Question

Sandy 的抽屉里有 $5$ 双袜子，每双颜色不同。星期一，Sandy 从抽屉中的 $10$ 只袜子里随机选出两只单袜。星期二，Sandy 从剩下的 $8$ 只袜子里随机选出 $2$ 只；星期三再从剩下的 $6$ 只袜子里随机选出两只。星期三是 Sandy 第一次选到同色袜子的概率为 $\frac{m}{n}$ ，其中 $m$ 与 $n$ 是互质的正整数。求 $m + n$ 。

In a drawer Sandy has $5$ pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the $10$ socks in the drawer. On Tuesday Sandy selects $2$ of the remaining $8$ socks at random and on Wednesday two of the remaining $6$ socks at random. The probability that Wednesday is the first day Sandy selects matching socks is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

答案

答案：341

知识点：基本概率无放回抽样对称性

难度评级：2510

解答：

想象把十只袜子按每天两只发完五天；把无序袜子对分配到各天的所有方式等可能，而交换天数不会改变分布。因此交换星期一与星期三可知，所求概率（不配对、不配对、配对）等于星期一配对且星期二、星期三不配对的概率。

这个模式按顺序很容易计算。星期一配对的概率为 $\frac{1}{9}$ （第二只必须是第一只的另一只）。剩下的 $8$ 只袜子构成 $4$ 个完整对，所以星期二不配对的概率为 $1 - \frac{4}{\binom{8}{2}} = \frac{6}{7}$ 。星期二的不配对会拆开两对，在剩下的 $6$ 只袜子中留下 $2$ 个完整对，所以星期三不配对的概率为 $1 - \frac{2}{\binom{6}{2}} = \frac{13}{15}$ 。

概率为 $\frac{1}{9} \cdot \frac{6}{7} \cdot \frac{13}{15} = \frac{26}{315}$ ，所以 $m + n = 26 + 315 = 341$ 。

Imagine dealing all ten socks out two per day for five days; every assignment of unordered pairs to days is equally likely, and permuting the days does not change this distribution. Swapping Monday and Wednesday therefore shows that the desired probability (mismatch, mismatch, match) equals the probability of a match on Monday followed by mismatches on Tuesday and Wednesday.

That pattern is easy to compute in order. Monday matches with probability $\frac{1}{9}$ (the second sock must be the first sock's mate). The remaining $8$ socks then form $4$ complete pairs, so Tuesday mismatches with probability $1 - \frac{4}{\binom{8}{2}} = \frac{6}{7}.$ Tuesday's mismatch breaks two pairs, leaving $2$ complete pairs among the $6$ remaining socks, so Wednesday mismatches with probability $1 - \frac{2}{\binom{6}{2}} = \frac{13}{15}.$

The probability is $\frac{1}{9} \cdot \frac{6}{7} \cdot \frac{13}{15} = \frac{26}{315},$ so $m + n = 26 + 315 = 341.$

2015 AIME I 第 5 题

答案：341

难度评级：2510

解答：

其他年份的第 5 题