点数减少 1：掷到偶数的概率

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专题: Probability / 概率
难度: L2
来源: QuantQuestion

题目详情

一枚公平六面骰点数为 1..6。先均匀随机选取其中一面，把该面的点数减 1（例如 4 变 3）。之后掷这枚改造后的骰子。

问：掷出偶数的概率是多少？

A fair $6$ -sided die has the values $1-6$ on the sides. The value of one of the sides (selected uniformly at random) is decreased by $1$ . Find the probability an even value is rolled on this die.

解析

若被减 1 的那面原本是偶数，则偶数面从 3 个变为 2 个，掷出偶数概率为 $2/6=1/3$ 。

若被减 1 的那面原本是奇数，则偶数面从 3 个变为 4 个，掷出偶数概率为 $4/6=2/3$ 。

两种情况各以 $1/2$ 发生，因此总体概率为

\frac12\left(\frac13+\frac23\right)=\frac12.

Original Explanation

It is equally likely to select an even or odd value to decrease the value by $1$ on. Subtracting one turns the side into one of opposite parity. If we select an even value to decrease, with probability $\dfrac{2}{6} = \dfrac{1}{3}$ we roll an even value. If we select an odd value to decrease, with probability $\dfrac{4}{6} = \dfrac{2}{3}$ we roll an even value. Therefore, the total probability is

\dfrac{\frac{1}{3} + \frac{2}{3}}{2} = \dfrac{1}{2}