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English Pages 160 [164] Year 2004
 BLACKJACK
ACE PREDICTION
DAvID McDowELL : SPUR OF THE MOMENT
PUBLISHING
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BLACKJACK ACE PREDICTION The art of advanced location strategies for the casino game of twentyone
For Flora
BLACKJACK ACE PREDICTION The art of advanced location strategies for the casino game of twentyone DAVID MCDOWELL
With a foreword by Michael Dalton
Copyright © 2004 by Spur of the Moment Publishing First edition / First printing ~ Printed in the United States of America 09 08 07 06 05 04
OSS]. Guat See) 1
ISBN 1879712105 Library of Congress Control Number: 2004110090
McDowell, David. Blackjack Ace Prediction: The art of advanced location strategies for the casino game of twentyone.
Includes appendices, bibliographical references and index.
All rights reserved. No part of this publication may be translated, reproduced, or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage and retrieval system, without the express written permission of the copyright owner. Inquiries should be addressed to Spur of the Moment Publishing, P.O. BOX 541967, Merritt Island, FL 329541967
USA. Visit our web site at www.BJRnet.com to order copies of this book and for possible updates, errata and to participate in our message forums. For wholesale and retail information email [email protected] or contact us at the address above.
Blackjack Review Network Gaining an edge in cards and lite!
PUBLISHER / EDITOR: Michael Dalton, Spur othe Moment Publishing, Merritt Island, Florida, [email protected] www.BJRnet.com.
COVER ARTWORK
CREDIT:
Copyright © Directional Publishing, Inc., “Ace of Spades” by Abigail T. Kamelhair.
PHOTOGRAPHIC
CREDITS:
Edward Thorp, courtesy of New Mexico State University Library Archives, Las Cruces. William
Earl
Walden,
courtesy
of Washington
State University
Library
Archives, Pullman.
Edward Thorp at University of California, Irvine, from I Have a Photographic Memory by Paul Richard Halmos, courtesy of the American Mathematical Society, Providence, Rhode Island.
Shoulder holster, courtesy of Warren of California, www. travelsecurityaccessories.com, 8009324465.
San
Diego,
DISCLAIMER: Although the author and publisher have exhaustively researched all sources to ensure the accuracy and completeness of the information contained in this book, we assume no responsibility for errors, inaccuracies, omissions or any other inconsistency herein. Any slights against people or organizations are unintentional. Gambling infers risk. The author and publisher accept no legal responsibility or financial liability whatsoever for any reader’s application of the information contained in this book.
TABLE OF CONTENTS Tables and Figures icc.::sscsaceesesranscese. cacseeqnrt Waccnacneceeeesueecerd 10 A CKNOWICASINENES: jc ccc.cc:ssaesnncecns sebccecsasdacarsenacecereecsevanrdiunes 12
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“How I Invented Ace Prediction ””—Ed Thorp ............... 30
Table of Contents
Chapter 2  NonRandom Shuffling ........ccccccscscsscsceeseeeees 32
Permutations by Cutting and Shuffling .....cccccccccccc0c0000000+. aL Thorp’s NonRandom Shuffling Analysis.........0cc0c00000000+. 34
Epstein’s Analysis of the Imperfect Shuffle..........c0ccc000. a7 Casino Card Shuffles: How Random Are They? .............. 39
inpormation Loss
Card Shujpline
Babiana; Mystery: Charles T. JOP
sates ena
40)
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Chapter 3  Analyzing Shuffles..........cccssssssecssscesssoveseareres 45
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rete
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Chapter 4  Exploiting Shufflles ............ssccssssscsssssereesscesees 66
Applying Hannum ’s Shuffling ANALYSIS .......2c00sc0evereere 66 SIC MleSIRO TEINGNAOMMES Sa
2 scapes opsondecwi sesrarer p 69
encerersseeec ace 72 cer car estnee VACHS SILI CW) COIICSSCS Sc cacoar HoweReliable Are the Weaknesses? t.nccusss.nvsosasevacsenncnn/242 74 Prearcine Exel POsitions Of CATAS 0. sse.ohticecnsncnseosntte 76 MCN
OTN OICI IU
Table of Contents
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Chapter 5  Locating ACES ........sssseseesesssssrsrenseerencensesensnees 79 eee
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79
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The Prediction Problemac
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Applying Epstein 's Prediction Theorem..w255..veae= 91
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Table of Contents
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MACOS FIrst CAN, 26.31 ea
108
How Often Does the Ace “Hit the Money”? .......cc00.000.:. Tt Skilled versus Unskilled Predictors ...ccccccccccccccscccsssssesees Li When ine Wealer Gets the ACen I QTIICMNOLICOL EX PECIAIION.,
0
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Chapter 8  How Much to Bet.............cccssssssssssssssscsssssece 120 CCIE
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120
122 126 NAT:
AD DCTUICES ¢cecessssorsecoceduessstncccesesoseonassoesvassausestcaase30%: TABLE 43. FOUR SHUFFLES BY HAND (%) PreShuffle Position Position
105156
1%
57208
261312
 209260
21%
8%
27% 105156
14%
16%
157208
16%
25%
CVShufjle was used to simulate four thousand iterations of BJAP 3 on computer (Table 44): TABLE 44. FOUR THOUSAND SHUFFLES BY COMPUTER PreShuffle Position
PostShuffle Position
53104 105156
157208
]
53104  105156  157208  209260
4%
 9%
 20%
 23%
T% 17%
16% 19%
27% 19%
22% 9%
28%
17%
13% 17% 26%
17% 8% 9
8% 15% 23%
25% 19%  18%
10%
The software identified the same two shuffle weaknesses. Variance”
between the two sets of results was average variance was two percent.
small.
The
* [FS] pp. 202203.
70
Exploiting Shuffles
Concordance among Shuffle Iterations
Kendall’s Coefficient of Concordance W can be used to assess the similarity of outcomes for repeated iterations of a shuffle procedure. ” Average postshuffle positions for halfdeck segments were computed (see Appendix E) for all four iterations and W was calculated using these halfdeck averages. Measures of W range between 0 and 1, with values close to 1 indicating a high level of agreement. For this series of shuffles W was 0.96 (p < 0.1) indicating a very high level of similarity among repeated iterations of the shuffle. With this level of agreement we can be sure the cards in preshuffle positions 53—104 will end up in postshuffle positions 209— 260 very consistently. Differences in Average PostShuffle Positions In a random shuffle the average postshuffle positions for each halfdeck would be roughly equal. The KruskalWallis” test statistict H was computed using equation 41:
H=12/N(N+1)(¥R7/n) 3(N+1)
4]
Where,
YR’ = squared sum of ranks of average postshuffle positions N=48 n=4
/4)  3(48 + 1) = 44.52 H = 12/ 48(48 + 1) X (150154 The observed chisquared value y? was 44.52 at 11 The average postshuffle degrees of freedom (p /
0.21
=> 0.70  =
0.60

