PUMaC 2013 · 组合（A 组） · 第 7 题

PUMaC 2013 — Combinatorics (Division A) — Problem 7

[ 7 ] The Miami Heat and the San Antonio Spurs are playing a best-of-five series basketball championship, in which the team that first wins three games wins the whole series. Assume that the probability that the Heat wins a given game is x (there are no ties). The expected value for the total number of games played can be written as f ( x ), with f a polynomial. Find f ( − 1).

解析

[ 7 ] The Miami Heat and the San Antonio Spurs are playing a best-of-five series basketball championship, in which the team that first wins three games wins the whole series. Assume that the probability that the Heat wins a given game is x (there are no ties). The expected value for the total number of games played can be written as f ( x ), with f a polynomial. Find f ( − 1). 3 Solution The chance of the Heat winning in three is x and the chance of them losing in three 3 3 is (1 − x ) . The chance of them winning in four is 3 x (1 − x ), and the chance of losing is 3 3 2 2 3 3 x (1 − x ) . The chance of them winning in five is 6 x (1 − x ) , losing is 6 x (1 − x ) . In short, 3 3 3 3 3 2 2 3 f ( x ) = 3( x + (1 − x ) ) + 4(3)( x (1 − x ) + x (1 − x ) ) + 5(6)( x (1 − x ) + x (1 − x ) ) 2 Subbing in x = − 1 gives f ( − 1) = 3( − 1 + 8) + 12( − 2 + − 8) + 30( − 4 + 8) = 21 − 120 + 120 = 21 So the answer is 21.