AMC10 2015 B

AMC10 2015 B · Q18

AMC10 2015 B · Q18. It mainly tests Probability (basic), Expected value (basic).

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?

Johann 有 64 个公平硬币。他翻转所有硬币。凡是落在反面的硬币再抛一次。第二抛落在反面的硬币第三次抛掷。现在正面朝上的硬币的期望数量是多少？

(A) 32 32

(B) 40 40

(D) 56 56

(E) 64 64

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): A coin can be tossed once, twice, or three times. View the problem as tossing each coin three times. If all three tosses are tails then the coin ends on a tail; however, if any of the three tosses is a head then the coin ends on a head (the subsequent tosses can be ignored). Thus each coin has a 7 out of 8 chance of landing on heads. Therefore the expected number of heads is $\frac{7}{8}\cdot 64=56$.

答案（D）：一枚硬币可以被抛掷一次、两次或三次。把这个问题看作对每枚硬币都抛掷三次：如果三次结果全是反面，则这枚硬币最终为反面；然而，只要三次中有一次是正面，则这枚硬币最终为正面（后续抛掷可忽略）。因此，每枚硬币最终为正面的概率是 $\frac{7}{8}$。所以正面的期望个数为 $\frac{7}{8}\cdot 64=56$。

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