outty wrote:There are 13 hearts in a full deck of 52 cards. In a certain game, you pick a card from a standard deck of 52 cards. If the card is a heart, you win. If the card is not a heart, the person replaces the card to the deck, reshuffles, and draws again. The person keeps repeating that process until he picks a heart, and the point is to measure how many draws it took before the person picked a heart and, thereby, won. What is the probability that one will pick the first heart on the third draw or later?
A 1/2
B 9/16
C 11/16
D 13/16
E 15/16
Since we want the probability of drawing the first heart on the third draw or later, we can do the "opposite" by determining the probability of drawing the heart on the first draw and that of drawing the first heart on the second draw. After we determine these probabilities, we will subtract the sum of their probabilities from 1. In other words, we will use the following:
1 = P(heart on the first draw) + P(heart on the second draw) + P(heart on the third or a later draw)
Let H = heart and N = non-heart, so P(H) = ¼ and P(NH) = ¾ x ¼ = 3/16. Thus, we know that the probability of a heart on the first draw is 1/4, and the probability of drawing a heart on the second draw is 3/16.
Therefore, the probability of drawing the first heart on the third draw or later is:
1 - 1/4 - 3/16 = 16/16 - 4/16 - 3/16 = 9/16
Answer: B
