EXAMPLE 27 Conditional probability for mutually exclusive events

Suppose two events and are mutually exclusive, with .

  1. Find .
  2. Are events and independent or dependent?

Solution

  1. Because and are mutually exclusive, . Then

    That is, if event A has occurred, then event B cannot occur. This is a natural consequence of events and 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 occurs, then the probability that event will occur is 0. Thus, we would expect events and to be dependent.

  2. We are given that . Thus, the product is also greater than 0. However, from (a), . Thus, , and from the alternative method for determining independence, we conclude that events and are dependent.

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

NOW YOU CAN DO

Exercises 85–92.