Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

Two eventsAandBarestatistically independentif Pr(A|B)=Pr(A) (or equiva-

lently if Pr(B|A)=Pr(B)). In words, the probability ofAgivenBis then the same

as the probability ofAregardless of whetherBoccurs. For example, if we throw

a coin and a die at the same time, we would normally expect that the probability

of throwing a six was independent of whether a head was thrown. IfAandBare

statistically independent then it follows that

Pr(A∩B)=Pr(A)Pr(B). (30.22)

In fact, on the basis of intuition and experience, (30.22) may be regarded as the

definitionof the statistical independence of two events.

The idea of statistical independence is easily extended to an arbitrary number

of eventsA 1 ,A 2 ,...,An. The events are said to be (mutually) independent if

Pr(Ai∩Aj)=Pr(Ai)Pr(Aj),

Pr(Ai∩Aj∩Ak)=Pr(Ai)Pr(Aj)Pr(Ak),

..

.

Pr(A 1 ∩A 2 ∩···∩An)=Pr(A 1 )Pr(A 2 )···Pr(An),

for all combinations of indicesi,jandkfor which no two indices are the same.

Even if allnevents are not mutually independent, any two events for which

Pr(Ai∩Aj)=Pr(Ai)Pr(Aj) are said to bepairwise independent.

We now derive two results that often prove useful when working with condi-

tional probabilities. Let us suppose that an eventAis the union ofnmutually

exclusiveeventsAi.IfBis some other event then from (30.17) we have

Pr(A∩B)=

∑

i

Pr(Ai∩B).

Dividing both sides of this equation by Pr(B), and using (30.19), we obtain

Pr(A|B)=

∑

i

Pr(Ai|B), (30.23)

which is theaddition law for conditional probabilities.

Furthermore, if the set of mutually exclusive eventsAiexhausts the sample

spaceSthen, from thetotal probability law(30.18), the probability Pr(B) of some

eventBinScan be written as

Pr(B)=

∑

i

Pr(Ai)Pr(B|Ai). (30.24)

A collection of traffic islands connected by a system of one-way roads is shown in fig-

ure 30.5. At any given island a car driver chooses a direction at random from those available.

What is the probability that a driver starting atOwill arrive atB?

In order to leaveOthe driver must pass through one ofA 1 ,A 2 ,A 3 orA 4 , which thus
form a complete set of mutually exclusive events. Since at each island (includingO)the
driver chooses a direction at random from those available, we have that Pr(Ai)=^14 for