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两罐 50 白 50 黑:最大化抽到白球

100 marbles

专题
Probability / 概率
难度
L4

题目详情

You are presented with two empty jars and 100 marbles on a table. There are 50 white marbles and 50 black marbles. You are to put all 100 of the marbles into the two jars in any way you choose. I will then blindfold you. I will shake the jars up to ensure good mixing, and I will rearrange the placing of the jars on the table so that you do not know which one is which. You may then request either the "left- hand" or the "right- hand" jar. You get to choose exactly one jar, you are allowed to withdraw at

most one marble from the jar, and you do not get a second chance if you are unhappy with your choice.

How many of each color marble should you place in each jar to maximize the probability that your blindfolded random draw obtains a white marble? 24

解析

最优分配:

  • 罐 A:放 1 个白球;
  • 罐 B:放剩余的 49 白 + 50 黑(共 99 个)。

然后随机选一个罐并从中随机取 1 个球,抽到白球概率为

121+124999=74990.7475.\frac12\cdot 1+\frac12\cdot\frac{49}{99}=\boxed{\frac{74}{99}}\approx 0.7475.