Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.2 PROBABILITY

O

B

A 1

A 2

A 3

A 4

Figure 30.5 A collection of traffic islands connected by one-way roads.

i=1, 2 , 3 ,4. From figure 30.5, we see also that

Pr(B|A 1 )=^13 , Pr(B|A 2 )=^13 , Pr(B|A 3 )=0, Pr(B|A 4 )=^24 =^12.

Thus, using the total probability law (30.24), we find that the probability of arriving atB
is given by

Pr(B)=

∑

i

Pr(Ai)Pr(B|Ai)=^14

( 1

3 +

1

3 +0+

1

2

)

= 247 .

Finally, we note that the concept of conditional probability may be straightfor-

wardly extended to several compound events. For example, in the case of three

eventsA, B, C, we may write Pr(A∩B∩C) in several ways, e.g.

Pr(A∩B∩C)=Pr(C)Pr(A∩B|C)

=Pr(B∩C)Pr(A|B∩C)

=Pr(C)Pr(B|C)Pr(A|B∩C).

Suppose{Ai}is a set of mutually exclusive events that exhausts the sample spaceS.IfB

andCare two other events inS, show that

Pr(B|C)=

∑

i

Pr(Ai|C)Pr(B|Ai∩C).

Using (30.19) and (30.17), we may write

Pr(C)Pr(B|C)=Pr(B∩C)=

∑

i

Pr(Ai∩B∩C). (30.25)

Each term in the sum on the RHS can be expanded as an appropriate product of
conditional probabilities,

Pr(Ai∩B∩C)=Pr(C)Pr(Ai|C)Pr(B|Ai∩C).

Substituting this form into (30.25) and dividing through by Pr(C)givestherequired
result.