EXAMPLE 24 Sampling with replacement

We draw a card at random from a shuffled deck, observe the card, and return it to the deck. The deck is then reshuffled, and we draw another card at random. What is the probability that both cards we select will be aces?

Solution

Define the following events:

We want to find ( and ), the probability of observing an ace on the first draw and an ace on the second draw. From the Multiplication Rule, . To find , recall that there are 4 aces in the deck of 52 cards. It is reasonable to assume that all cards are equally likely to be selected, so using the classical method, . Similarly, .

Next, we need to find , the probability of observing an ace on the second draw, given that we observe an ace on the first draw. Because the deck of 52 cards has not changed (except for shuffling), there are still 52 cards—4 of which are aces. Therefore, . Thus, the probability that both cards we select will be aces is .

Note that . Thus, by the alternative method for deter-mining independence, and are independent events when sampling with replacement.