HMMT 二月 2007 · 冲刺赛 · 第 17 题

英文解析

17. [ 10 ] During the regular season, Washington Redskins achieve a record of 10 wins and 6 losses. Computethe probability that their wins came in three streaks of consecutive wins, assuming that all possiblearrangements of wins and losses are equally likely. (For example, the record LLWWWWWLWWLWWWLL contains three winning streaks, while WWWWWWWLLLLLLWWW has just two.)
    Answer: . Suppse the winning streaks consist of w , w , and w wins, in chronological order,45
    1 2 3
    where the first winning streak is preceded by l consecutive losses and the i winning streak is imme-286
    diately succeeded by l losses. Then w , w , w , l , l > 0 are positive and l , l ≥ 0 are nonnegative.0
    i 1 2 3 1 2 0 3
    The equationsw + w + w = 10 and ( l + 1) + l + l + ( l + 1) = 8 ,
    1 2 3 0 1 2 3
    ( ) ( )
    9 7
    are independent, and have and solutions, respectively. It follows that the answer is
    2 3
    ( )( )
    9 7
    2 3315
    ( ) = .
    200216 6