题目

A box contains 3 pairs of blue socks and 2 pairs of green socks. Each pair consists of a left-hand and a right-hand sock.

Each of the socks is separate from its mate and thoroughly mixed together with the others in the box.

3 socks are randomly selected from the box.

Quantity A

Probability that a matched set (i.e., a left- and right-hand sock of the same color) will be among the 3 selected socks

Quantity B

选项

A.

Quantity A is greater.


B.

Quantity B is greater.


C.

The two quantities are equal.


D.

The relationship cannot be determined from the information given.


解析

我们来计算无法凑成配对的对立事件概率,再用1减去这个值。 出现该情况(无法凑成配对)的条件只能是: 抽到的3张全为同一只手的蓝色牌;抽到2张同一只手的蓝色牌+任意1张绿色牌;或抽到2张同一只手的绿色牌+任意1张蓝色牌 全蓝(BBB):$\frac{6}{10} \times \frac{2}{9} \times \frac{1}{8} = \frac{1}{60}$(抽到1张蓝色牌的概率为$\frac{6}{10}$,之后剩余9张牌中,同一只手的蓝色牌仅剩2张,即概率为$\frac{2}{9}$,以此类推) 2蓝1绿(BBG):$\left( \frac{6}{10} \times \frac{2}{9} \times \frac{4}{8} \right) \times 3 = \frac{12}{60}$,乘以3是因为该情况有3种不同排列方式:BBG、BGB、GBB 2绿1蓝(GGB):$\left( \frac{4}{10} \times \frac{1}{9} \times \frac{6}{8} \right) \times 3 = \frac{6}{60}$ $P = 1 - \left( \frac{1}{60} + \frac{12}{60} + \frac{6}{60} \right) = \frac{41}{60}$ 答案应该选A
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