never have to ask for a comp. Your play earns points which may be used in any manner you wish. The only problem is crowding. If your table is cold, switching is difficult unless you want to play high. Being retired, I limit myself to $10 or $20 bets, occasion¬ ally betting higher only when the count makes it irre¬ sistible.

APPENDIX H:

10 MILLION HANDS OF BLACKJACK

T

he

following computer

JL output contains the results of running 10 million hands of Blackjack. Summaries are presented for a. Basic Strategy: Expectation was -0.2337 percent b. Running Count Adjustments for bet size and strategy for two different counting methods (see bl and b2 below): Multiply the basic bet-unit of $20 by one-half the run¬ ning count, if the running count exceeds +3. If the running count exceeded +2, do not hit 16 vs 10, including 8-8; do not hit 12 vs 2 or 3. If the running count is less than -1, hit 12 versus 4. If the running count is less than -4, do not soft double.

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BLACKJACK

BOOK

The two counting methods shown below were analyzed: bl. Plus 1 for 2 thru 7; Minus 1 for 9, Ten, Ace Expectation: +0.4007 b2. Plus 1 for 4 thru 7; Minus 1 for Ten Expectation: +0.3012 The summaries contain: Player expectation, percent Net amount won or lost Number of games won Number of games lost Number of games tied Number of “regular ’ games (no double, split, or BJ) Number of wins, losses, ties for regular games Number of Blackjacks: dealer, player, pushes Number of Double-downs: won, lost, tied Number of Splits: won, lost, tied Number of Doubles after Split: won, lost, tied

The strategies used do not contain any insurance bets. d. Dealer results with upcards of Ace thru 10: proba¬ bility of reaching 17, 18, 19, 20, 21, and Bust e. For every upcard from Ace thru 10 versus all 2-card player combinations: Number of Games and number won, lost, tied Number of Doubles and number won, lost, tied Number of Splits and number won, lost, tied f. Comparison of results for each of two counting methods (bl) and (b2) cited above:

THE

ULTIMATE

BLACKJACK

BOOK

/

121

Running count from -19 thru +20 in unit increments Number of Games, Percent Expectation for each count g. Basic Strategy used for the 10,000,000 hand run: Player action for upcard vs first two player cards Player action for upcard vs player total after draw: Hard 12 thru 17; Soft 17 thru 19 h. Number of hands Won and Lost when player or deal¬ er was dealt any card from Ace thru Ten. The 10,000,000 games run was done on a Gateway 90MHz Pentium with 16Meg RAM. Running time was approximately five hours. The composition of the deck was displayed at intervals of about 500,000 games apart to check for random distrib¬ ution of the cards. The computer program was prepared by the author, using PC Cobol. The program is input-driven, and very flexible. The object deck is available on a floppy disk for a fee. Contact the author.

122

/

THE

ULTIMATE

BLACKJACK

BOOK

COMPUTER OUTPUT: RESULTS WITH NO COUNT ADJUSTMENT UNITS ARE GAMES PLAYER EXPECTATION, PCT

-0.2337 10000000 GAMES

NET AMOUNT WON WITH BET UNIT OF 20 WAS 520270.00NG

=10000000

W

=04474397

L

=04895818

T

:00853960

REG =08005864

WR

=03315632

LR

=03935658

TR

:00754574

RBJ =00927919

RBD

=00452764

RBP =00453599

RBT

:00021556

NDB =00842042

WDB =00468405

LDB =00314455

TDB

:00059182

NSP =00224175

WSP

LSP

TSP

:00018648

SPD =00060927

SPDW=00036678

=00236761

=00192941

SPDL =00022020

SPDT :00002229

VARYING STRATEGY (BELOW) IS AS FOLLOWS: STAND IF COUNT > 2 AND YOU HOLD 16 VS 10 (including eights) HIT 12 VS 4 IF COUNT IS LESS THAN -1 NO SOFT DOUBLING IF COUNT IS LESS THAN -4. VARY BETS AS FOLLOWS: MULTIPLY BY COUNT-FACTOR (0.5 * count) IF COUNT EXCEEDS 3.

