Question

Cards are drawn from a standard 52 card deck. After each card is drawn, it is put back in the deck and the cards are reshuffled so that each card drawn is independent of all others. Let N be the random variable that represents the number of cards that are drawn before the second appearance of an Ace. For example, if the sequence of cards drawn was {2, 5, K, 7, A, 5, 3, J, A, ...} then N would take on a value of N=8. Find the probability that N is equal to 17. Show all work! Thanks

Answer 1

The probability p of drawing an Ace during any given draw is

.

This probability will remain the same for each draw as the card is being replaced and the deck is reshuffled after each draw.

Let X be a random variable which indicates the number of draws required to get s=2 aces (which is 2 successes). Let the probability of sucess (getting an ace) be p=1/13.

We know that X has a Negative Biniomial distribution with the following probability

We have to remember here that x is the number of cards drawn including the second ace. But the random variable N represents the number of cards that are drawn before the second appearance of an Ace. That means N=X-1

Replacing we get the distribution of N as

We want the probability that N=-17