Elements of Finite Probability

Table of contents :
Preface......Page 5
Contents......Page 7
1.1 RANDOM EXPERIMENTS......Page 9
1.2 EMPIRICAL BASIS OF PROBABILITY......Page 11
1.3 SIMPLE EVENTS AND THEIR PROBABILITIES......Page 17
1.4 DEFINITION OF PROBABILITY MODEL......Page 22
1.5 UNIFORM PROBABILITY MODELS......Page 27
1.6 THE ALGEBRA OF EVENTS......Page 31
1.7 SOME LAWS OF PROBABILITY......Page 37
2.1 A MODEL FOR SAMPLING......Page 41
2.2 THE NUMBER OF SAMPLES......Page 47
2.3 ORDERED SAMPLING......Page 52
2.4 SOME FORMULAS FOR SAMPLING......Page 57
3.1 PRODUCT MODELS FOR TWO-PART EXPERIMENTS......Page 61
3.2 REALISM OF PRODUCT MODELS......Page 66
3.3 BINOMIAL TRIALS......Page 74
3.4 THE USE OF RANDOM NUMBERS TO DRAW SAMPLES......Page 79
4.1 THE CONCEPT OF CONDITIONAL PROBABILITY......Page 83
4.2 INDEPENDENCE......Page 89
4.3 TWO-STAGE EXPERIMENTS......Page 96
4.4 PROPERTIES OF TWO-STAGE MODELS; BAYES' LAW......Page 103
4.5 APPLICATIONS OF PROBABILITY TO GENETICS......Page 108
4.6 MARRIAGE OF RELATIVES......Page 115
4.7 PERSONAL PROBABILITY......Page 118
5.1 DEFINITIONS......Page 124
5.2 DISTRIBUTION OF A RANDOM VARIABLE......Page 128
5.3 EXPECTATION......Page 132
5.4 PROPERTIES OF EXPECTATION......Page 136
5.5 LAWS OF EXPECTATION......Page 140
5.6 VARIANCE......Page 146
6.1 THE BINOMIAL DISTRIBUTION......Page 154
6.2 THE HYPERGEOMETRIC DISTRIBUTION......Page 160
6.3 STANDARD UNITS......Page 166
6.4 THE NORMAL CURVE AND THE CENTRAL LIMIT THEOREM......Page 172
6.5 THE NORMAL APPROXIMATION......Page 176
6.6 THE POISSON APPROXIMATION FOR np = 1......Page 181
6.7 THE POISSON APPROXIMATION: GENERAL CASE......Page 184
6.8 THE UNIFORM AND MATCHING DISTRIBUTIONS......Page 189
6.9 THE LAW OF LARGE NUMBERS......Page 193
6.10 SEQUENTIAL STOPPING......Page 197
7.1 JOINT DISTRIBUTIONS......Page 202
7.2 COVARIANCE AND CORRELATION......Page 208
7.3 THE MULTINOMIAL DISTRIBUTION......Page 215
TABLES......Page 221
INDEX......Page 229
INDEX OF EXAMPLES......Page 235

Citation preview

ELEMENTS OF FINITE PROBABILITY

HOLDEN-DAY SERIES IN PROBABILITY AND STATISTICS E. L. Lehmann, Edi/or

ELEMENTS OF FINITE PROBABILITY

J. L. Hodges, Jr. and E. L. Lehmann University of California, Berkeley

1965

HOLDEN-DAY, INC. San Francisco, London, Amsterdam

© Copyright 1964, 1965, by HOLDEN-DAY, INC. 728 Montgomery Street, San Francisco

All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. Printed in the United States of America. Library of Congress Catalog Card Number: 64-66419

Preface

After publication of our book Basic Concepts of Probability and Statistics, it was suggested to us that "Part I. Probability' be made available sepa'

rately, to serve as a text for a

one quarter or one semester course in prob-

ability at the precalculus level.

