Question

Larry 和 Julius 正在玩游戏，轮流向放在台沿上的瓶子扔球。Larry 先扔。第一个把瓶子击落的人获胜。每一轮中，玩家把瓶子击落的概率都是 $\dfrac12$ ，且与之前发生的情况相互独立。Larry 赢得游戏的概率是多少？

Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is $\dfrac12,$ independently of what has happened before. What is the probability that Larry wins the game?

$\dfrac12$

$\dfrac35$

$\dfrac23$

$\dfrac34$

$\dfrac45$

答案：C

知识点：递推概率等比数列

难度评级：1540

小提示：

Larry 立即获胜的概率是 $\dfrac12$ 。

Larry wins immediately with probability $\dfrac12$

大提示：

如果两人都没击中（概率 $\dfrac14$ ），游戏回到起点；解 $x = \dfrac12 + \dfrac14 x$ 。

If both miss (probability $\dfrac14$ ), the game resets; solve $x = \dfrac12 + \dfrac14 x$

解答：

设 $x$ 为 Larry 获胜的概率。他以概率 $\dfrac12$ 立即获胜；或者两名玩家都没击中（概率 $\dfrac14$ ），游戏重新开始。

所以 $x = \dfrac12 + \dfrac14 x$ ，得 $\dfrac34 x = \dfrac12$ ，因此 $x = \dfrac23$ 。

因此，正确选项是 C。

Let $x$ be the probability Larry wins. He wins right away with probability $\dfrac12,$ or both players miss (probability $\dfrac14$ ) and the game restarts.

So $x = \dfrac12 + \dfrac14 x,$ giving $\dfrac34 x = \dfrac12$ and $x = \dfrac23.$

Thus, the correct answer is C.

2015 AMC 12B 第 9 题

答案：C

难度评级：1540

小提示：

大提示：

解答：

其他年份的第 9 题