HMMT 十一月 2012 · THM 赛 · 第 7 题

HMMT November 2012 — THM Round — Problem 7

[ 4 ] The game of rock-scissors is played just like rock-paper-scissors, except that neither player is allowed to play paper. You play against a poorly-designed computer program that plays rock with 50% probability and scissors with 50% probability. If you play optimally against the computer, find the probability that after 8 games you have won at least 4.

解析

[ 4 ] The game of rock-scissors is played just like rock-paper-scissors, except that neither player is allowed to play paper. You play against a poorly-designed computer program that plays rock with 50% probability and scissors with 50% probability. If you play optimally against the computer, find the probability that after 8 games you have won at least 4. 163 Answer: Since rock will always win against scissors, the optimum strategy is for you to always 256 play rock; then, you win a game if and only if the computer plays scissors. Let p be the probability n that the computer plays scissors n times; we want p + p + p + p + p . Note that by symmetry, 0 1 2 3 4 p = p for n = 0 , 1 , . . . , 8, and because p + p + · · · + p = 1, p + · · · + p = p + · · · + p = (1 − p ) / 2. n 8 − n 0 1 8 0 3 5 8 4 Our answer will thus be (1 + p ) / 2. 4 ( ) 8 If the computer is to play scissors exactly 4 times, there are ways in which it can do so, compared 4 ( ) 35 1+ 8 8 8 128 to 2 possible combinations of eight plays. Thus, p = / 2 = 35 / 128. Our answer is thus = 4 4 2 163 . 256