2014 AIME I 第 2 题

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

把取到两个绿球的概率和取到两个蓝球的概率相加

Add the probabilities of green-green and blue-blue draws

解方程

\frac{4}{10} \cdot \frac{16}{16+N} + \frac{6}{10} \cdot \frac{N}{16+N} = \frac{29}{50}

，即可求得

N

Solve

\frac{4}{10} \cdot \frac{16}{16+N} + \frac{6}{10} \cdot \frac{N}{16+N} = \frac{29}{50}

for

N

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}\text{。}

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.

其他年份的第 2 题