The

present book, the result of these

suggestions, contains all the material of Part I of the earlier book, supple-

mented by two new

A

sections at the end of Chapter

6.

careful treatment of probability without calculus

attention

is

restricted to finite probability models,

senting experiments with a finite

number

is

i.e.,

of outcomes.

basic notions of probability, such as probability model,

possible only

if

to models repre-

Fortunately, the

random

variable,

expectation and variance, are in fact most easily introduced with

finite

Furthermore, many interesting and important applications lead to finite models, such as the binomial, hypergeometric and sampling models. More than most branches of mathematics, probability theory retains its models.

connections with the empirical is

phenomena from which

it

grew.

While

it

possible to present probability in a purely abstract manner, divorced

from its empirical origins, the subject thereby loses much of its vitality and interest. On the other hand, if the theory is developed in its empirical context, it becomes essential to make clear the distinction between the real world and its mathematical representation. Our treatment is therefore centered on the notion of a probability model and emphasizes its relation to, and distinction from, the underlying random experiment which it represents.

One aspect of this approach is a careful treatment of the independence assumption in building probability models. Many complex experiments are made up of simpler parts, and it is frequently possible to build models for such experiments by first building simple models for the parts, and then combining them into a "product model' for the experiment as a whole. Particular attention is devoted to the realism of this procedure, and in general to the empirical meaning of the independence assumption, which '

so often

is

applied invalidly.

While basically we have included only concepts that can be denned and can be proved without calculus, there is one exception: we give approximations to certain probabilities whose calculation we have explained in principle, but whose actual computation would be too laboriThus, in particular, we discuss the normal approximation to the ous. binomial and hypergeometric distributions and the Poisson approximation Since the limit theoto the binomial and Poisson-binomial distributions. rems which underlie these approximations require advanced analytical techniques and are hence not available to us, we give instead numerical results that

illustrations to develop

The book

some

feeling for their accuracy.

some of which provide simple and results of the text.* Six of at the end of the book as Tables

contains over 400 problems,

exercises while others extend the ideas

the more important tables are collected

A-F.

A

decimal numbering system

is used for sections, formulas, examples, Thus, Section 5.4 is the fourth section of Chapter 5. To simplify the writing, within Chapter 5 itself we omit the chapter and refer to this section simply as Section 4. Similarly Example However, within Section 5.4 5.4.2 is the second example in Section 5.4. we omit the chapter and section and refer to the example as Example 2; in other sections of Chapter 5 we omit the chapter and refer to the example

tables, problems,

as

Example

and

so forth.

4.2. J.

L.

Hodges, Jr.

E. L. Berkeley

January, 1965

*

Instructors

may

obtain answer books by writing to the publisher.

Lehmann

Contents

l.

1. 2. 3. 4. 5. 6. 7.

PROBABILITY MODELS

Random experiment Empirical basis of probability . Simple events and their probabilities . Definition of a probability model . Uniform probability models The algebra of events Some laws of probability 2.

1.

2. 3. 4.

16 21

25

31

35 41 46 51

PRODUCT MODELS

1. Product models for two-part experiments 2. Realism of product models. 3. Binomial trials 4. The use of random numbers to draw samples 4.

5 11

SAMPLING

A model for sampling The number of samples Ordered sampling Some formulas for sampling 3.

3

55 60 68 73

CONDITIONAL PROBABILITY

1. The concept of conditional probability 2. Independence 3. Two-stage experiments . 4. Properties of two-stage models; Bayes' Jaw

77 83

90 97

CONTENTS 102

5.

Applications of probability to genetic

6.

Marriage of relatives

109

7.

Personal probability

112

5.

RANDOM VARIABLES 118

1.

Definitions

2.

Distribution of a

3.

Expectation

126

4.

Properties of expectation

130

5.

Laws

134

6.

Variance

140

7.

Laws

143

random

of expectation

of variance

6.

1.

2. 3.

4. 5.

6. 7. 8. 9.

10.