©
0.50

a. 0.30

8 0.40 0.20
 
0.10 
0.00

0
1
2
3
Additional Aces
FIGURE 68. ADDITIONAL ACES FOR SIX DECKS
* [FS] pp. 148149. 98
Predicting Aces
In moving from one to six decks, the probability of one or more additional Aces has increased from 0.1694 to 0.2064. From a practical standpoint, if a player can pinpoint an Ace to within a few cards, it pays to know its suit. Memorizing the target Ace, as well as the key card that precedes it, will increase your chances of receiving additional random Aces within the same few cards. STEERING THE ACES
In years gone by, players would spread their bet across multiple boxes to steer the Ace to the money. By carefully calculating the number of boxes needed, players could “‘catch” the Ace as their own first card, force a “bad” card (4, 5 or 6) to
the dealer,”° or increase their chances of getting a Tenvalue as their second
card.
Such
tactics,
when
successful,
led to
substantially increased advantages for the player, even when the “hit rate’ was much less than 100%. In most of today’s games it is difficult to steer cards with such accuracy. When it goes wrong, for example, if a player accidentally steers a Five to his own hand, it can be very costly. Also, Snyder (2003) has warned that the “steering hands,” i.e. those hands that do not catch the Ace, can cost the player up to 2% per hand in negative decks. He advised: “players who sequence aces would often find it more profitable in the long run to play a single hand when the key card(s) predict an ace is coming, not multiple hands in an attempt to catch it.”*’ Situations still arise, however, where it can pay to deviate from basic strategy. For example, suppose you see the cards in Fig. 69:
Predicting Aces
O95
q

a6
*
sy
*
ce +
ao
*
pekatse
a
He
ipl
BOXx3
BOX 2
Box 1
ee
SEA
A
FIGURE 69. STEERING THE ACES
According to basic strategy, the player at box 2 should stand with hard 15 against dealer up card Four. However, an Ace tracker may deliberately hit the hand until it busts so that, at the end of the round, the Ace of Clubs and the Ace of
Hearts are scooped up and placed together in the discard tray, with the Seven of Clubs above them as the key card. With a minimum bet, the cost of the “wrong” basic strategy play is relatively small (about 0.20 of the bet, for this example) compared to the large gain that will be made later when the Aces hit the big money. Note also, that if you memorize a key card in a hand next to an Ace (for example, the Five of Spades in Figure 69) and the hand busts, the key card goes to the discard tray without the Ace. Conversely, if the hand with the Ace busts, the Ace goes to the discard tray without the key card. Sometimes, an Ace will be on top of the discards and more discards will be placed on top of it. You can use a discard as a key card if you are sure it is placed immediately on top of the Ace. FALSE KEY CARDS
A false key card is one of identical suit and value to a genuine key card but with no Ace following. Since any false key card will not fool us occurring after a real key/target card sequence, we need only concern ourselves with one or more
false key cards that appear in isolation and before the genuine 100
Predicting Aces
key card. /f you observe two identical key cards next to each other, or separated by one or more cards, do not bet. The equation for the hypergeometric distribution: P(X= i) =h(i, n, M, N)
62
was used to calculate the probability P;; of one false key card occurring in isolation before the real key card within 26 cards. Since half the time, on average, the false key card will occur after the genuine sequence, the size of the sample n is assumed to be 12 cards (26 cards minus the genuine twocard sequence, divided by two). The number of successes in the population M is 5 (there are five duplicates of the real key card in a sixdeck shoe) and the population size N is 310 (the number of cards in sixdecks minus the two cards in the genuine sequence): Piz; = hd, 12, 5,310) =0.17 And, for all possible values of P(i): TABLE 63. PROBABILITY OF A FALSE KEY CARD I
AND
f.
032
0.80 ce 0.70
1.60}
1
O77
= 0.60 

