HMMT 二月 2013 · COMB 赛 · 第 3 题
HMMT February 2013 — COMB Round — Problem 3
题目详情
英文原题
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 retur n to a previous marble; he receives 3 points for keeping a red marble, loses 2 points for k eeping a blue marble, and gains 0 points for passing. All distributions of colors are equally li kely 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
英文解析
- On a game show, Merble will be presented with a series of 2013 marbles, each of which is either red orblue on the outside. Each time he sees a marble, he can either keep it or pass, but cannot return to aprevious 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 seethe 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?
Answer: First, we note that if all the marbles are red or all are blue, then it is impossible 1
2012
for Merble to win; we claim that he can guarantee himself a win in every other case. In particular, his 2
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
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