HMMT 二月 2007 · 冲刺赛 · 第 17 题

HMMT February 2007 — Guts Round — Problem 17

[ 10 ] During the regular season, Washington Redskins achieve a record of 10 wins and 6 losses. Compute the probability that their wins came in three streaks of consecutive wins, assuming that all possible arrangements of wins and losses are equally likely. (For example, the record LLWWWWWLWWL- WWWLL contains three winning streaks, while WWWWWWWLLLLLLWWW has just two.)

解析

[ 10 ] During the regular season, Washington Redskins achieve a record of 10 wins and 6 losses. Compute the probability that their wins came in three streaks of consecutive wins, assuming that all possible arrangements of wins and losses are equally likely. (For example, the record LLWWWWWLWWL- WWWLL contains three winning streaks, while WWWWWWWLLLLLLWWW has just two.) 45 Answer: . Suppse the winning streaks consist of w , w , and w wins, in chronological order, 1 2 3 286 where the first winning streak is preceded by l consecutive losses and the i winning streak is imme- 0 diately succeeded by l losses. Then w , w , w , l , l > 0 are positive and l , l ≥ 0 are nonnegative. i 1 2 3 1 2 0 3 The equations w + 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 7 315 2 3 ( ) = . 16 2002 6