Question

将标有 $1$ 、 $2$ 、 $3$ 的三个球放入一个罐中。抽出一个球，记录其编号，再将球放回罐中。这个过程再重复两次，每次抽取各球的可能性都相同。若记录的三个编号之和为 $6$ ，则三次都抽到编号为 $2$ 的球的概率是多少？

Three balls marked $1,$ $2$ and $3$ are placed in an urn. One ball is drawn, its number is recorded, and then the ball is returned to the urn. This process is repeated and then repeated once more, and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is $6,$ what is the probability that the ball numbered $2$ was drawn all three times?

$\frac1{27}$

$\frac18$

$\frac17$

$\frac16$

$\frac13$

答案：C

知识点：条件概率系统列举有放回抽样

难度评级：1690

小提示：

列出由 $\{1,2,3\}$ 中元素组成且和为 $6$ 的有序三元组

List the ordered triples from $\{1,2,3\}$ whose sum is $6$

大提示：

所有可能是 $(1,2,3)$ 的排列以及 $(2,2,2)$

The possibilities are the permutations of $(1,2,3)$ together with $(2,2,2)$

解答：

和为 $6$ 的有序三元组为 $(1,2,3)$ 的六个排列和三元组 $(2,2,2)$ 。它们的可能性相同，七个三元组中恰有一个含有三个 $2$ 。因此条件概率为 $\frac{1}{7}$ 。

所以正确答案是 C。

The ordered triples with sum $6$ are the six permutations of $(1,2,3)$ and the triple $(2,2,2).$ They are equally likely, and exactly one of the seven has three $2$ ’s. The conditional probability is therefore $\frac{1}{7}.$

Therefore, the correct answer is C.

1983 AMC 12 第 15 题

答案：C

难度评级：1690

小提示：

大提示：

解答：

其他年份的第 15 题