单骰与双骰

Solo or Pair

专题: Probability / 概率
难度: L4
来源: QuantQuestion

个人笔记

题目详情

你可以从两个游戏里选一个：

游戏 1： 掷一枚公平六面骰 4 次，要求至少出现一次 6。

游戏 2： 掷一对公平六面骰 24 次，要求至少出现一次双 6（66）。

哪个游戏的获胜概率更高？若游戏 1 更高答 1，游戏 2 更高答 2，若相等答 3。

英文原题

You are given the option between two games involving fair standard $6$ -sided dice. Which one gives you a higher probability of winning? Answer $1$ for the first game, $2$ for the second game, or $3$ if both give you equal probability of winning.

Game 1: You are given $4$ rolls of a single die and you must roll $6$ at least once.

Game 2: You are given $24$ rolls of a pair of dice and you must roll $66$ at least once.

解析

游戏 1 获胜概率：

1-\left(\frac{5}{6}\right)^4\approx 1-0.4823=0.5177.

游戏 2 获胜概率：单次出 66 的概率为 $1/36$ ，所以

1-\left(\frac{35}{36}\right)^{24}\approx 1-0.5134=0.4866.

因此游戏 1 更高，答案为 $\boxed{1}$ 。

英文解析

Probability of winning Game 1:

1-\left(\frac{5}{6}\right)^4\approx 1-0.4823=0.5177.

Probability of winning Game 2: The probability of a single 66 is $1/36$ , so

1-\left(\frac{35}{36}\right)^{24}\approx 1-0.5134=0.4866.

So game 1 is higher and the answer is $\boxed{1}$ .