Probability Without Tears 0684179946, 0684182521

Table of contents :
Contents
1 Simple Probabilities
Choice and Chance
Equally Likely Outcomes
A ‘Scale’ of Probability
Probability p's and q's
Theoretical Probability
Judging by Experience
Practical Probability
Gambling, Probability and Belief
What are the Odds?
Experiments in Probability
2 Combined Probabilities
Total Probability
Addition Rule
Joint Probability
Probability Tree
Multiplication Rule
When Results are NOT Independent
Using Both Probability Rules Together
One Only v. At Least One
Review
Examination Questions
Answers to Examination Questions
3 Combinations and Permutations
A Reminder
Combining and Permuting
How Many Ways?
Filling Places
Restricted Neighbours
Review Questions
Answers to Review Questions
Formulae for Permutations
Using Factorials
Permutations of Things that are NOT All Different
From Permutations to Combinations
The Combinations Formula
Forming Groups
Combining Combinations
Review Questions
Answers to Review Questions
4 Probability by Combinations
Probability Again
The Combinations Method
Review
Experiments in Probability and Sampling
Combinations and the Binomial Distribution
Binomial Probabilities
The Term and its Coefficient
Coefficients from Pascal’s Triangle
Review
Coefficients from Combinations
Calculating the Coefficient
Expected Frequency Distributions
Samples and Populations
Review Questions
Answers to Review Questions
Review
Practice
Examination Questions
Answers to Examination Questions

Citation preview

Probability Without Tears Derek Rowntree

Probability Without Tears Derek Rowntree

Charles Scribner’s Sons • New York

Copyright © 1984 Derek Rowntree Library of Congress Cataloging in Publication Data Rowntree, Derek. Probability without tears. 1. Probabilities. I. Title. QA273.R8665 1984 519.2 ISBN 0-684-17994-6 ISBN 0-684-18252-1 (pbk.)

84-5510

Copyright under the Berne Convention All rights reserved. No part of this book may be reproduced in any form without the permission of Charles Scribner’s Sons.

1 3 5 7 9 11 13 15 17 19 1 3 5 7 9 11 13 15 17 19

FC FP

20 18 16 14 12 10 8 6 4 2 20 18 16 14 12 10 8 6 4 2

P R IN T E D IN T H E U N IT E D STATES O F A M E R IC A

Contents

1 Simple Probabilities Choice and Chance Equally Likely Outcomes A ‘Scale’ of Probability Probability p's and q's Theoretical Probability Judging by Experience Practical Probability Gambling, Probability and Belief W hat are the Odds? Experiments in Probability

1 1 5 7 9 13 17 21 23 24

2 Combined Probabilities Total Probability Addition Rule Joint Probability Probability Tree Multiplication Rule When Results are NOT Independent Using Both Probability Rules Together One Only v. At Least One Review Examination Questions Answers to Examination Questions

26 26 28 32 36 39 44 51 59 69 71 74

3 Combinations and Permutations A Reminder Combining and Permuting How Many Ways? Filling Places Restricted Neighbours Review Questions

76 76 77 79 84 89 91

6

vi

contents Answers to Review Questions Form ulae for Perm utations Using Factorials Perm utations of Things that are NO T All Different From Perm utations to Com binations The Com binations Form ula Form ing G roups Combining Com binations Review Questions Answers to Review Questions

4 Probability by Combinations Probability Again The Com binations M ethod Review Experiments in Probability and Sampling Com binations and the Binomial Distribution Binomial Probabilities The Term and its Coefficient Coefficients from Pascal’s Triangle Review Coefficients from Com binations Calculating the Coefficient Expected Frequency Distributions Samples and Populations Review Questions Answers to Review Questions Review Practice Examination Questions Answers to Exam ination Questions

93 94 94 102 107 109 111 111 114 114 115 115 116 123 125 126 126 136 137 146 146 148 152 156 160 161 163 164 165 169

1 Simple Probabilities

Choice and Chance Because it plays such a big part in our lives, most o f us have some idea o f what probability is about. We know it involves weighing up the chances or likeli­ hood o f something or other taking place. Let’s plunge straight into just such a probability situation. The table shows the wages earned by all the men in a particular factory. ($245 < $250 m eans ‘at least $245 but less than $250 per w eek’; or $245.00 to $249.99.) Thus, we see that 2 m en earn at least $245 but less than $250, 7 m en earn at least $250 but less than $255, and so on. Weekly wage $

No. of men

245 250 255 260 265 270 275

2 7 18 40 21 11 1

< < < < < <
>) \ 4

(iii)

4

2

(iv) \ 4

(v) ? = 0 4

A ‘Scale’ of Probability In fact, a probability fraction can never be less than 0 (absolute impossibility). N or can it be greater than 1 (absolute certainty). This is because the num ber of outcomes giving the result we are looking for can never exceed the total number o f possible outcomes. Thus, when we measure probability we are using a ‘scale’ that runs from ZERO to UNITY:

Choice and Chance

1

At which end o f the probability scale would you put the probability that (i) one day you will die? (ii) I can swim round the world in 30 minutes?