2
0.01
2 0.40 ee

@ 0.50
302.00 4
0,00
5
0.00
Total
1.00
>/
0.18

0.90


f


0.20  0.10
0.00

0
1
: 2

 3
4
%
5
False Key Cards
FIGURE 610. PROBABILITY OF A FALSE KEY CARD
Summing for one or more false key cards: P(i) = 0.17 + 0.01 = 0.18. Since we are tracking four key cards per shoe, we must also calculate the probability Pj) of none, one or more different key cards being duplicated. The probability P(1) of one key card being duplicated is 0.18. The probability of two key cards being duplicated P(2) is 0.18 X 0.18 = 0.03,
Predicting Aces
101
(P(3) and P(4) = 0). Finally, 1  (P(1) + P(2) + PB) yields the probability P(0) for zero key cards:
PA)
TABLE 64. PROBABILITY OF FALSE KEY CARDS j
AND
0
0.79
cal
1 2 3 4
0.18 0.03 0.00 0.00
Sal ao  std oa
Total
=f
1.00 
1.00
    :
021
False zsCards
FIGURE 611. PROBABILITY OF FALSE KEY CARDS
For one or more different key cards P(j) = 0.18 + 0.03 = 0.21. POINTER CARDS
Another subtle principle from card magic, pointer cards, can reduce the number of bets made on false key cards. Note that 22 cards. in’ every deck (Ace, 3; 5,6, 7, 8; 9of Clubs,
Spades, Hearts, and 7 of Diamonds) have an asymmetric face design:
FIGURE 612. ASYMMETRIC FACE DESIGNS
In Fig. 612, for example, the genuine key card (left) has the majority of its pips facing up; therefore a duplicate card would not fool us with the majority of pips facing down. If you examine Bee® casino decks closely you will see (Fig. 613) that every card has a small but discernable difference in the width of the white space between the corner pips and the edge of the card. A sharpeyed sequence tracker can utilize this, also. Memorizing a card and its alignment 102
Predicting Aces
halves the number of bets made because of false key cards. P(j) is reduced to ~0.10. im Wide Space> ¢
4
+
q © Narrow Space
FIGURE 613. ASYMMETRIC WHITE SPACE
Asymmetric Back Designs
Steve Forte, an expert in gambling /egerdemain, offers this method to “read” Aces from the back: “the vast majority of decks are cut slightly offcenter. At first glance the backs all look the same, but when you carefully examine the edges, the small ‘triangles’ around the sides vary dramatically in size.”** Figure 614 shows an example using the back of an Ace from a new, standard 52card Bee® deck (No. 92 Club Special, Match Blue Diamond Back No. 67): \x
KOK. RS +
KKK WY + XY x
XXYY4°4°OO)
tatatatatets $, *
+4 + + + XXX
OK
¢ AY" %* %y)ne% ’ % wsDX)fe + ry x)PKS ate’ + ny + afore
4 4,
FIGURE 614. ASYMMETRIC BACK DESIGN
The card has big triangles running along the top edge and little triangles along the bottom edge (this edge is visible when the card is about to be dealt from a dealing shoe). Trackers can also use this to predict whether their first card, the dealer’s hole card, or their own draw card, will be “high” or “low.” Playing the turn applies to handheld games only. Forte explained: “As he receives his cards, the advantage player simply turns them to the desired alignment. Eventually, after a few deals, he’s got the deck ‘marked.’ All the little cards have the little triangles on top; all the big ones have big Predicting Aces
R
103
triangles on top.”*’ One very advanced play for games involves noting the backalignment of tracked. If an Ace is due on first base the player alignment of the next card to be dealt. If it matches
shoedealt any Aces checks the that of the
tracked Ace he bets on box 1. If not, he bets on box 2.
Reading the Edges
Here is Laurent Bujold’s secret for cutting Aces. Note how the diamond pattern on the back of a Bee® card is “bled off” at the edges. This causes stripes along the sides of the deck as shown in Figure 615:
FIGURE 615. “READING” THE EDGES OF A BEE® DECK
Look closely and you will see one card (reversed in direction) with edge markings out of line with the others. A row of dashes is visible in the white spaces between the stripes. Only one such readable Ace is required to gain a huge advantage. PUTTING IT ALL TOGETHER
Predicting Aces at the table is easier than the theory makes it look. Here we show how to predict an Ace in a game with the twopass, combo shuffle as described in Chapter 4. In this example, you have already seen the first four rounds of play. Having counted the cards as they were dealt, you know there are exactly 49 cards in the discard tray. Therefore, most of the cards in the next round will fall into the “weakness” segment identified in Chapter 4. 104
Predicting Aces
According to Table 52, two out of three times (0.654) the cards will include at least one Ace.
If two Aces are dealt, the
probability they will be next to each other is 0.154 (Table 53). On the fifth round, an Ace appears:
HT fober[4 BOXx4
BOXx3
BOX 2
BOXx1
FIGURE 616. AN ACE IN A TRACKABLE POSITION
You note the key card that will be under the Ace when the dealer scoops up the cards. You know the sequence lies within a sixcard slug ranging from positions 56—61 in the discard tray. From Chapter 4, you know this sequence is very likely to be in the third halfdeck from the top after the shuffle. However, to make sure, you eyeball the shuffle closely, and track the sequence to its final location. You are offered the cut card, and cut to bring the Ace straight to the top of the six decks.
On the first round of the new shoe ...
i" Oe oe 
BOX 4
BOX 3
FIGURE 617.
Flee
BOX2
BOX1
A FALSE KEY CARD APPEARS
Predicting Aces
105
... a key card comes out, but you realize it is false because its pips are facing the opposite way to those of the real key card! A few cards later, the genuine key card appears. You count the number of cards following the real key card (two) and predict the Ace will appear on box 4  2 = 2 in the next round. You make a big bet on box 2...
in ee BOXx4
BOXx3
BOX 2
BOX 1
FIGURE 618. THE ACE FALLS ON THE PREDICTED BOX
... the Ace falls on box 2! This is an extremely elegant and incredibly powerful technique.
THE CURSE OF WANG
‘
J ang was an old Chinese guy who could be found most evenings sitting at third base. All the dealers called him the “Zen Master” because win or lose, he never lost his temper and, regardless of whether he got good cards or bad, he never complained. However, Wang was very superstitious. He kept a little spider in a matchbox. Whenever he was dealt a stiff he would peek inside the box to consult the spider. Depending on which
106
Predicting Aces
side of the box the insect was sitting on, Wang would hit or stand. One night Wang was dealt a Two and a Ten for a hard total of 12, while the dealer’s card was
a Three.
The old man
looked inside the box and the position of the spider told him he should stand. Wang stood. The dealer drew a Seven and a Ten for a total of 20 and all five players at the table lost. It was too much for one high roller from New York. He stood up and yelled at Wang: “Learn how to play the game, buddy!” before storming off. Wang never even blinked. On the next round Wang got a Four and a Queen for a total of 14. Again, the spider signaled: “stand.” The dealer got 19 and everyone lost again. A man and his wife were completely disgusted and got up to leave. The husband pointed angrily at Wang and said to the dealer: “That old fool has put a curse on the whole table!” Wang didn’t flinch. The old man’s play was so pathetic even the pit boss was shaking his head. Eventually, he said to the dealer: “Let him play any darned fool way he pleases,” before wandering off. I was catching plenty of Aces and, with no pit bosses watching, betting the table maximum without any heat whatsoever. The winnings were piling up fast. After a while, Wang quit. Since I’d filled my boots I decided to leave too. Outside, I noticed Wang lurking in the dark alleyway across the street. I walked over and whispered: “Wang, here’s your hundred dollars. See you again tomorrow.” Without saying a word, the “Zen Master” took his share of the winnings and disappeared into the night.
Predicting Aces
107
7 MATHEMATICAL
EXPECTATION
BEAT THE DEALER
In his bestselling blackjack book Beat the Dealer, Edward Thorp wrote: “Suppose there are five cards left, mostly Aces and Tens, and that you decide to take five hands. Then you get all five of the cards in this favored group, and the dealer gets none of them, for he runs out of cards and must
shuffle before dealing the first card to himself. If you now get a Ten as your first card this gives you a 15 to 20 per cent advantage; starting with an Ace gives youa 35 to 40 per cent advantage.”” VALUE OF AN ACE AS FIRST CARD
In Casino Holiday (1970), Jacques Noir gave +51% as the player’s percentage with Ace as first card. He noted: “Thorp gives an estimate of this number as 3540%. The 5% spread in his estimate indicates it is just a crude guess, and this should not be interpreted as a real difference of opinion.” Griffin (1981) stated: “the player’s expectation as a function of his own initial card [is 52% for an Ace].””* Snyder 108
Mathematical Expectation
(2003) observed: “Depending on the number of decks in play and, especially, on the rules, the ace may be worth anywhere
from about 51% to 53% to the player’s hand.””
Julian
Braun’s blackjack simulation data” enable us to produce Table 71, the player’s precise expected value E(X)* with Ace as first card, for one and four decks: TABLE 71. EXPECTATION WITH ACE AS FIRST CARD Ace
1deck
4decks
2
0.005
0.022
3
0.031
0.050
4
0.069
0.077
5
0.108
0.103
6
0.057
0.074
T
+0.052
+0.038
8
+0.276
+0.270
o
+0.594
+0.582
10
+1.445
+1.433
J
+1.445
+1.433
Q
+1.445
+1.433
K
+1.445
+1.433
A
+0.406
+0.332
Totals
+6.84/13
+6.63/13
E(X)
= +0.526
= +0.510
Expected value for all first cards for one and four decks are shown in Table 72: TABLE 72. EXPECTATION FOR ALL FIRST CARDS FirstCard
A
2
3
4
5
6
i
8
y)
10
One Deck
+0.53
0.13
0.15
0.16
0.21
0.20
0.18
0.08
0.00
+0.15
Four Decks +051
.043,
0.15.017
021
0.21.
0.18.
0.09..
0.01
+0.14
With an Ace as first card, expectation ranges from +0.51 to
* TFS] p. 161. Mathematical Expectation
109
+0.53. Rules giving the player more options when holding an Ace, such as soft doubling, resplitting Aces, and doubling after splits, increase the value of the Ace to the player. If a hand with Ace as first card is played many thousands of times the expectation in a fourdeck game is +0.51 regardless of the value of your second card. However, expectation for any individual hand depends on the value of the card that falls on the Ace. There are equal numbers of each second card in neutral decks. There are slightly less Aces because one is used up as first card. “Good” second cards for the player are 7, 8, 9, 10, J, Q, K, and Ace. “Bad” second cards are 2, 3, 4,5, and 6. Note: Ifa 26 value falls on an Ace, expectation is negative!
For a detailed analysis of how to play the hand if the player does not receive a Tenvalue as second card on the Ace, see
the article “More on the Ace in Hand ...” by James Grosjean and Previn Mankodi.”° Probability of a Ten as Second Card
Eq. 71 gives the conditional probability of receiving a Tenvalue as second card, given that the first card is an Ace:
P(T) p(A) = p(A) X p(T) /p(A)
Fel
For six decks:
P(Z) p(A) = (24/312 K 96/311) / (24/312) =0.31 To increase the chance of getting a natural keep a running count of each round individually and key Aces only in rounds that have a negative count.
* [FS] p. 149. 110
Mathematical Expectation
HOW
OFTEN DOES THE ACE “HIT THE MONEY”?
Player's Expectation—100% Correct Predictions If the Ace landed on the predicted betting box every time, the player’s expectation would be +0.51. Each one hundredunit bet would earn 51 units! Unfortunately this is unrealistic. The exact number of Aces that will hit the money can vary greatly, depending on the number of cuts and riffles in the shuffle. Player’s Expectation—38% Correct Predictions
According to the tworiffle shuffling results reported earlier, 38% is a more realistic estimate for predicting the correct box. The Ace will land on the expected box in 38 out of every one hundred predictions made: @ @
100% expectation = +0.51 38% expectation = 0.38 X +0.51 = +0.1938
Recall from Chapters 3 and 6 that broken sequences and false key cards further reduce the player’s expectation. In Chapter 3 the probability of broken sequences in a simple riffleandrestack shuffle was estimated to be 0.15. In other words, fifteen times in every one hundred predictions the Ace will fail to land where expected because it has been cut away from the key card in the shuffle. In Chapter 6, the probability of false key cards was calculated as 0.10. Thus, false key cards will fool us into betting on phantom Aces a further ten times. This means we must reduce the 38% hit rate by a further 25% to leave 13%. Lastly, since the Ace hits the predicted box only 13 times in every hundred bets, we must deduct the cost of playing the other 87 hands using basic strategy.
Mathematical Expectation
111
SKILLED VERSUS UNSKILLED PREDICTORS
A useful way to view “hit rate” is to compare the expected results of a skilled predictor, such as a professional Ace tracker, with an unskilled player—that is, someone who predicts Aces merely by guessing. For instance, in our Sixdeck game, the unskilled “guesser” finds that, on average, he = predicts successfully one Ace in every thirteen attempts: 13 X 24/ (24 + 288) = 1, yielding an overall hit rate of 0.07. The skilled Ace tracker, on the other hand, predicts an average of almost two Aces in thirteen attempts: p = 13 X 42 / (24 + 288) = 1.75, with a hit rate of around 0.13. In other words, the
skilled player is nearly twice as good at predicting Aces as the unskilled player (Table 73): TABLE 73. SKILLED VS. UNSKILLED PREDICTORS
Aces 0
Skilled 0.15
Unskilled 0.35
i}
0.31
0.39
2
0.29
0.19
3
0.17
0.06
>,
eed aes ees 0.70 } 
ees te nie)
= 0.60  9g 0.50
© 0.40  Eaves)

