无法辨认的骰子
Illegible Dice
题目详情
给你两个公平骰子：一个是普通六面骰，另一个是一个无法辨认点数的六面骰。每次同时掷这两个骰子时，两者点数和在 1 到 12 之间的每个整数都等概率出现。求这个无法辨认的骰子六个面的点数之和。
You are given two fair dice: one is a normal six-sided die and the other is an illegible, six-sided die. Whenever you toss the two dice, it is equally likely for the sum of the faces to be any integer between 1 and 12, inclusive. What is the sum of the sides of the illegible die?
解析
两个六面骰一共存在 36 种可能的点数组合。若这 12 个和都等概率出现，那么每个和必须恰好对应 3 种组合。
为了让和为 1 出现 3 次，无法辨认的骰子上必须有三个 0。进一步地，为了让和为 12 出现 3 次，无法辨认的骰子上必须有三个 6。
因此，我们已经确定这个骰子的六个面，其点数总和为 18。
Original Explanation
There are 36 possible face combinations of two six-sided dice rolls. In order for each of the twelve outcomes to be equally likely, each must be able to occur exactly three times. In order for 1 to occur three times, there must be three 0's on the illegible die. Furthermore, in order for 12 to occur three times, there must be three 6's on the illegible die. Hence, we have solved the sides of the illegible die and the sum of its sides is 18.