Question

有三个罐子。三枚硬币中的每一枚都随机且相互独立地放入三个罐子之一。装有最多硬币的罐子中的硬币数的期望是多少？

There are three jars. Each of three coins is placed in one of the three jars, chosen at random and independently of the placements of the other coins. What is the expected number of coins in a jar with the most coins?

$\dfrac{4}{3}$

$\dfrac{39}{27}$

$\dfrac{5}{3}$

$\dfrac{17}{9}$

$2$

答案：D

知识点：期望值分类讨论

难度评级：1630

解答：

共有 $3^3 = 27$ 种等可能的放法。其中 $3$ 种把所有硬币放在同一个罐子中，此时最大值为 $3$ ； $6$ 种把三个罐子各放一枚硬币，此时最大值为 $1$ 。其余 $18$ 种为 $2$ - $1$ 分布，此时最大值为 $2$ 。所以最大硬币数的期望为 $\frac{3 \cdot 3 + 2 \cdot 18 + 1 \cdot 6}{27} = \frac{51}{27} = \frac{17}{9}$ 。因此正确答案是 D。

There are $3^3 = 27$ equally likely placements. Of these, $3$ pile all coins into one jar (max $3$ ), and $6$ put one coin in each jar (max $1$ ). The other $18$ split $2$ - $1$ (max $2$ ). So the expected maximum is $\frac{3 \cdot 3 + 2 \cdot 18 + 1 \cdot 6}{27} = \frac{51}{27} = \frac{17}{9}.$ Therefore, the answer is D.

2025 AMC 10A 第 16 题

答案：D

难度评级：1630

解答：

其他年份的第 16 题