0.20  0.10
4
0.06
0.01
5
0.02
0.00
6
0.00
0.00
Totals >/
1.00 0.85
1.00 0.65
0.00
0 eV
eeee ee
No. of Aces predicted successfully in 13 attempts
FIGURE 71. SKILLED VS. UNSKILLED PREDICTORS
Table 73 reveals the skilled predictor makes one or more correct predictions (for every thirteen attempts) 85% of the time while the guesser manages the same only 65% of the time. However, consider how slight the difference is between the expert and the guesser. The expert predicts just six more Aces in every hundred than the guesser but, because the Ace is the most important card in blackjack, these six hands give the expert a significant advantage over the house—the source of the expert’s lucrative profits. 112
Mathematical Expectation
WHEN THE DEALER GETS THE ACE
The player expects to get the Ace “on purpose” 13 times in every one hundred attempts. This will be true at a full table, where the dealer has little chance of getting the Ace “by accident.” sometimes
However, at less than full tables, the dealer will get the Ace accidentally instead of the player, further reducing the expectation of the player. The probability of the dealer getting the Acecan be calculated fairly precisely, given the number of riffles, the dealer’s riffle “signature” and the number of betting boxes covered. But, for a more general “formula,” we need a more general rule. Snyder (2003) proposed a “rule of thumb” solution that says, in the long run, the player and the dealer will share the Aces 50/50.”° This works for any number of riffles and any number of boxes covered. It is a very conservative rule, since you are effectively saying the dealer will get one tracked Ace by accident for every extra Ace the player catches using skill, but it has the great virtue of simplicity. When the dealer gets the Ace, it costs the player 0.34 with multiple decks (0.37 if the Ace falls as the dealer’s hole card), but this is more than compensated for by the fact that, when the player gets the Ace, it earns +0.51 per bet. The player's net expectation is +0.51  0.34 = +0.17. However, this is before any deductions are made for broken sequences, false key cards or basic strategy disadvantage. MATHEMATICAL
The
author’s
EXPECTATION
formulas,
given
below,
can
when
be
used
to
predicting
calculate mathematical expectation E(X) Aces. Eq. 72 can be used to calculate the probability P(h) that the Ace will Ait the money: 72
P(h)=a(b+f) Where,
Mathematical Expectation
113
a = probability of an average number of cards separating initially adjacent cards b = probability of broken sequences f= probability of false key cards Assuming a = 0.38,
b= 0.15,f= 0.10
P(h) = 0.38  (0.15 + 0.10) = 0.13 Therefore, the probability P(m) that the Ace will miss the money =   A = 0.87. At this point we invoke Snyder’s rule of thumb—for every additional Ace the player successfully predicts using skill, the dealer gets an Ace by accident. In this case that means the dealer gets six additional Aces. P(m) is reduced from 0.87 to 0.81, while P(d), the probability of the dealer getting the Ace by accident, becomes 0.06. Eq. 73 can now be used to calculate mathematical expectation E(X): E(X) = E,h +
E,d +
E3m
73
Where,
E = player’s expectation if the Ace hits the money E, = player’s expectation if the dealer gets the Ace by accident E; = player’s expectation if the Ace misses the money h = probability that the Ace will hit the money d = probability that the dealer will get the Ace by accident m = probability that the Ace will miss the money Assuming £, = +0.51, E> = 0.34, E; = 0.005 h = 0.13, d= 0.06 m= 0.81
E(X) = (0.51 X 0.13) + (0.34 X 0.06) + (0.005 X 0.81) = 0.0663  0.0204  0.0040 = +0.042 Against the tworiffle R&R shuffle, Ace Prediction achieves singletrial win probability p ~0.52 X (1  0.083)” =
0.48, loss probability gq~0.48 X (1  0.083) ~0.44.°% 114
For
Mathematical Expectation
simplicity, the increase or decrease in fortune for each play is assumed to be one unit. Under these assumptions, a = 1 and B = 1. Expected value E(X) of a single play, X = 1, is given by
E(X) = ap  B:
E(1) = (1 X 0.48)  (1 X 0.44) =+0.04 With lots of plugs, strips and box cuts, the probability of broken sequences was estimated as 0.23. Substituting this into equation 72 yields:
P(h) = 0.38  (0.23 + 0.10) = 0.05 Assuming E; = +0.51, E> = 0.34, E3 = 0.005 h = 0.05, 0.06 m= 0.88
d=
E(X) = (0.51 X 0.05) + (0.34 X 0.06) + (0.005 X 0.89) = 0.0255  0.0204  0.0044 = +0.00 The additional cuts have destroyed the profitability of this game completely. This highlights the extreme importance of finding shuffles with as few passes, riffles, and cuts, plugs, strips or box cuts as possible. PREDICTABILITY OF CASINO SHUFFLES
Do not believe the above expectation figures to be precise. They are simply my best guesses based on probability theory. Before predicting Aces in a real casino, you should test the actual predictability of the shuffle using statistics. Ultimately, as Epstein pointed out, mathematical expectation depends on accuracy of prediction. John May, a professional gambler, advised: “Many pro players will try mentally to locate an Ace over a hundred times before actually playing it ... Some teams will send in a minimum bettor to ‘clock’ the dealer. This informant will come away and communicate the information to his Mathematical Expectation
115
confederates, who then analyze the data and send in a big money team.””” To collect the required data, the “clocker’ mentally predicts an Ace one hundred times, betting the minimum. A tally is kept of how many times the Ace lands on the predicted box. This is compared to the number of successes predicted by theory. If the success rate is higher than one in thirteen, you have a betterthanguessing chance of predicting when an Ace will be dealt. In the sixdeck, tworiffle R&R shuffle analyzed earlier 13 correct predictions are expected for every hundred made. However,
the actual
number
may
be less, or more,
than
expected. If it is less, but the game is still marginally profitable, then you have two options: find a more predictable shuffle, or play knowing you have a lower expectation. If you opt for the latter, make sure you reduce your bet size to a safe level. The Exposed Ace Occasionally a dealer will accidentally expose an Ace. This usually happens when the dealer anticipates a hit, then realizes the card is not required to finish the round.’ A common casino practice is to place the exposed card under the shoe, ready for the first hand of the next round.'®! If you are sitting at first base, and know for sure your first card will be an Ace, you should make a very large bet! Using conventional basic strategy to play out the hand, Equation 74 yields the “optimal” fraction fof bankroll to wager:
f=EN
74
Where,
J= fraction of bankroll E(X) = expectation on the hand v = variance on the hand
116
Mathematical Expectation
Calculating the Variance
The variance’ for the outcome of this hand is greater than a normal blackjack hand (average variance ~1.2)!% because of the higher likelihood of a 3 to 2 payoff for a natural. For expectation = +0.51, variance is calculated as follows: TABLE 74. VARIANCE FOR THE EXPOSED ACE eats
Payoff
eee
Square
Prob.
Product 0.47
Win
1.50
0.99
0.98
0.48
Loss
1.00
1.51
2.28
0.44
1.00
Push
0.00
0.51
0.26
0.08
0.03
Totals
‘1.00
Thus,
f=+0.51/1.50=0.34 In other words, if you know an Ace will be your first card, you should risk 34 percent of your bankroll. However, do this only if you are certain you will get the Ace. Unless an Ace is exposed as above, unpredictability caused by card riffling, broken sequences and false key cards means you cannot be absolutely certain where it will appear. It could go to other players, or worse, the dealer. In such cases, bet
size should be much smaller.”
* See Chapter 8 of this book for a detailed discussion of optimal bet size when other players or the dealer can get the Ace.
Mathematical Expectation
117
THE COUNT OF CIAMPINO
