Example 27

EXAMPLE 27 Conditional probability for mutually exclusive events

Suppose two events $A$ and $B$ are mutually exclusive, with $P (A) > 0 and P (B) > 0$ .

Solution

Because $A$ and $B$ are mutually exclusive, $P (B and A) = 0$ . Then

$P (B given A) = \frac{P (A and B)}{P (A)} = 0$

That is, if event A has occurred, then event B cannot occur. This is a natural consequence of events $A$ and $B$ being mutually exclusive.

What Results Might We Expect?

Two events are independent if the occurrence of one does not affect the probability that the other will occur. However, as we saw in (a), if event $A$ occurs, then the probability that event $B$ will occur is 0. Thus, we would expect events $A$ and $B$ to be dependent.
We are given that $P (A) > 0 and P (B) > 0$ . Thus, the product $P (A) \cdot P (B)$ is also greater than 0. However, from (a), $P (A and B) = 0$ . Thus, $P (A) \cdot P (B) \neq P (A and B)$ , and from the alternative method for determining independence, we conclude that events $A$ and $B$ are dependent.

In other words, if two events are mutually exclusive, then they are dependent.

NOW YOU CAN DO

Exercises 85–92.