Question

一个罐子里有 $10$ 颗红糖和 $10$ 颗蓝糖。Terry 随机取出两颗糖，然后 Mary 从剩下的糖中随机取出两颗。已知他们取到的颜色组合（不考虑顺序）相同的概率为 $\frac{m}{n}$ ，其中 $m$ 和 $n$ 是互质正整数。求 $m + n$ 。

A jar has $10$ red candies and $10$ blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $m + n.$

答案

答案：441

知识点：无放回抽样组合分类讨论

难度评级：2180

解答：

颜色组合相同恰好发生在两人都取两红、两人都取两蓝，或两人都取一红一蓝。Terry 取两红的概率为 $\frac{\binom{10}{2}}{\binom{20}{2}} = \frac{45}{190} = \frac{9}{38}$ 。此后剩下 $8$ 颗红糖和 $10$ 颗蓝糖，所以 Mary 取两红的概率为 $\frac{\binom{8}{2}}{\binom{18}{2}} = \frac{28}{153}$ 。该情形的概率为 $\frac{9}{38} \cdot \frac{28}{153} = \frac{14}{323}$ ，由对称性，两人都取两蓝也是 $\frac{14}{323}$ 。

对混合颜色，Terry 成功的概率为 $\frac{10 \cdot 10}{\binom{20}{2}} = \frac{10}{19}$ ，此后两种颜色各剩 $9$ 颗，Mary 成功的概率为 $\frac{9 \cdot 9}{\binom{18}{2}} = \frac{9}{17}$ ，所以该情形概率为 $\frac{10}{19} \cdot \frac{9}{17} = \frac{90}{323}$ 。

总概率为 $\frac{14 + 14 + 90}{323} = \frac{118}{323}$ 。因为 $118 = 2 \cdot 59$ ， $323 = 17 \cdot 19$ ，该分数已最简，所以 $m + n = 118 + 323 = 441$ 。

The combinations match exactly when both draw two reds, both draw two blues, or both draw one candy of each color. The probability that Terry draws two reds is $\frac{\binom{10}{2}}{\binom{20}{2}} = \frac{45}{190} = \frac{9}{38},$ after which $8$ reds and $10$ blues remain, so Mary draws two reds with probability $\frac{\binom{8}{2}}{\binom{18}{2}} = \frac{28}{153}.$ That case has probability $\frac{9}{38} \cdot \frac{28}{153} = \frac{14}{323},$ and by symmetry two blues each is also $\frac{14}{323}.$

For mixed draws, Terry succeeds with probability $\frac{10 \cdot 10}{\binom{20}{2}} = \frac{10}{19},$ leaving $9$ of each color, and Mary with probability $\frac{9 \cdot 9}{\binom{18}{2}} = \frac{9}{17},$ for $\frac{10}{19} \cdot \frac{9}{17} = \frac{90}{323}.$

The total is $\frac{14 + 14 + 90}{323} = \frac{118}{323}.$ Since $118 = 2 \cdot 59$ and $323 = 17 \cdot 19,$ the fraction is in lowest terms, and $m + n = 118 + 323 = 441.$

2004 AIME II 第 2 题

答案：441

难度评级：2180

解答：

其他年份的第 2 题