‘[s exiled King of Italy, Umberto II, waved his last goodbye to a crowd of loyal supporters at Rome’s Ciampino airport. As he boarded the plane, he whispered to his financial attaché: “fa i conti” (do the accounts). An appointments secretary, standing behind him, thought he said: “make them counts.” Thus, two hundred new Italian noblemen were created that day, all of them named after the airport from which the King left Italy. In 1973 Alessandro de Luca, Count of Ciampino, watched in horror as his entire portfolio of stocks turned sour. He was ruined. No longer able to afford his elegant rooms at Monaco’s Hotel de Paris, he turned in desperation to the card tables of MonteCarlo, where, to his surprise, he discovered an
easy way to beat the blackjack game. Nearly every day for the next 25 years, he exploited the same sweet secret. At the Grand Casino, where croupiers spend 25 to 30 years perfecting their skills, Alessandro knew every dealer by name and, of course, they all knew him, but he had a special fondness for the apprentice, an earnest, bespectacled young man named Henri Renaud. Alessandro would reserve seat No. 1 at Henri’s table when it opened at 4 p.m. He had noticed Henri would slide a card forward, out of the lip of the shoe, and then retract it when he
realized another card was not needed to complete the round. Henri would place the card drawn in error underneath the shoe, ready for the first hand of the next round. If it was an Ace,
Alessandro
would
bet
10,000
French
francs,
the
maximum his pocketbook allowed. On the following round, with 100% certainty, Alessandro would receive the accidentally exposed Ace, and was often thrilled to see a Ten appear as his second card. Although he had no idea how big his percentage was, his meticulously kept accounts told him it was approximately half of everything he bet.
118
Mathematical Expectation
For Alessandro this single bet, of which he made around two hundred per year, constituted his entire day’s labor.
Win,
or lose, he would gather up his chips, stroll through the passage souterrain linking the casino to the hotel, and put on a fresh shirt before dinner. Occasionally, if he had had a winning day, he would allow himself a quiet smile, and toast Henri with a bottle of Veuve Clicquot 1959.
Mathematical Expectation
119
How MUCH TO BET
EXPECTED RETURN
Expected return is the mean number of bets 4 you expect to win over a given number of hands, based on the mathematical expectation of the game: j= mathematical expectation X number of bets (n)
81
For example, for one bet, n = 1: «= +0.04 X 1 bet = +0.04 bets. For a series of n bets, multiply by n. For example, for four bets, n = 4: uw = +0.04 X 4 bets = +0.16 bets. Frequency of Betting Opportunities
Blackjack expert Stanford Wong conducted a study of sixdeck shoe games to determine the average speed of a game. On average, one player and the dealer take 12 seconds to play out a single hand. Each additional player adds another 7 seconds to a single round. The average sixdeck shuffle takes 100 seconds.'** Wong concluded that one player at a sixdecker gets an average of 248 rounds per hour, two players get
120
How Much to Bet
158, three players get 116, four get 91, five get 76, six get 64 and seven players get 56.'”° In a sixdeck game with 66% penetration and seven players (excluding the dealer) you see an average of ten rounds per shoe. The number of betting opportunities possible per shoe is equal to the number of rounds per shoe minus one = 9. Fewer players mean more rounds per shoe and more betting opportunities. The random probability of a betting opportunity depends on the number of riffles and whether the sequence falls in the playable part of the shoe. With seven players and the dealer, the minimum number of cards possible for one round is 16. In a game where four of the six decks are dealt, 208
cards
are
used
before
the cut card is reached.
However, when the cut card comes out in the middle of play the round is completed. Thus, on average, another ((8 X 2.7) 1) /2 = 10 cards will be seen. One card is deducted because, if the cut card is poised to come out, the next hand is not dealt.'°° Thus, a realistic estimate of the “upper” number of cards possible for ten rounds is 218. This gives 218  16 = 202 cards dealt out as “playable” and 201 possible points where the end of a round can occur. Actual betting opportunities occur when the end of a round coincides with the appearance of a key card but not the Ace that follows it. The probability of an actual betting opportunity goes up if you hold in memory more than one sequence per shoe. Holding four sequences per shoe, the probability of a betting opportunity in a tworiffle shuffle (with average threecard separations of the type ABBBA) is given by 16 /201 = 0.08. Table 81 below shows the average number of bets 7 you will make per hour in a game with a tworiffle shuffle, assuming: e
A sixdeck game with 66% of cards dealt before a
e e e
reshuffle. An average of 2.7 cards per hand. Number of players remains constant. Holding four sequences in your head from shoe to shoe.
How Much to Bet
121
TABLE 81. NUMBER OF BETS PER HOUR A
Players at table (including the dealer)
B
Cards per hand
8
DS
C
Cards per round (including the dealer)(4 XB)
21
D_
Cards dealt per shoe
218
E
Rounds per shoe (D/ C)
F
Shoes per hour
6
G_
Rounds per hour (E X F)
60
H_
Possible bets per shoe (£ — 1)
9
I
Possible bets per hour (F X #)
54
J
ACTUAL BETS PER HOUR (/ X 0.08)
4
10
Expected Return per Hour n=4,
p= 0.48, q = 0.44
u=+0.04 X 4 bets per hour = 0.16 ofa bet
RISK The number of bets we will actually win or lose over a series of wagers is more realistically predicted by the standard deviation o, a measure of variability from the expected number. For x trials of blackjack, standard deviation is given bye”
o=1.1Vn
82
Where, m = number of bets For n = 1, standard deviation per unit bet, o= ivn=1,3
Standard deviation per hour For n = 4 bets, hourly standard deviation s, is:
S= 1,1N 4=999 bets
122
How Much to Bet
Dr. John L. Kelly Jr.'°* (19231965) proposed betting a fraction of bankroll (total amount available for investment) that maximizes bankroll growth rate and (in theory at least) reduces risk of ruin to zero. Kelly advised betting a decimal fraction k = p  q of bankroll equal to the expectation on the next bet.'° The player resizes the bet according to the current bankroll as often as practically possible, and all winnings are reinvested. Equation 83''° was used to calculate r the exponential rate of growth of a bankroll for various fractions of k:
p= (k1 / 200 Ly)
83
Where,
k = fraction of Kelly Bet j= expected return per unit bet = +0.04 y = variance per unit bet = o°= 1.21 For example, for k=  (full Kelly)
r=(11°/2) X (0.047 / 1.21) = 0.000661157 TABLE 82. RELATIVE BANKROLL GROWTH RATE fraction”
k
0.040
l
RGR
I
0.036
0.9
0.98
0.032
0.8
0.94
0.028
0.7
0.88
0.024
0.6
0.80
0.020
0.5
0.71
0.016
0.4
0.60
0.012
0.3
0.48
0.008
0.2
0.34
0.004
0.1
0.18
* Fraction of bankroll wagered on a single bet.
How Much to Bet
123
Equation 84''' was used to compute the probability P(d) of the bankroll doubling before halving:
Pia
84
ea o
Where, a=0.5
k = fraction of Kelly Bet TABLE 83. DOUBLING BEFORE HALVING fraction
k
Prob.
0.040
1
0.67
0.036
0.9
0.70
0.032
0.8
0.74
0.028
0.7
0.78
0.024
0.6
0.83
0.020
0.5
0.89
0.016
0.4
0.94
0.012
0.3
0.98
0.008
0.2
0.998
0.004
0.1
0.999
Equation 85''* was used to compute the probability P/h) of ever losing half of the bankroll:
HO
Ge
85
Where,
a k = fraction of Kelly Bet
124
How Much to Bet
TABLE 84. EVER LOSING HALF fraction
k
0.040
1
Prob. 0.5
0.036
0.9
0.43
0.032
0.8
0.35
0.028
0.7
0.28
0.024
0.6
0.20
0.020
0.5
0.13
0.016
0.4
0.06
0.012
0.3
0.02
0.008
0.2
0.002
0.004
0.1
0
Richard Epstein, in “An Optimal Gambling System For Favorable Games,” (1964)''’ advised betting a fraction of bankroll equal to half the expectation on the next bet (“4%
Kelly”). Thorp commented: “My gambling and _ investment experience, as well as reports from numerous blackjack players and teams, suggests that most people strongly prefer the increased safety and psychological comfort of ‘half Kelly’ (or some nearby value), in exchange for giving up 4 of their growth rate.”''* For the professional Ace tracker playing at very high stakes, hard experience suggests “’4 Kelly” (fraction of Kelly Bet = 0.3) is most appropriate:
aD
Bsaa
Probability o & 8
3 ee— i
o &6