122

variable

SPECIAL DISTRIBUTIONS

The binomial distribution The hypergeometric distribution

148

Standard units The normal curve and the central limit theorem The normal approximation The Poisson approximation for np = 1

160

The Poisson approximation: general case The uniform and matching distributions The law of large numbers

178

Sequential stopping

191

7.

154

166 170

175 183

187

MULTIVARIATE DISTRIBUTIONS

1.

Joint distributions

196

2.

Co variance and correlation The multinomial distribution

202 209

3.

TABLES A.

B-C. D. E. F.

Number

of combinations Binomial distributions Square roots Normal approximation Poisson approximation

216 217 219 220 221

INDEX

223

EXAMPLE INDEX

229

CHAPTER

1

PROBABILITY MODELS

1.1

RANDOM EXPERIMENTS

The theory

of probability is a mathematical discipline which has found important applications in many different fields of human activity. It has extended the scope of scientific method, making it applicable to experiments whose results are not completely determined by the experimental

conditions.

The agreement among

most scientific on the fact that the experiments on which the theories are based will yield essentially the same results when they are repeated. When a scientist announces a discovery, other scienscientists regarding the validity of

theories rests to a considerable extent

tists in different

Sometimes the turns out to

same ment

in the is

parts of the world can verify his findings for themselves.

results of

mean

two

two workers appear to

disagree, but this usually

that the experimental conditions were not quite the

cases.

If

the same results are obtained

repeated under the same conditions,

we may say

when an

experi-

that the result

is

determined by the conditions, or that the experiment is deterministic. It is the deterministic nature of science that permits the use of scientific theory for predicting what will be observed under specified conditions. However, there are also experiments whose results vary, in spite of all Familiar examples efforts to keep the experimental conditions constant. are provided by gambling games: throwing dice, tossing pennies, dealing from a shuffled deck of cards can all be thought of as "experiments" with unpredictable results. More important and interesting instances occur in many fields. For example, seeds that are apparently identical will produce plants of differing height, and repeated weighings of the same object with a chemical balance will show slight variations. A machine which sews together two pieces of material, occasionally for no apparent reason

—

—

will

miss a

stitch.

If

we

are willing to stretch our idea of "experi-

PROBABILITY MODELS

4

ment," length of

life

may

be considered a variable experimental

since people living under similar conditions will die at different

predictable ages.

[CHAP.

1

result,

and un-

We shall refer to experiments that are not deterministic,

and thus do not always yield the same result when repeated under the same conditions, as random experiments. Probability theory (and the theory of statistics, which is based on it) are the branches of mathematics that have been developed to deal with random experiments. Let us now consider two random experiments in more detail. As the first

we take the experiment of throwing a die. This is one of random experiments and one with which most people are acquainted. In fact, probability theory had its beginnings in

example,

the simplest personally

the study of dice games.

Example

Throwing a die. Suppose we take a die, shake it vigorously and throw it against a vertical board so that it bounces onto a table. When the die comes to rest, we observe as the experimental result the number, say X, of points on the upper face. The experiment is not deterministic: the result X may be any of the six numbers 1, 2, 3, 4, 5, or 6, and no one can predict which of the values will be obtained on any partic1

.

in a dice cup,

ular performance of the experiment.

We may make every effort to control

by always placing the die in number of times, always throwing it against the same spot on the backboard, and so In spite of all such efforts, the result will remain variable and on.

or standardize the experimental conditions,

the cup in the same position, always shaking the cup the same

unpredictable.

Example 2.

Measuring a distance. It between two points a few miles apart. distance by use of a surveyor's chain.

is

desired to determine the distance

A

surveying party measures the

found in practice that the measured distance will not be exactly the same if two parties do the job, In spite of or even if the same party does the job on consecutive days. the best efforts to measure precisely, small differences will accumulate and the final measurement will vary from one performance of the experiment It is

to another.

