HMMT 二月 2013 · COMB 赛 · 第 3 题

HMMT February 2013 — COMB Round — Problem 3

On a game show, Merble will be presented with a series of 2013 marbles, each of which is either red or blue on the outside. Each time he sees a marble, he can either keep it or pass, but cannot return to a previous marble; he receives 3 points for keeping a red marble, loses 2 points for keeping a blue marble, and gains 0 points for passing. All distributions of colors are equally likely and Merble can only see the color of his current marble. If his goal is to end with exactly one point and he plays optimally, what is the probability that he fails?

解析

On a game show, Merble will be presented with a series of 2013 marbles, each of which is either red or blue on the outside. Each time he sees a marble, he can either keep it or pass, but cannot return to a previous marble; he receives 3 points for keeping a red marble, loses 2 points for keeping a blue marble, and gains 0 points for passing. All distributions of colors are equally likely and Merble can only see the color of his current marble. If his goal is to end with exactly one point and he plays optimally, what is the probability that he fails? 1 Answer: First, we note that if all the marbles are red or all are blue, then it is impossible 2012 2 for Merble to win; we claim that he can guarantee himself a win in every other case. In particular, his strategy should be to keep the first red and first blue marble that he encounters, and to ignore all the 2 1 others. Consequently, the probability that he cannot win is = . 2013 2012 2 2