Page 28 - 4660

Conditional probability

We now derive two results that often prove useful when working with conditional probabilities.
Let us suppose that an event A is the union of n mutually exclusive events A i . If B is some other
event then from (2.15) we have

∑
P(A ∩ B) = P(A i ∩ B).
i

Dividing both sides of this equation by P(B), and using (3.2), we obtain

∑
P(A|B) = P(A i |B), (3.11)
i
which is the addition law for conditional probabilities. Furthermore, if the set of mutually
exclusive events A i exhausts the sample space S then, from the total probability law, the
probability P(B) of some event B in S can be written as

P(B) = P(A i )P(B|A i ). (3.12)

Example 3.7. A collection of traffic islands connected by a system of one-way
roads is shown in Fig. 3.1. At any given island a car driver chooses a direction
at random from those available. What is the probability that a driver starting at
O will arrive at B? ,

A 4

A 3

O

A 1

A 2
B

Figure 3.1 – A collection of traffic islands connected by one-way roads

Solution. In order to leave O the driver must pass through one of A 1 , A 2 , A 3 or A 4 , which thus
form a complete set of mutually exclusive events. Since at each island (including O) the driver
chooses a direction at random from those available, we have that P(A i ) = 1 for i = 1, 2, 3, 4.
4
From Fig. 3.1, we see also that

1 1 2 1
P(B|A 1 ) = , P(B|A 2 ) = , P(B|A 3 ) = 0, P(B|A 4 ) = = .
3 3 4 2
Thus, using the total probability law (3.12), we find that the probability of arriving at B is given
by
( )
∑ 1 1 1 1 7
P(B) = P(A i )P(B|A i ) = + + 0 + = .
4 3 3 2 24
i

28