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兰多教授

Professor Rando

专题
Probability / 概率
难度
L6

题目详情

Rando 教授有 4 名研究生，Daphne、Max、Mindy 和 Sam。教授 提出以下游戏：他独立生成两个随机整数并且 统一从 1 到 5（含），然后告诉每个学生一个 关于这两个数字的不同事实。他告诉：

  • Daphne 整数的绝对差，

  • Max 整数的最大值，

  • Mindy 是整数中的最小值，并且

  • Sam 整数之和。

然后，每天直到比赛结束，他都会聚集他的学生并问他们， 按字母顺序排列，表示两个整数的同一性。每个学生都有 每天只有一次回答的机会，即当她或他被要求时。每个学生 仅当他或她明确知道答案时才回答，否则 那天没有给出答复。所有的学生都知道这一点，并且没有 勾结。一旦学生给出答案（这将是正确的），该学生 获胜，游戏结束。

每个学生获胜的可能性有多大？

Professor Rando has 4 grad students, Daphne, Max, Mindy, and Sam. The professor proposes the following game: he generates two random integers independently and uniformly from 1 to 5 (inclusive), and then tells each of his students a different fact about the two numbers. He tells:

  • Daphne the absolute difference of the integers,
  • Max the maximum of the integers,
  • Mindy the minimum of the integers, and
  • Sam the sum of the integers.

Then, each day until the game ends, he congregates his students and asks them, in alphabetical order, for the identity of the two integers. Each student has only one chance to answer each day, when she or he is called upon. Each student answers ONLY when the answer is definitively known to him or her, and otherwise gives no answer that day. All of the students know this, and there is no collusion. Once a student gives an answer (which will be correct), that student wins and the game ends.

How likely is each student to win?

解析

这 4 名学生获胜的机会是：

  • 达芙妮：1/5

  • 最大：6/25

  • 明迪：6/25

  • 山姆：8/25

总会有人获胜，尽管这可能需要不止一天的时间。请注意 兰多教授只抽一次号码，而不是每天，并且还要注意 兰多教授抽出任意给定的 2 个不同数字的可能性是其两倍 比他画任何给定的一组 2 个相同的数字。我们将省略更多 现在彻底解释一下，因为一月份的谜题也与教授有关 兰多。

恭喜本月提交正确解决方案的所有人， 尤其是尼尔·阿尔伯格 (Neil Alberg)，本月随机选出的简街 (Jane Street) 获胜者 T 恤！

Original Explanation

The chances that the 4 students win are:

  • Daphne: 1/5
  • Max: 6/25
  • Mindy: 6/25
  • Sam: 8/25

Someone always wins, although it might take more than one day. Note that Professor Rando only draws his numbers once, not each day, and also note that Professor Rando is twice as likely to draw any given set of 2 different numbers than he is to draw any given set of 2 identical numbers. We’ll leave out a more thorough explanation for now, since January’s puzzle also relates to Professor Rando.

Congratulations to everyone who submitted a correct solution this month, especially Neil Alberg, this month’s randomly-selected winner of a Jane Street t-shirt!