2015 AMC 10B 第 18 题

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

小提示：

一枚硬币最终为反面，当且仅当前三次都是反面。

A coin is tails at the end only after three tails

大提示：

使用期望的线性性。

Use linearity of expectation

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\text{。}

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

小提示：

大提示：

解答：

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