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
P(1st card is ace and 2nd 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(2nd card is heart | 1st card is heart) = 12/51