Question

Bob 和 Alice 各有一个袋子，袋中各有蓝、绿、橙、红、紫五种颜色的球各一个。Alice 随机从她的袋子中选一个球放入 Bob 的袋子。然后 Bob 随机从自己的袋子中选一个球放入 Alice 的袋子。这个过程结束后，两个袋子中内容相同的概率是多少？

Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process the contents of the two bags are the same?

$\dfrac{1}{10}$

$\dfrac{1}{6}$

$\dfrac{1}{5}$

$\dfrac{1}{3}$

$\dfrac{1}{2}$

答案：D

知识点：基本概率对称性

难度评级：1460

解答：

Alice 将一个球移给 Bob 后，Bob 的袋子中有 $6$ 个球，其中恰好一种颜色出现两次。

两个袋子最终相同，当且仅当 Bob 还回这两个同色球中的一个。六个球中有两个符合条件，所以概率为 $\tfrac26=\tfrac13$ 。

所以正确答案是 D。

Alice moves one ball to Bob, so Bob's bag holds $6$ balls with exactly one color appearing twice.

The two bags end up identical exactly when Bob returns one of that duplicated pair. Two of the six balls qualify, so the probability is $\tfrac26=\tfrac13.$

Thus, the correct answer is D.

2006 AMC 10B 第 17 题

答案：D

难度评级：1460

解答：

其他年份的第 17 题