Tree diagrams are often useful for determining compound probabilities and conditional probabilities. Although you will not do a card problem on the final, it is useful for demonstrating how tree diagram are constructed. For this problem you need to remember that there are 52 card in a deck, 13 cards in each suit, and 4 cards of each denomination.

**Problem 13** Assume that two cards are drawn without replacement from a standard 52-card deck, find the following:

a. What is the probability that two aces are drawn?

**Solution** The probability that the first card is an ace and the second card is an ace is the product of the probabilities along the branch containing the aces in a tree diagram

So

P(1^{st} card is ace and 2^{nd} card is ace) = ^{4}/_{52} ∙ ^{3}/_{51} = ^{12}/_{2652} = ^{1}/_{221}

b. What is the probability of drawing a second heart given that you have already drawn a heart?

**Solution** To find the probability that the second card is a heart, given that the first card is a heart, we examine a tree diagram.

Alternately, we can realize that if one heart has been drawn, there are twelve hearts left in the remaining 51 cards so

P(2^{nd} card is heart | 1^{st} card is heart) = ^{12}/_{51}