Question

Tom、Dick 和 Harry 正在玩一个游戏。他们同时开始，各自不断投掷一枚公平硬币，直到第一次出现正面时停止。三人投掷次数都相同的概率是多少？

Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which point he stops. What is the probability that all three flip their coins the same number of times?

$\dfrac18$

$\dfrac17$

$\dfrac16$

$\dfrac14$

$\dfrac13$

答案：B

知识点：几何分布独立事件

难度评级：1910

解答：

单个玩家第一次正面出现在第 $n$ 次的概率为 $\left(\tfrac12\right)^n$ 。三人都在第 $n$ 次停止的概率为 $\left(\left(\tfrac12\right)^n\right)^3=\left(\tfrac18\right)^n$ 。对 $n\ge1$ 求和，得 $\displaystyle\sum_{n=1}^\infty\left(\tfrac18\right)^n =\frac{1/8}{1-1/8}=\frac17$ 。

所以正确答案是 B。

A player's first head comes on flip $n$ with probability $\left(\tfrac12\right)^n.$ All three stopping on the same flip $n$ has probability $\left(\left(\tfrac12\right)^n\right)^3=\left(\tfrac18\right)^n.$ Summing over $n\ge1,$ $\displaystyle\sum_{n=1}^\infty\left(\tfrac18\right)^n =\frac{1/8}{1-1/8}=\frac17.$

Thus, the correct answer is B.

2016 AMC 12B 第 19 题

答案：B

难度评级：1910

解答：

其他年份的第 19 题