How

is it

on indeterminacy? by an empirical observation: while the result on

possible for a scientific theory to be based

The paradox

is

resolved

any particular performance

of such

an experiment cannot be predicted,

a long sequence of performances, taken together, reveals a stability that

can serve as the basis for quite precise predictions. The property of longrun stability lies at the root of the ideas of probability, and we shall examine it in more detail in the next section.

EMPIRICAL BASIS OF PROBABILITY

1.2]

EMPIRICAL BASIS OF PROBABILITY

1.2

To

obtain some idea about the behavior of results in a sequence of repetirandom experiment, we shall now consider some specific examples.

tions of a

Example

Throwing a

1.

throws of a

number

value of the

Figure

die.

In this experiment,

die.

X of points showing,

n=10

(a)

shows certain

1

we were not

results of 5000

interested in the actual

but only in whether

(b) tt=50

1.0

(c)

1.0

1.0

X was less 7i=250

r

•

-OS 0.5

•

—

•

•

0.5-

0.5 •

0

•

•

•••

•

-

•

.

• •

•

•

1

-

•

1

1

A

1

i

1

20

10

Number

three

(X ^


2. We shall employ the same terminology in the model. as is

the

number

of points

X

Definition.

The complement

of

an event

E is the event E which

consists

just of those simple events in 8 that do not belong to E. It follows from this definition that the sets E and E have no common member, and that between them they contain all simple events of the

event set

8.

The

situation

Figure the simple events in 8,

namely

e 3 , eb

e h e2

,

e4

,

is

illustrated in Figure

1.

and

1,

where

E

consists of

Complementary events

E

consists of the remaining simple events

.

An interesting special case is the complement of the event set 8 itself. By definition, 8 consists of all simple events of the model under discussion, so that 8

is

a set without members.

This set

is

known

as the empty

set.

THE ALGEBRA OF EVENTS

1.6]

27

we think in terms of results, 8 corresponds to a result that cannot occur, = 7 when rolling a die. such as the result Related to any two given results R, S is the result which occurs when both the result R and the result S occur, and which will be called the result If

X

U

R

For example, if with two-child families R denotes the result is a boy and S that the second child is a boy, then "R and S" denotes the result that both children are boys. Similarly, if R indicates that a card drawn from a deck is a heart while S indicates that it is a face card, then the result "R and S" will occur if a heart face card is drawn. Suppose the results R, S are represented in the model by the events E, F respectively. What event in the model then corresponds to the result "R and S" ? To answer this question notice that the result U R and *S" occurs whenever the experiment yields a simple result, which is a simple Therefore the event in the model corresponding result both of R and of S. to "R and S" will consist of those simple events that belong both to E and The situation is illustrated in Figure 2a, where E = {e h e 3 e4 ee} to F.

and S."

that the

first

child

,

,

(a)

Figure

(b)

2.

Intersection and union

corresponds to the result R consisting of the simple results r h r3 n, r 6 and where F = {e 2 e 4 e*, e-i) corresponds to the result S consisting of the simple Here "R and £" will occur whenever the experiment results r 2 r 4 r 6 r 7 ,

,

,

.

,

yields one of the simple results r 4 or r6

.

The event corresponding

"R and S" consists therefore of the simple events common part or intersection of the two events E, F.

result

Definition.

which consists It

may

The

intersection

(E and F)

of

e4 , e6

-

to the

It is the

two events E, F is the event both to E and to F.

of all simple events that belong

of course

common.

,

,

,

The

happen that two events E, F have no simple events and F) is then the empty set.

intersection (E

in

PROBABILITY MODELS

28 Definition.

Two

belongs to both

E

events E,

and

F

[CHAP.

are said to be exclusive

if

1

no simple event

to F.

The results corresponding to two exclusive events will be such that not both of them can happen on the same trial of the random experiment. For example, when throwing a die, the results > 4 are ^ 2 and exclusive, but the results "X is even" and "X is divisible by 3" are not, = 6. Again, for two-child families, the results since both occur when "first child is a boy" and "both children are girls" are exclusive, but the results "first child is a boy" and "second child is a boy" are not, since both

X

X

X

children

may

The

be boys.

trated in Figure

case of two exclusive events

E

and

F

is illus-

3.

