两骰条件概率

Two Dice Sum

专题: Probability / 概率
难度: L2
来源: QuantQuestion

个人笔记

题目详情

掷两枚公平六面骰（点数 1..6）。在不看结果的情况下，你被告知“至少有一枚骰子点数为 6”。

问：两枚骰子点数之和为 7 的概率是多少？

英文原题

You roll two six-sided dice, each with faces showing 1 through 6. Without looking, you are told that one die shows a 6. What is the probability that the sum of the two dice is 7?

解析

在“至少有一枚为 6”的条件下，等可能的有序结果为 11 种：

(6,1),(6,2),\dots,(6,6),(1,6),(2,6),\dots,(5,6).

其中点数和为 7 的只有 $(6,1)$ 与 $(1,6)$ 两种。

因此条件概率为 $\boxed{\frac{2}{11}}$ 。

（若题意改为“恰好一枚为 6”，则答案会是 $2/10=1/5$ 。）

英文解析

Under the condition that "at least one is 6", there are 11 possible ordered results:

(6,1),(6,2),\dots,(6,6),(1,6),(2,6),\dots,(5,6).

Only $(6,1)$ and $(1,6)$ have sum 7.

So the conditional probability is $\boxed{\frac{2}{11}}$ .

(If the condition were changed to “Exactly one is 6,” the answer would be $2/10=1/5$ .)