0.10
0.00 
P(h) = 0/02


 

  
lead
y
0.1 0.2 0.3 04 05 06 0.7 08 09 10 Optimal Fraction of Kelly Bet
FIGURE 81. OPTIMAL FRACTIONS OF A KELLY BET
How Much to Bet
125
Fig. 81 shows the bankroll will grow at 0.48 of the rate of betting “full Kelly,” but the probability of the bankroll doubling before halving is 0.98, while the chance of ever losing half of it is only 0.02.
OPTIMAL BET SIZE
Equation 86''° was used to calculate x, the number of units of bank required for any given fraction of a Kelly Bet: x=
s?/ ku
56
Where,
s = standard deviation per hour in bets k = fraction of Kelly Bet j4= expected return per hour in bets For the “4 Kelly” bettor:
x=
2.27/03X0.16
=100 units
Optimal bet size For E(X) = +0.04: e Bankroll = 100 units
e e¢
+0.04 X 0.30 =+0.012 Optimal Bet Size ~0.01 X 100 units ~1 unit
Therefore, divide your bankroll by one hundred to get one betting unit. For example: Bankroll = US$40,000
126
How Much to Bet
Betting unit = US$40,000 = US$400 100 Betting units can be in any currency. One unit = US$40 0 or one unit = UK£400. Betting units can be of any value. One
unit = US$40 or one unit = US$400. Average Hands to Double or Halve
Equation 87''° yields the average number of hands N before you either double the bankroll or lose half of it:
N=
_1_ In(b*/a*")
87
r
Where,
r = exponential bankroll growth rate* q = probability of doubling before halving b=2 a=05
N=
_1_
1n(2°?8/0.5°"*')
= 2051 hands
r
“DO NOT FORGET”
[:was after midnight when we left the casino, absolutely giddy with excitement. Another big win! Predicting Aces in Casino BadenBaden’s American Salon had turned out to be both easy and profitable. It was a simple tworiffle R&R
shuffle, with no “heat.”
We’d
won
nearly $20,000. It was time to leave town and bank the loot. The next morning Dan and I went to the bahnhof to catch the train to Ziirich. As it happened, every seat onboard was
* For this example, r= 0.000324433, k = 0.3 How Much to Bet
127
booked that day, but Dan, ever the optimist, had a clever, if
rather pricey, solution: “We’ll travel in the dining car.” So, we found ourselves speeding to Switzerland, eating gourmet food, sipping wine, reliving the exciting coup we'd pulled off in Germany’s oldest and grandest gambling hall. On the floor, under the table, sat Dan’s battered old black attaché case. Inside was $60,000 in crisp hundreddollar
bills—our $20,000 winnings plus the original $40,000 “joint bankroll.” We pulled into Ziirich about noon, got a tram to ParadePlatz, then walked the few streets to Furst & Fuchs
Bank, where we found the graysuited and bespectacled manager, Herr Gruber, waiting for us in the marble lobby. “Gentlemen,” he said, gesturing towards an oakpaneled door. Sinking into Gruber’s plush office chairs, we relaxed for the first time on the trip. Dan placed the case on the antique desk in front of him, twirled it round to face Gruber, pressed
back the clasps and flicked open the lid. “We'd like to deposit this in our numbered account,” he said. Gruber stared for a moment or two, and then raised his eyebrows. Dan jumped up and looked inside. His face froze. Nothing! “Oh my God, I’ve left the money in the hotel room!” “Phone them up!” I almost shouted, but Dan shook his head “No.” “What then?” While Gruber sat openmouthed, we figured the odds, from “best” to “worst”: A. The money was still in the hotel room. B. A chambermaid had found it. C. The next person to occupy the room had found it. Ever the pessimist, my bet was “C.” “We'll have to go back,” cried Dan. “Right now!” Several hours later, after swallowing a tale about a forgotten passport, the kindly clerk at our BadenBaden hotel turned the key in the door of Dan’s old room.
128
How Much to Bet
My nerves were in shreds as Dan groped gingerly into the hole in the sink pedestal where he’d hidden the banknotes. Eventually, he pulled out a black leather shoulder holster. Inside, in three neatly packed wads, was $60,000.
Attached to
the holster was a luggage ticket on which he’d carefully printed in big red letters the words: “DO NOT FORGET.”
How Much to Bet
129
APPENDICES APPENDIX A: SHUFFLING TERMINOLOGY Discards
Segment Slug Stack Pile
Break Grab Riffle Strip Shuffle Box Cut Player Cut
Cutoffs Topped Plugged Bottomed
Riffle & Restack Stepladder
Combo Shuffle OnePass TwoPass CrissCross
130
The cards placed in the tray at the end of a round. Any identified portion of the discards. A segment with extra high or low cards. Groups of shuffled cards placed one on top of the other. Any portion of a stack. To divide a stack or pile. To pick up a pile of cards with one hand. To interleave two grabs. The cards from the top of a grab are dealt on to the table, reversing their order. The dealer makes a lopsided cut by hand. The player inserts the cut card into the final stack. The cards that end up behind the cut card after the dealer’s cut. The cutoffs are placed on top of the discards. The cutoffs are inserted into the middle of the discards. The cutoffs are placed under the discards. A shuffle where grabs are made on/y from the original piles. A shuffle where grabs are made from the final stack as well as the original piles. A combination of two types of shuffles. The shuffle is over when the original piles are gone and the final stack is complete. The final stack is rebroken for a second shuffle. Piles are married using alternating grabs in an “X” pattern.
Appendices
APPENDIX B: CARD MOVEMENTS IN PERFECT SHUFFLES
One perfect
Two perfect
jie 24 29.) 42) SG Se ts) ot 44 46 + 20°33 S092 35 468
2 4 “6G 8
1S 17 19 21
28 30 32. 0934
AT 243 45 47
i)es)
wWlon
aS‘©
S14 16 118. 20s 27, 24
1 4 28) 15 38 48, 80, 17 3° 45 326 H9 2h T4734: 949" 36 23. SiP e251
4 6 28 10 12
N lon
ee we
eS No
De
ww a+
No on nN Ww
Two modified perfect
Aes? ae, jameey Aeneas ageess 50 37 NnNO
Ww\©
Appendices
we
Three perfect
Louesseel 4 Te 345 3°35 16. 4 36 17 Sue 371s Ge PE ICR ji 39" 20.
Aue l, Kae eed Ago oe Oral 22 48° 29 10 a 49°30 11 43 4 B50 kerries ees OR ibs eos 4.52
Three modified perfect
2940. DIP PA, 2242) 23g AS. DAN AA 25° 45°
132
7 27 AT 8.928) ASS 9 329 TA 10. $30 “SOR 11 73 ih eS) 42 32°59)
1k 34, isso: Ge6 igs 1838 1G 30"
he a3 es es 6
Appendices
APPENDIX C: MEAN OF D;
MEAN OF D; (ONERIFFLE SHUFFLE) D;
Frequency
0 1 2 3 4
271 1029 176 21 3 1500
Differences
1 0 1 2 3}
FXD
DX (F XD)
271 0 176 42 9 44
271 0 176 84 27 558
MEAN OF D; (TWORIFFLE SHUFFLE) D;
Frequency
Differences
FXD
DX (F XD)
0 1 2
74 184 253
3 2 1
222 368 253
666 736 253
3} 4
550 211
0 1
0 Dall
0 DIN
5 6 7
M5, 47 iL)
2 3 4
190 141 68
380 423 272
& 9
4 4
5 6
20 24
100 144
10
0
7
0
0
ipl
] n= 1440
8
8 A= 181
64 B= 3249
Appendices
133
MEAN OF D, (THREERIFFLE SHUFFLE) D; Frequency
0
45
Differences 7
FXD 315 660 865 1444
DX (F XD) 2205 3960