Figure

3.

Exclusive events

Another result related to the given results R, S is the result "R or S" which occurs when either the result R occurs or the result S occurs, or both If for example R denotes the result that the first child of a twooccur. child family is a boy while S denotes that the second child is a boy, then "R or S" will occur if at least one of the children is a boy. Similarly, if R denotes that a card drawn from a deck is a heart, while S denotes that it is a face card, then

"R or S" will occur if the card when it is a heart face card.

is

a heart or a face card,

including the case

What event in the model corresponds to the result "R or S" ? To answer this question, notice that the result U R or S" occurs whenever the experiment yields one of the simple results of R or of S or both. Therefore the event in the model corresponding to "R or S" will consist of those simple events belonging to in Figure 2b,

E or to F or E = {e h e

where as before

both. 3,

e4

,

e6}

The

F =

consisting of the simple results n, r 3 r4 r 6 while ,

,

situation

illustrated

is

corresponds to the result

,

{e 2 , e if

e&,

R

e 7 } corre-

sponds to the result S consisting of the simple results r2 n, r 6 r7 Here "R or S" will occur whenever the experiment yields one of the simple The composite event corresponding to "R or S" results n, r 2 r 3 r 4 r 6 r 7 It is obtained by consists therefore of the simple events e h e 2 e h e 4 e & e 7 combining or uniting the simple events of E with those of F, producing the set (E or F) enclosed by the dashed line in Figure 2b. ,

,

,

,

,

.

,

,

,

.

,

.

THE ALGEBRA OF EVENTS

1.6]

Definition.

The union (E

or F) of

29

two events E, F

is

the event which

consists of all simple events that belong to E, or to F, or to both

Example

1.

E

5.3, let

Two

E and F.

In the model for throwing two dice of Example "first die shows one point," so that E row of tableau (5.2). Similarly, let F correspond to

dice.

correspond to the result

consists of the first

"second die shows one point," consisting of the first column of (5.2). or F) consists of the eleven simple events in the top and left mar-

Then (E

gins of (5.2),

On

point."

and corresponds

(1, 1)

Since

By

result "at least one die

shows one

to the results "both dice

show one point."

Consider an event E and its complement have no simple event in common, they are exclusive. the definition of complement, any simple event not in E belongs to E,

Example T$.

and corresponds to the

the other hand, (E and F) consists only of the simple event

2.

E

Complementation.

and

E

so that every simple event belongs either to

union of

E and E is the entire event set 8:

The reader should be warned

E or to E.

(E or E)

of a confusion that

=

It follows that the 8.

sometimes

arises

due to

connotations of the word "and" in other contexts. From the use of "and" as a substitute for "plus," it might be thought that (E and F) should represent the set obtained by putting together or uniting the sets E and F. However, in the algebra of events (E and F) corresponds to the result il R and S" which occurs only if both R and S occur, and thus (E and F) is the common part or intersection of E and F. F) is used for (E or F) while (E O F) In many books the notation (E

U

is

used for (E and F).

PROBLEMS 1.

An

of the

experiment consists of ten tosses with a coin. of each of the following events:

(i)

at least six heads

(ii)

(iii)

2.

most six heads no heads. at

In Problem 5.6 give a verbal description of the complements of the events

described in parts 3.

Give a verbal description

complement

(i),

(ii),

(v),

and

(viii).

In Problem 5.9 give verbal descriptions of the complements of the events

described in parts (i)-(v). 4.

In Problem 5.6 give verbal descriptions of the intersections of each of the follow-

ing pairs of events: (i)

the events

(i)

(ii)

the events

(i)

and and

(ii)

(iii)

the events

(ii)

(iii)

(iv)

the events

(i)

and (iii) and (iv)

:

:

PROBABILITY MODELS

30

and (v) and (vi) the events (ii) and (vi) events

(i)

(viii)

the events

(i)

(ix)

the events (iv) and

(vii)

(x)

the events (vi) and

(vii).

