EXAMPLE 6 Dependent or Independent?

For many sweepstakes, the consumer is automatically entered into the drawing after making a purchase. However, often sweepstake rules state that “no purchase is required to enter,” and the consumer is given the option to enter by completing an online form or mailing in a postcard. Let event be winning the sweepstakes and event be making a purchase. According to the sweepstake rules, your chance of winning the contest is not affected by whether or not you make a purchase. If we use the notation as shorthand for “the probability of winning, given a purchase is made,” then according to the rules, . In this case, events and are independent.

For the next example, return to Figure 8.7 (page 347) in Example 3. Consider the following events:

Are events and independent or dependent? We know . But what about the probability that the sum on the dice is 7, given both dice show values less than 5—in other words, ? In Figure 8.13 the outcomes in event are outlined by a blue rectangle. Each outcome in , of which there are 16, is equally likely and there are only two outcomes in for which the sum of the dice is 7. Hence, . (The proportion of sums of 7 in the outcomes for differs from the proportion of sums of 7 in the sample space.) In this case, knowledge that event has occurred makes it less likely that occurs. Therefore, events and are dependent.