RESULTS VARYING BET AND STRATEGY WITH COUNT 1 UNITS ARE COUNT-FACTOR PLAYER EXPECTATION, PCT 0.4007 21848300 GAMES NET AMOUNT WON WITH BET UNIT OF 20 WAS 1940480.00 NG

=21848300

W

=09808192

L

=10626939

T

=01876217

REG =17436154

WR

=07244416

LR

=08532833

TR

=01658905

RBJ =02170816

RBD . =01057965

RBP =01058764

RBT

=00054087

NDB =01778282

WDB

=01004858

LDB =00649513

TDB =00123911

NSP =00463048

WSP

=00500154

LSP

TSP

SPD =00122680

SPDW =00074558

=00386628

SPDL=00043514

=00039314

SPDT =00004608

THE ULTIMATE BLACKJACK BOOK

/

123

RESULTS VARYING BET AND STRATEGY WITH COUNT 2 UNITS ARE COUNT-FACTOR PLAYER EXPECTATION, PCT 0.3012

19160681 GAMES

NET AMOUNT WON WITH BET UNIT OF 20 WAS 1279610.00 NG

=21848300

W

=09808192

L

=10626939

T

=01876217

NG

=19160681

w

=08596828

L

=09322284

T

=01643826

REG =15345951

WR

=06378165

LR

=07511061

TR

=01456725

RBJ =01848573

RBD

=00900654

RBP =00903419

RBT =00044500

NDB =01563900

WDB

=00882419

LDB =00572544

TDB =00108937

NSP =00402257

WSP

=00432825

LSP

TSP

SPD =00109637

SPDW =00066674

=00338025

SPDL =00038822

=00033664

SPDT =00004141

DEALER PROBABILITIES OF REACHING 17 THRU BUST: (BJ NOT INCLUDED) REACH (Rows represent upcards from Ace thru Ten) 20

21

BUST

17

18

19

A

0.1881

0.1893

0.1889

0.1897

0.0769

0.1668

2

0.1398

0.1348

0.1297

0.1241

0.1183

0.3531

3

0.1337

0.1309

0.1251

0.1207

0.1147

0.3746

4

0.1305

0.1236

0.1211

0.1167

0.1121

0.3958

5

0.1217

0.1226

0.1172

0.1121

0.1078

0.4183

6

0.1657

0.1064

0.1063

0.1020

0.0975

0.4218

7

0.3679

0.1382

0.0786

0.0791

0.0737

0.2621

8

0.1285

0.3597

0.1293

0.0687

0.0698

0.2437

9

0.1202

0.1168

0.3522

0.1201

0.0604

0.2299

T

0.1212

0.1205

0.1214

0.3676

0.0378

0.2311

\lk

/ THE ULTIMATE BLACKJACK BOOK

SUMMARY OF GAME RESULTS: UP= 1 Cl C2

GMS

w

L

T

DBL

-w

DBL -L

DBL

DBL -T

SPL

SPL

-w

SPL -L

SPL -T

A

1

1

6479

3094 3144

241

0

0

0

0 5252 3094 1917

1

2

8719

2503 5543

673.