(v) the (vi) (vii)

[CHAP.

the events

(iii)

and

1

(vi)

In Problem 5.6 determine for each of the following pairs whether they are

5.

exclusive (i)

(ii)

(iii)

(iv)

and (iii) and (iii) the events (iii) and (v) the events (iv) and (v) the events

(i)

the events

(ii)

(v)

the events (iv) and

(vi)

the events (v) and

(vi)

the events (v) and

(vii)

(vii) (viii)

the events

(vi)

(vi)

and

(vii).

In Problem 5.9 pick out three pairs from the events described in parts

6.

(i)-(vi)

that are exclusive. 7.

Give verbal descriptions of the unions of each of the pairs of events of Problem 4.

8.

In Problem

give verbal descriptions of the unions of each of the following

5.9,

pairs of events (i)

the events

(i)

(ii)

the events

(ii)

For each and G.

9.

F,

If E,

(i)

If

(ii)

E,

If E,

(iii)

From a

tion;

if

Let

etc.

E

of the following statements

class of ten,

first

first

an instructor

student does not

Suppose that the event

Is it true that then

JDraw diagrams any sets E and F (i)

(ii)

(Ei

selects a student to

first

know

all

E,

answer a certain ques-

F

student knows the answer, and

the

the answer but the second one does.

F

broken up into exclusive pieces E h

E

2,

.

.

.

,Ea

can be written as

and F) or {E2 and F) or

to illustrate the facts

E or F) = E&ndF (E and F) = E or F. (

set £ is

any event

F =

(1)

12.

correct for

E and F exclusive?

11.

for

it is

student cannot answer, he selects one of the remaining students;

denote the event that the

event that the

Are

determine whether

F are exclusive, and F, G are exclusive, then E, G are exclusive, F are exclusive, and E, G are exclusive, then E, (F or G) are exclusive, (F or G) are exclusive, then E, F are exclusive.

10.

the

and (iii) and (iii).

.

.

.

(Ea and F)?

(known as De Morgan's Laws)

that,

.

SOME LAWS OF PROBABILITY

1.7]

31

SOME LAWS OF PROBABILITY

1.7

In the preceding section we have denned three operations on events: complementation, intersection, and union. We shall now consider how probabilities behave under union and complementation. The probabilities of intersections will be studied in Chapters 3 and 4. A law connecting the probability of the union (E or F) with the probabilities of E and of F is obtained most easily in the special case that E and F are exclusive. If R and S are two exclusive results, then not both can happen on the same trial of the experiment, and therefore in a sequence of trials

KR or S) = For

example, in

an evening

player will be the

diamond

sum

number

of the

f(R or S)

=

f(R)

#08).

number

of red flushes dealt to a

of heart flushes

and the number

we

trials

+ /(£)

see that

R,

if

S

are exclusive.

Since probability is supposed to represent long-run frequency, gests that in the model we should have the equation

P(E

(1)

The proof Figure so

if

E,

F

this sug-

are exclusive.

may

be illustrated on

of the five simple events e h e h e 6 e 6 e 8 ,

,

,

definition of the probability of a composite event (Section 4),

right side

or F)

may

just

P(E)

= P(eO

+

P(e 8 )

+

P(e B )

+

P(e«)

+

Pfo).

be written as

[P(e 8 ) is

P(F)

There (E or F) consists

P(E

which

+

= P(E)

of this addition law for exclusive events

6.3.

by the

The

or F)

of

Dividing both sides of the displayed

flushes that he receives.

equation by the total number of

+

#(«)

of poker, the

+

+

+

P(*)

P(cg)]

+

[P(ei)

It is clear that the

P(F).

+

P(ee)]

argument would hold

in

general.

Example

1.

