HMMT 二月 2013 · 冲刺赛 · 第 15 题

英文解析

15. [ 8 ] Tim and Allen are playing a match of tenus . In a match of tenus , the two players play a seriesof games, each of which is won by one of the two players. The match ends when one player has wonexactly two more games than the other player, at which point the player who has won more gameswins the match. In odd-numbered games, Tim wins with probability 3 / 4, and in the even-numberedgames, Allen wins with probability 3 / 4. What is the expected number of games in a match?
    Answer: Let the answer be E . If Tim wins the first game and Allen wins the second game 16
    2 23
    or vice versa, which occurs with probability (3 / 4) + (1 / 4) = 5 / 8, the expected number of additionalgames is just E , so the expected total number of games is E + 2. If, on the other hand, one of Tim and
    Allen wins both of the first two games, with probability 1 − (5 / 8) = 3 / 8, there are exactly 2 games inthe match. It follows that
    3 5
    E = · 2 + · ( E + 2) ,
    8 8
    and solving gives E = .16
    Guts Round 3
    ° °