Question

一个罐子中有 $4$ 个绿球和 $6$ 个蓝球。第二个罐子中有 $16$ 个绿球和 $N$ 个蓝球。从每个罐子中各随机取出一个球。两个球颜色相同的概率为 $0.58$ 。求 $N$ 。

An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58.$ Find $N.$

答案

答案：144

知识点：基本概率独立事件一次方程

难度评级：1750

解答：

两个球都是绿色的概率为 $\frac{4}{10} \cdot \frac{16}{16+N}$ ，两个球都是蓝色的概率为 $\frac{6}{10} \cdot \frac{N}{16+N}$ 。条件给出 $\frac{64 + 6N}{10(16 + N)} = \frac{29}{50}.$

清除分母，得到 $5(64 + 6N) = 29(16 + N)$ ，所以 $320 + 30N = 464 + 29N$ ，从而 $N = 144$ 。

Both balls are green with probability $\frac{4}{10} \cdot \frac{16}{16+N},$ and both are blue with probability $\frac{6}{10} \cdot \frac{N}{16+N}.$ The condition is $\frac{64 + 6N}{10(16 + N)} = \frac{29}{50}.$

Clearing denominators, $5(64 + 6N) = 29(16 + N),$ so $320 + 30N = 464 + 29N,$ giving $N = 144.$

2014 AIME I 第 2 题

答案：144

难度评级：1750

解答：

其他年份的第 2 题