Two

Suppose that

dice.

in

throwing two dice, the 36 simple If T denotes the sum of the

events displayed in (5.2) are equally likely. points on the

The

two

dice, then, as

shown

probabilities of the other values of

Table t

P{T =

t)

1.

in

T

Example

5.3,

P{T =

7)

=

?%.

are as follows (Problem 5.4).

Probabilities of values of

T

2

3

4

5

6

7

8

9

10

11

12

&

&

A

&

A

A

A

A

A

A

&

PROBABILITY MODELS

32

Let us

now by

divisible 10,

= P(E)

P(G)

event set

TH oi

S, it

sum T

will

be

if

+ P(F) law

special case of the addition

complement

that the

and only if T is either equal to 5 or to the exclusive events E: T = 5 and F: T = 10. Hence

Since this occurs

G is the union of

A

G

find the probability of the event 5.

[CHAP. 1

(1) is

E

Since i? and

E.

A + A = A-

=

obtained by taking for

and

are exclusive

union

their

F

the

is

the

follows that

+

P(E)

P(E)

= P(E

=

E) = P(S)

or

1.

This establishes the important law of complementation

=

P(E)

(2)

As we that

shall see later,

-

1

P(E).

the main usefulness

of this relation resides in the fact

often easier to calculate the probability that something does not

it is

happen than that it does happen. As an illustration of (2), consider once more the model

of

Example

suppose we wish to determine the probability of the event E: the of the points of

E is the

showing on the two dice is three or more. The complement E T = 2, which has probability -jV Hence

event

:

P(E)

The concept

1-^=35.

=

of exclusive events extends readily to

The events E h E2 event belongs to more than one Definition.

The addition law

.

,

.

of

.

.

.)

=

P(E2 )

F

no simple

manner:

+

.

if

E,

if

them.

+

P(ft)

more than two events.

are said to be exclusive

.

also extends in the obvious

P(Ei or E* or

If

and

1

sum T

.

.

Eh E

2,

.

.

.

are exclusive.

are not exclusive, law (1) will in general not be correct, as

be seen by examining Figure 6.2b.

P(E) = P( ei ) P(F)

+

Pfe)

+

=

may

Here P(e 4 ) P(e 4 )

+ +

P(e 6 ) P(e 6 )

+

P(e 2 )

+

P(e 7 )

so that

P(E)

+

= Pfe)

P(P)

+

P(e 3 )

+

2P(e 4 )

+

2P(e 6 )

+

P(e 2 )

+

P(e7 )

while

P(E

or F)

=

P{e,)

+

P(*)

+

P(e 4 )

+

P(e»)

+

P(e2)

+

P(e7 ).

+

P(P) but only once in Since P(e 4 ) and P(e 6 ) occur twice in P{E) P(P) is greater than P{E or P) (except P(E or P), it follows that P(E)

+

in the trivial case

A

correct law

when both P(e 4 ) and

must allow

P(ee) are equal to zero).

for the double counting of the probabilities of

those simple events which belong both to

E

and

to F, that

is

to the inter-

SOME LAWS OF PROBABILITY

1.7]

section

(E and F).

33

In Figure 6.2a the event (E and F) consists of

e4

and

that

e 6 , so

F)

=

P(e 4 )

+

= P(E)

+

P(F)

- P(#

P(E and

P(ee).

It follows that

P{E

(4)

or F)

This addition law holds for

all

whether or not they are excluin Figure 6.3), (E and F) will be

sive. If they happen to be exclusive (as empty and have probability 0, in which case

Example two

Two

1.

dice (continued).

F

dice the events E,

point'

'

and "second

die

The

(4)

reduces to

In Example

6.1

corresponding to the results

P(F)

fa,

= &,

likely, it is

we considered If

for

shows one

the 36 simple

seen that

P(E and

and

(1).

"first die

shows one point" respectively.

events displayed in (5.2) are equally

P(E) =

and F).

