Question

星期六下雨的概率是 $40\%$ ，星期天下雨的概率是 $30\%$ 。不过，如果星期六下雨，星期天下雨的可能性是星期六不下雨时的两倍。这个周末至少有一天会下雨的概率是 $\frac{a}{b}$ ，其中 $a$ 与 $b$ 是互质的正整数。求 $a + b$ 。

There is a $40\%$ chance of rain on Saturday and a $30\%$ chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Find $a + b.$

答案

答案：107

知识点：条件概率对立事件概率

难度评级：2070

解答：

设星期六不下雨时星期天下雨的概率为 $p$ ；星期六下雨时，这个概率为 $2p$ 。由星期天下雨的总概率得 $0.4(2p) + 0.6p = 0.3,$ 所以 $1.4p = 0.3$ ，从而 $p = \frac{3}{14}$ 。

周末完全不下雨，恰好是星期六不下雨且随后星期天也不下雨，其概率为 $\frac{3}{5}\left(1 - \frac{3}{14}\right) = \frac{3}{5} \cdot \frac{11}{14} = \frac{33}{70}$ 。因此至少一天下雨的概率是 $1 - \frac{33}{70} = \frac{37}{70}$ ，已经是最简分数，所以 $a + b = 37 + 70 = 107$ 。

Let $p$ be the probability that it rains on Sunday given a dry Saturday; given a rainy Saturday it is $2p.$ The overall Sunday chance gives $0.4(2p) + 0.6p = 0.3,$ so $1.4p = 0.3$ and $p = \frac{3}{14}.$

The weekend is completely dry exactly when Saturday is dry and then Sunday is dry: $\frac{3}{5}\left(1 - \frac{3}{14}\right) = \frac{3}{5} \cdot \frac{11}{14} = \frac{33}{70}.$ So the probability of rain on at least one day is $1 - \frac{33}{70} = \frac{37}{70},$ which is in lowest terms, and $a + b = 37 + 70 = 107.$

2016 AIME II 第 2 题

答案：107

难度评级：2070

解答：

其他年份的第 2 题