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
选项
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