2015 AMC 12B 第 9 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
9.
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 ，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 independently of what has happened before. What is the probability that Larry wins the game?
小提示：
Larry 立即获胜的概率是 。
Larry wins immediately with probability
大提示：
如果两人都没击中（概率 ），游戏回到起点；解 。
If both miss (probability ), the game resets; solve
解答：
设 为 Larry 获胜的概率。他以概率 立即获胜；或者两名玩家都没击中（概率 ），游戏重新开始。
所以 ，得 ，因此 。
因此，正确选项是 C。
Let be the probability Larry wins. He wins right away with probability or both players miss (probability ) and the game restarts.
So giving and
Thus, the correct answer is C.
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