110
2
173
3
361
4
278
2502
5)
439
1756
6
638
7
830
8
507
9
326
10
241
11
176
12
110
HS
59
14
38
US
31
16
13
17
5
18
9
LS)
4
20
2
21
2
BY
1
23
0
4325 5776
fF Nn NHN ery CO WN —
DLT
Ssa& oe xe eS gS
24 25)
0
26
0
27
0
28
0
29
1 n = 4400
134
toi)
A =1208
484 B= 41624
Appendices
APPENDIX D: x TEST OBSERVED PostShuffle Position
152 53104 105156 157208 209260 261312
EXPECTED Diff?/Exp.
Appendices
PreShuffle Position
152 64 38 30 313) 13 30 34.67 24.82 0.32 0.63 0.08 13.54 0.63 DOSa/i 25 0.0000
53104 10
105156 20
157208 44
30 42 30 36 50
56
13
52
6.21 0.63 Ls) 0.63 0.05 6.78
De 13.13 0.16 0.93 0.93 13.54
10.78 0.82 4.63 12.32
37 29 29
209260 54 40 22 14 26
PEMA 8.67
261312
16 20 43 49 33 47
10.05 6.21 2.00 S293 0.08 4.39
135
APPENDIX E: KENDALL’S CONCORDANCE
COEFFICIENT
Data table for 2 deck averages Iteration
A
B
Cc
D
je)
F
H
I
K
L
1
94.92
155.85
197.12
172.66
118.69
221.12
141.92
240.92
222.42
119.46
2
102.00
142.62
171.54
190.56
134.62
228.27
120.62
221.46
209.00
161.54
3
94.73
152.92
188.65
175.49
122.08
23131
84.35
247.23
231.12
149.58
4
95.00
165.96
186.73
162.95
129.42
225.92
99.65
238.15
217.50
145.04
Ranks
I
Iteration
il
K 11
11
1
10
12
1
10
We = il
10
47
41
20 2 441
136
J
Zl
4
52026 = 22
el
484
225
Appendices
APPENDIX F: KRUSKALWALLIS TEST
ud
A
B
YR= YR2= N=48
31 961 4
Ol) 10201 4
ase GSS O76 AT kT SO 16084 17689 15876 3364 28561 5776 2209 32761 100 25600 7056 4 ey eee oe Be ee eae ee
240
2550
4422
0.0051
X 37538
147
DR/n =
Cc
DE)
3969
841
ee?
7140
GP
1444
mo
552
8190
See
25
6400
1764
rou 44.52 zy df=
l1 p KNAPP, Cards. 1996.
THOMAS
R., Learning Statistics
Thousand Oaks, California: SAGE
Through Playing Publications, Inc.,
* SIEGEL, SIDNEY, Nonparametric Statistics for the Behavioral Sciences. New York: McGrawHill Book Company, 1956. > SNYDER, ARNOLD, The Blackjack Shuffle Tracker’s Cookbook. Las Vegas, Nevada: Huntington Press, 2003. 2 , Jrade Secrets of the Shuffle Trackers Part III,” Blackjack Forum, Vol. XV, #1, March, 1995, pp. 1929.
” SNYDER, 2003, pp. 7477. : Morsk, DAVID H., Blackjack Reality: The David H. Morse
Method for Winning at Blackjack. Self published, 1998, pp. 7986.
» MAY, JOHN, Get the Edge at Blackjack. Chicago: Illinois: Bonus Books, Inc., 2000, pp. 4759.
2 ROBERTS, STANLEY, et al., The Gambling Times Guide to Blackjack. Secaucus, New Jersey: Lyle Stuart, September, 146
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'' CATLIN, DON, “Remembering John Gwynn,” The Rolling Good Times Online, www.scoblete.com, July 1, 2001.
'* GWYNN, JOHN M. JR. AND ARNOLD SNYDER, “Man vs. Computer: Does Casino Blackjack Differ From ComputerSimulated Blackjack?,” Blackjack Forum, Vol. VIII #1, March, 1988, p. 6.
'° SNYDER, ARNOLD, “Ruffled by the Shuffle,” Blackjack Forum, Vol. X, #1, March , 1990, p. 5.
'4 ZENDER, BILL, Card Counting for the Casino Executive. Las Vegas, Nevada: Self published, 1990, p. 93. '° O’BRIEN,
DOMINIC,
How
to Develop a Perfect Memory.
London: England: Pavilion Books Limited, 1993, p. 144.
'© KONIK, MICHAEL, “One Step Ahead: Sophisticated Gamblers Use Legal Techniques to Gain Small Advantages at Casino Games,” www.cigaraficionado.com, Wednesday, March 1, 1995.
Archives,
'’ DANESE, ROSEANN, “Card pros count more than money,” Windsor, Ontario: The Windsor Star, Friday, July 21, 1995.
'8 SNYDER, 1995, p. 19. '9 SCOBLETE, FRANK, Best Blackjack. Chicago, Illinois: Bonus Books, Inc., 1996, p. 175.
20 ANGELI, MICHAEL, “Fleecing Las Vegas; blackjack team,” Esquire, May 1997, No. 5, Vol. 127, p. 63.
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2 MORSE, 1998, p. 1. *3 MAY,
JOHN,
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Secaucus,
New
Jersey: Carol Publishing Group, 1998, p. 68.
, Get the Edge at Blackjack. Chicago: Illinois:

