Two players, A and B , alternately draw a card from a well-shuffled standard deck of 52 cards with replacement. Player A wins the game if he draws a face card, and player B wins if he draws a heart. If A makes the first draw, the probability that A wins the game is

Step-by-step solution

Let P(A) be the probability that A wins on a single draw, and P(B) be the probability that B wins on a single draw. A face card is a Jack, Queen, or King. There are 12 face cards in a deck. P(A) = 12 52 = 3 13 P( A ) = 1 - 3 13 = 10 13 There are 13 hearts in a deck. P(B) = 13 52 = 1 4 P( B ) = 1 - 1 4 = 3 4 Player A can win on the 1st draw, or the 3rd draw (if both A and B fail on their first attempts), or the 5th draw, and so on. This forms an infinite geometric progression: P(A wins ) = P(A) + P( A )P( B )P(A) +