Question

Jen 参加抽奖，从 $S = \{1, 2, 3, \ldots, 9, 10\}$ 中选择 $4$ 个不同元素。随后从 $S$ 中随机抽出四个元素。如果 Jen 选的数中至少有两个被抽中，她就中奖；如果她选的四个数全部被抽中，她就中头奖。在 Jen 已经中奖的条件下，她中头奖的概率为 $\frac{m}{n}$ ，其中 $m$ 和 $n$ 是互质正整数。求 $m + n$ 。

Jen enters a lottery by selecting $4$ distinct elements of $S = \{1, 2, 3, \ldots, 9, 10\}.$ Then four elements of $S$ are drawn at random. Jen wins a prize if at least two of her numbers were drawn, and wins the grand prize if all four of her numbers were drawn. The probability that Jen wins the grand prize given that Jen wins a prize is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

答案

答案：116

知识点：条件概率组合

难度评级：2230

解答：

所有 $\binom{10}{4} = 210$ 种抽法等可能。与 Jen 的彩票恰好有 $k$ 个数重合的抽法数为 $\binom{4}{k}\binom{6}{4-k}$ ，所以中奖的抽法数为 $\begin{aligned} &\binom{4}{2}\binom{6}{2} + \binom{4}{3}\binom{6}{1} \\ &\quad {}+ \binom{4}{4}\binom{6}{0} \\ &= 90 + 24 + 1 = 115, \end{aligned}$ 其中正好 $1$ 种抽法中头奖。

因为中头奖必然中奖，条件概率为 $\frac{1/210}{115/210} = \frac{1}{115}$ ，所以 $m + n = 1 + 115 = 116$ 。

All $\binom{10}{4} = 210$ draws are equally likely. The number of draws sharing exactly $k$ numbers with Jen's ticket is $\binom{4}{k}\binom{6}{4-k},$ so the number winning a prize is $\begin{aligned} &\binom{4}{2}\binom{6}{2} + \binom{4}{3}\binom{6}{1} \\ &\quad {}+ \binom{4}{4}\binom{6}{0} \\ &= 90 + 24 + 1 = 115, \end{aligned}$ and exactly $1$ of these wins the grand prize.

Since the grand prize implies a prize, the conditional probability is $\frac{1/210}{115/210} = \frac{1}{115},$ so $m + n = 1 + 115 = 116.$

2024 AIME I 第 4 题

答案：116

难度评级：2230

解答：

其他年份的第 4 题