0

0

0

0

0

0

0

0

1

3

8695

2459 5615

621

0

0

0

0

0

0

0

0

1

4

8550

2169 5743

638

0

0

0

0

0

0

0

0

1

5

8698

2215 5905

578

0

0

0

0

0

0

0

0

1

6

8520

2089 5647

784

0

0

0

0

0

0

0

0

1

7

8551

2253 5551

747

0

0

0

0

0

0

0

0

1

8

8591

3183 4336

1072

0

0

0

0

0

0

0

0

1

9

8696

4498 3071

1127

0

0

0

0

0

0

0

0

0 10667

0

0

0

0

0

0

0

0

1 10 34486 23819

241

2

2

4390

1011

3101

278

0

0

0

0

0

0

0

0

2

3

9132

1994 6527

611

0

0

0

0

0

0

0

0

2

4

9330

1973 6790

567

0

0

0

0

0

0

0

0

2

5

9199

1795 6610

794

0

0

0

0

0

0

0

0

2

6

9103

2098 6171

834

0

0

0

0

0

0

0

0

2

7 9369

2552 5976

841

0

0

0

0

0

0

0

0

2

8

9168

2923 5416

829

0

0

0

0

0

0

0

0

2

9

9266

3387 5170

709

0

0

0

0

0

0

0

0

2 10 37055

7179 27673

2203

0

0

0

0

0

0

0

0

3

3

4457

928 3250

279

0

0

0

0

0

0

0

0

3

4

9278

1828 66/ /

773

0

0

0

0

0

0

0

0

3

5

9318

2082 6373

863

0

0

0

0

0

0

0

0

3

6

9297

2648 5748 '

901

0

0

0

0

0

0

0

0

3

7

9345

2949 5472

924

0

0

0

0

0

0

0

0

3

8

9365

3349 5304

712

0

0

0

0

0

0

0

0

3

9

9240

1778 6894

568

0

0

0

0

0

0

0

0

3 10 36726

6664 28034

2028

0

0

0

0

0

0

0

0

4

1057 2848

435

0

0

0

0

0

0

0

0

4

4340

THE ULTIMATE BLACKJACK BOOK

Cl C2

GMS

w

4

5

9224

2561

4

6

9379

4

7

4 4

L

DBL

DBL

-w

DBL -L

DBL -T

SPL

SPL

-w

125

SPL -L

SPL -T

845

0

0

0

0

0

0

0

0

3079 5402

898 ,

0

0

0

0

0

0

0

0

9251

3343 5227

681

0

0

0

0

0

0

0

0

8

9284

1757 6994

533

0

0

0

0

0

0

0

0

9

9189

1693 6996

500

0

0

0

0

0

0

0

0

4 10 36562

6161 28450

1951

0

0

0

0

0

0

0

0

5

5

4532

1519 2612

401

0

0

0

0

0

0

0

0

5

6

9125

3326 5102

697

0

0

0

0

0

0

0

0

5

7

9181

1753 6888

540

0

0

0

0

0

0

0

0

5

8

9021

1560 6971

490

0

0

0

0

0

0

0

0

5

9

9433

1541

7411

481

0

0

0

0

0

0

0

0

5782 29432

1740

0

0

0

0

0

0

0

0

5 10 36954

5818

T

/

6

6

4452

911

3294

247

0

0

0

0

0

0

0

0

6

7

9237

1609

7143

485

0

0

0

0

0

0

0

0

6

8

9122

1487 7130

505

0

0

0

0

0

0

0

0

6

9

9172

1402

7367

403

0

0

0

0

0

0

0

0

5206 29783

1610

0

0

0

0

0

0

0

0

719 3416

209

0

0

0

0

0

0

0

0

6 10 36599 7

7

4344

7

8

9153

1459

7282

412

0

0

0

0

0

0

0

0

7

9

9294

1407 7486

401

0

0

0

0

0

0

0

0

7 10 36787

4209 27693

4885

0

0

0

0

0

0

0

0

8

8

7414

2465 4444

505

0

0

0

0 5994 2465 3024

505

8

9

9141

1060 6849

1232

0

0

0

0

0

0

0

0

8 10 36639

9085 22724

4830

0

0

0

0

0

0

0

0

9

1046 2797

586

0

0

0

0

0

0

0

0

9 10 36711 14025 17761

4925

0

0

0

0

0

0

0

0

10 10 72385 36921 25745

9719

0

0

0

0

0

0

0

0

9

4429

126

/ THE ULTIMATE BLACKJACK BOOK

UP = 2 Cl C2

1

1

GMS

6910

w

L

T

3867 2846

DBL

197

0

DBL,

-w

DBL -L

0

0

DBL, -T

SPL

SPL -W

SPL -L

0 4886 3098 1591

SPL -T

197

*