F

events E,

F)

=

fa.

probability of the event (E or F) corresponding to the result that at

one of the dice shows one point is then, by (4), equal to fa + fa — This can also be seen directly from the fact that the event -g-J. fa (E or F) consists of 1 1 simple events each having probability fa It is important to keep in mind that law (3) requires the assumption that the events are exclusive. When they are not, it is necessary to correct for multiple counting, as is done in law (4) for the case of two events. The formulas for nonexclusive events become rapidly more complicated least

=

as the

number

of events

is

increased (see Problem 12).

PROBLEMS 1.

In Problem

scribed in parts 2.

5.6, find (i),

In Problem

(ii),

the probabilities of the complements of the events de-

(v),

5.9, find

and

(vii).

the probabilities of the complements of the events de-

scribed in parts (i)-(v). 3.

In Problem

5.8,

use the law of complementation to find the probabilities of the

following events. (i)

(ii)

4.

At At

least one of the least

two

digits is greater

one of the two digits

In Example

1,

is

than zero,

even.

use the results of Table

1

and the addition law

(3) to find

the

probabilities of the following events: (i)

(ii)

(hi)

T ^ 10 T < 5 T is divisible by

3.

be the number of boys. Make a similar table to Table 1 showing the probabilities of B taking on its various possible values 0, L 2, and 3. 5.

In Problem 5.6 let

B

,,

PROBABILITY MODELS

34 Use the table

6.

of the preceding

[CHAP.

problem and the addition law

(3) to find

1

the

probabilities of the following events: (i) (ii)

(iii)

B^

2

B > B ^

2 1.

A

manufacturer of fruit drinks wishes to know whether a new formula (say B) has greater consumer appeal than his old formula (say A) Each of four customers is presented with two glasses, one prepared from each formula, and is asked to state which he prefers. The results may be represented by a sequence of four letters; thus ABAB would mean that the first and third customers preferred A, while the second and fourth preferred B. Suppose that the 2 4 = 16 possible outcomes of the experiment are equally likely, as would be reasonable if the two formulas are If S denotes the number of consumers preferring the in fact equally attractive. new formula, make a table similar to Table 1, showing the probabilities of S taking on its various possible values. 7.

.

Use the table

8.

of the preceding

problem and the addition law

(3) to find

the

probabilities of the following events.

At least two of the customers prefer the new product. At most two of the customers prefer the new product, More than two of the customers prefer the new product.

(i) (ii)

(iii)

Let S consist of the six simple events

9.

=

P{ez)

.2,

=

P(e 4 )

.25,

E=

=

P(e 5 )

{e h e 2 },

.3,

F =

eh

=

P(««)

.

{e 2 e 3 e 4} ,

.

.

,

let

P(eO

=

P(e 2 )

=

.1,

If

G =

,

and

e6,

,

.05.

{e 2 e s e b } ,

,

find

(ii)

P(E) P(F)

(iii)

P(E

(i)

10. .05,

P(F P(E

(iv)

(v)

Let 8 consist of the six simple events e h = .14, P(e 4 ) = .1, P(e b ) = .2, P(e 6 )

E=

{e h e 4 e 6 } ,

F =

,

{e 3

,

.

.

.

=

,

(i)

(iii)

and left-hand

11.

or G).

e4, e 5 },

let P(ei)

=

.01,

P(e 2)

=

If

G = it is

{e 2 e 5 } ,

true

by computing both the

+ P(F) + P{G) + P(G).

results.

Check equation

(4) for

the events

E and F

Check the following generalization 12. when E, F, G are the events of Problem

P(E

and

sides:

P(E or F) = P(E) P{E or G) = P{E) P{F or G) = P{F)

Explain your

e6,

.5.

check for each of the following equations whether

(ii)

F

or

or F)

P(e 3 )

right-

or G)

or

F

or G)

=

[P{E)

-

+

of

Problem

of equation (4)

9.

by computing both sides

9:

P(F)

[P{E and F)

+ P(G)] + P{E and G) + P(F and