Question

Johann 有 $64$ 枚公平硬币。他抛所有硬币。任何反面朝上的硬币都再抛一次。第二次仍为反面的硬币再抛第三次。现在正面朝上的硬币的期望数量是多少？

Johann has $64$ fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads?

$32$

$40$

$48$

$56$

$64$

答案：D

知识点：期望值对立事件概率

难度评级：1070

解答：

一枚硬币最后仍为反面，当且仅当它连续 $3$ 次都是反面，概率为 $\frac 18$ 。因此任意一枚硬币最后为正面的概率为 $\frac 78$ 。

由于一枚硬币为正面的概率是 $\frac78$ ，且一共有 $64$ 枚硬币，所以现在正面朝上的硬币期望数为 $64 \cdot \dfrac 78 = 56.$

所以正确答案是 D。

A coin ends as tails if and only if it has $3$ flips that are tails, which happens with probability $\frac 18.$ Thus, the probability of any coin being heads is $\frac 78 .$

Since each of the $64$ coins ends as heads with probability $\frac78,$ linearity of expectation gives the expected number of coins that are now heads: $64 \cdot \dfrac 78 = 56.$

Thus, the correct answer is D .

2015 AMC 10B 第 18 题

答案：D

难度评级：1070

解答：

其他年份的第 18 题