; That 1 can swim round the ; world in 30 minutes

That a playing card will be a diamond

That a fair coin will turn up heads

That a month is NO T April

That you will die one day

It’s certain you will die one day and impossible for me ever to swim round the world. But few of life’s possibilities are as clear-cut: rarely does probability equal 0 or 1. Mostly we are dealing with the fractions between 0 and 1— like the probability of - that a tossed coin will land ‘heads’, or of - that a randomly-chosen playing card will be a ‘diam ond’. (See the probability scale.) In which ha lf o f the scale would you put the probability that: (i) your entry in the R ead er’s Digest Sweepstakes will arrive safely by mail? (ii) you will win a huge prize? (i) The probability of safe arrival m ust be close to 1. (ii) The probability o f your winning a huge prize m ust be very close to 0.

Probability P's and q's To save space, we often use the letter ‘p' to stand for a given probability. Thus, in talking about my chances o f swimming round the world, p = 0. But p = 1 that you will die some day. Suppose I ask you to m eet m e at a New Y ork area airport, but I forget to tell you which one. You m ight work out that I could equally well arrive at Kennedy, L aG uardia, or Newark. So what would be your chance of m eeting me (p ) if you went to wait at one of these three airports? In this case, p = ? . p = j (In this case, out o f three equally likely outcom es— your waiting at any one of the three airports— only one will give you the looked-for result.)

8

Simple Probabilities

If ‘p ’ is the probability o f a result happening, the probability o f the alternative— failure to happen— is often written as ‘q \ p = probability o f ‘success’ q = probability o f ‘failure’* In the case of your problem with the three airports, q = ? .

2

q = - (There are two outcom es leading to failure and only one that will give you ‘success’.) All right then: p = probability the looked-for result does happen q = probability the looked-for result fails to happen Since the result we are looking for in any probability ‘situation’ must either happen or else fail to happen, the two fractions m ust add up to 1. T hat is: p + q = 1In our last example, p =

q = | , and ^ + - = 1-

Suppose the chances o f an astronaut returning safely from a mission to the 9

moon are p = — . W hat would q mean? And what would be its value? q would be the probability o f the astronaut failing to return. Since, in this case, * = n ’ « = n - < T h a tis ’ 1 - £ • > Note that, if: p = probability o f ‘success’ and q = probability o f ‘failure’ p + q = 1, therefore q = 1 — p and p = 1 — qA tourist is walking along a country lane looking for the local church. He comes to a cross-roads and there are five roads to choose from. He has no way o f knowing which road will pass the church, all look equally likely. W hat is p (probability o f passing church) and what is q ? (Check with the diagram.)

Choice and Chance

9

p (probability of success) = q (failure) = (Two roads lead to the church and three do not. So, out o f the five possible outcomes, two would give the successful result, three a failure. Notice that, of course, p + q = 1 .) If the probability o f a particular result happening is: (>) ^

(») I

(«0 \

(iv) \

(v) 0-3

(vi) 0 95

what is the probability o f its failing to happen? Using your knowledge that p + q = 1: (i) \

(ii) |

(«i) |

(iv) \

(v) 0.7

(vi) 0.05

So we have got used to the idea that probability can be measured on a scale, and we have worked out the probability that a few imaginary results will happen or fail to happen. But how do we actually decide probabilities in real life? There are two main ways— theoretical and practical. W e’ll look at each in turn. Theoretical Probability Sometimes we feel we can state the probability of a result simply by examining the circumstances— ‘weighing up the possibilities’. For instance, consider the two captains tossing a coin to decide which of their football team s shall kick off. This m ethod is generally thought to be a fair way to begin a game. But what do we m ean by ‘fair’? Which o f the following is the theory on which we base this idea o f fairness? (a) Heads on one toss will be followed by tails on the next, and so on? or (b) The coin is as likely to come down heads as tails? or (c) In a large num ber o f tosses, the num ber o f heads and tails would be exactly the same? (b) The coin is as likely to come down heads as tails. The possibility o f the coin remaining up in the air or landing on its edge are so remote that only two outcomes remain: heads or tails. Since, in theory, there is no reason why a coin should land on one side rather than the other, we say they are both equally likely (and the two teams have an equal chance of winning the toss). However, this doesn’t mean we should expect exactly half and half in a large num ber o f tosses; nor has the coin a memory to enable it to give heads this time if it gave tails last, and vice versa.

10

Simple Probabilities

It is the idea o f equally likely outcom es that allows us to come up with a theoretical probability for the coin landing, say, heads. Since this is one o f two equally likely outcomes, /?(head) = i The probability o f the other outcom e, ^(tail) =

?

.