Bonus Books, Inc., 2000, p. 47.
25
TAMBURIN,
Techniques—Card
HENRY,
“New
Sequencing,”
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Winning
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26 MCGARVEY, ROB, Aces & Faces Blackjack. Self published, May, 2001, p. 83.
*7 “ACE”, “Clump Reading and Shuffle Tracking,” www.aceten.com, 2001. *8 NEWMAN,
SANDRA, “How to win at blackjack,” Observer
Sport Monthly, http://observer.guardian.co.uk, T2002;
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zs , ‘Reader of the pack,” Observer Sport Monthly, http://observer.guardian.co.uk, Sunday, April 7, 2002. °° MEZRICH, BEN, Bringing Down the House: The Inside Story of Six MIT Students Who Took Vegas for Millions. New Y ork, New York: The Free Press, 2002, “How to Count Cards and Beat Vegas,” p. 257.
“ , “Hacking September, 2002.
148
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*° SNYDER, 2003, pp. 7477. ** ETLING, LEAH, “He had the upper hand, now all bets are off,” Santa Barbara NewsPress, Sunday, November 9, 2003.
*° JACOBSON, ELIOT, “The Mayor’s Podium: Ace location techniques,” www.cardcounter.com, May, 2004. O’NEIL, PAUL, “The Professor Who Breaks the Bank,” Life, Chicago, Illinois: Time, Inc., March 27, 1964, p. 84.
a CALDERBANK, ROBERT AND NEIL J. A. SLOANE, “Obituary: Claude Shannon,” Nature, Vol. 410, No. 768, 2001.
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*° O’NEIL, 1964, p. 91. 40
THORP,
EDWARD
O.,
“Nonrandom
Shuffling
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Applications to the Game of Faro,” Journal of the American Statistical Association, Vol. 68, No. 344, December, 1973, p. 844. 4! THORP,
EDWARD
O. AND
WILLIAM
E. WALDEN,
“The
Solution of Games by Computer,” (unpublished manuscript), 1963.
” THORP, 1973, p. 464. "
, The Mathematics of Gambling.
Secaucus, New
Jersey: Lyle Stuart, March, 1985.
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SOLOMON
W., “Permutations
by Cutting and
Shuffling”, SIAM Review, Vol. 3, No. 4, October,
1961, p.
293: 4S THOMAS, ROBERT MCG., JR., “Peter Griffin, Solver of Blackjack, Dies at 61,” The New York Times, Obituaries,
Monday, November 2, 1998. © GRIFFIN, PETER A., The Theory of Blackjack (Revised and expanded). Las Vegas, Nevada: Gambler’s Book Club Press,
Oeed BOS ape 30. “7 MACALUSO, PAT, Learning Simulation Techniques on a Microcomputer Playing Blackjack and Other Monte Carlo Games, Blue Ridge Summit, Pennsylvania: TAB BOOKS Mesos
Sp Os
“8 THORP, 1962, p. 171. *° MACTUTOR HISTORY OF MATHEMATICS ARCHIVE, THE, www.standrews.ac.uk, School of Mathematics
“Biographies—Feller,  William,” and Statistics, University of St.
Andrews, Fife, Scotland.
°° FELLER, WILLIAM K., An Introduction to Probability Theory and Its Applications, Vol. 1, New York: John Wiley & Sons Inc., 1957, p. 368.
>! THORP AND WALDEN, 1963. °? RUCHMAN, PETER, “Not Fade Away: An Appreciation of Julian Braun (19292000), Part II,” www.casinogaming.com, Sunday, March 4, 2001.
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Massachusetts:
AddisonWesley
Publishing
Company, Inc., 1969, pp. 124151.
> THORP, 1973, p. 844.
a
, 1973, p. 845.
°° EPSTEIN, RICHARD A., The Theory of Gambling and Statistical Logic. San Diego, California: Academic Press, L977, ps 164.
”” Ibid. °8 GWYNN AND SNYDER, 1988, p. 7. >? HANNUM, 2000, pp. 5152. 6° EMANUEL, DAVID C. AND KENNETH H. SUTRICK,
“Non
random Shuffling Strategies in Blackjack,” Communications in Statistics: Theory and Methods, Vol. 17, Issue 9, 1988, p.
2954.
°" [bid. 6? GILBERT, EDGAR N., “Theory of shuffling,” Technical Memorandum, Bell Laboratories, Murray Hill, New Jersey, 1955.
© EPSTEIN, 1977, p. 167. 64 TREFETHAN, LLOYD N., AND LLOYD M. TREFETHAN, “How many shuffles to randomize a deck of cards?” Proceedings of the Royal Society of London, Series A, 1999.
6 EPSTEIN, 1977, pp. 167168.
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66 MAY, 2000, p. 51. 6? FULVES, KARL, Charles Jordan’s Best Card Tricks. Toronto, Ontario: General Publishing Company, Ltd., 1992,
pp. v—vi.
68 BAYER,
DAVID,
AND
PERSI
DIACONIS,
Dovetail Shuffle to its Lair,’ The Annals Probability, Vol. 2, No. 2, May, 1992, p. 299.
“Trailing
the
of Applied
© JORDAN, CHARLES T., “Trailing the Dovetail Shuffle to Its Lair,” in Thirty Card Mysteries, reprinted Oakland, California:
Magic Limited—Lloyd E. Jones, 3" ed., 1974, p. 1. 7°
MONTICUP,
PETER,
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Virginia: www.magictricks.com, 2003.
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Inc.,
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Library, Biographies,
1996—
™ FULVES, 1992, p. 114. ” LEwIS, ANGELO, “Scraps from My Note Book, by Professor Hoffman, No.l—Card Reading Extraordinary,” London,
England: Gamage’s The Magician Monthly, 8, 1912, p. 67.
” BAXTER, THOMAS, AND “MARKO”, “True Men of Mystery,” The Learned
Pig Magic
eZine,
Vol.
1, No.
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November 30, 1999.
™ THORP, 1973, p. 845.
SIEGEL, 1956, pp. 229238.
”® Ibid. 77
, 2003, p.97.
’8 EPSTEIN, 1977, p. 218. 152
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o
, “Card Prediction for A Shuffled Deck,” Hughes
Aircraft Co., El Segundo, OP68, July, 1964, p. 1. 81
California, Report TP641911,
OTIS pal67:
*? BAYER AND DIACONIS, 1992, p. 294. 8° ALDOUS, DAVID, AND PERSI DIACONIS, “Shuffling Cards and Stopping Times,” American Mathematical Monthly, Vol.
93, May, 1986, p. 345.
8* EPSTEIN, 1977, pp. 167171. =
, 1977, p. 179.
8° ZENDER, 1990, pp. 9495.
87 SNYDER, 2003, p. 75.
88 KONIK, 1995.
8°Ibid. °° THORP, 1962, p. 98. °! Norr, JACQUES, Casino Holiday. Berkeley, Oxford Street Press, 2" ed., 1970, p. 127.
California:
°2 GRIFFIN, 1981, pp. 145146. °3 SNYDER, 2003, p. 76.
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°4 BRAUN,
JULIAN
Chicago, Illinois: 1980, pp. 82, 83.
H., How Data
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°° GROSJEAN, JAMES, AND PREVIN MANKODI, “More on the Ace
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°° SNYDER, 2003, p. 76. °7 EPSTEIN, 1977, p. 234. °8 WILSON, ALLAN N., The Casino Gambler’s Guide. York: Harper & Row, Publishers, Inc., 1970, p. 289.
New
”” MAY, 2000, pp. 5556. '° GROSJEAN, JAMES, Beyond Counting. Oakland, California: RGE Publishing, 2000, p. 93.
'°' MAY, 2000, p. 63. '0? WONG, STANFORD, Professional Blackjack. New York, New York: William Morrow and Company, Inc., 1981, p. 91.
ne
, 1981, p. 93.
104
, Professional Blackjack.
Las Vegas, Nevada: Pi
Yee Press 19945 pa245:
_
, 1994, p. 237.
°° SCHLESINGER, DONALD, Blackjack Attack: Playing the
Pros’ Way.
Las Vegas, Nevada:
gl
, 2004, p. 16.
ed., 2004, p. 18.
154
RGE Publishing, Ltd., 3"
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°° BERLEKAMP, ELWYN, R., “Shannon Lecture,” Institute of
Electrical and Electronics Engineers: — International Symposium on Information Theory, San Antonio, Texas, January 17—22, 1993.
' KELLY, JOHN L. JR., “A New Interpretation of Information Rate,” Bell System Technical Journal, 35, No. 4 (July, 1956), Page:
''° CHIN, WILLIAM AND Marc J. INGENOSO, “Risk Formulas for Proportional Betting,” www.bjmath.com, March 13, 2004, p. 4.
He
92004165,
ao Ibid. ''S EPSTEIN, RICHARD A., “An Optimal Gambling System For Favorable Games,” Hughes Aircraft Co., Report TP641926, OP61, July, 1964, pp. 2—S.
''4 THORP, EDWARD O., “The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market,” paper presented at the Tenth International Conference on Gambling and Risk Taking, Montreal, June 1997; revised Friday, May 29, 1998, p. 32.
''S CHIN AND INGENOSO, 2004, p. 9.
116
References
, 2004, p. 5.
155
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