1981 AMC 12 第 26 题
先试着解答 1981 AMC 12 第 26 题，然后核对你的答案与精心整理的解答，解答来自 LIVE by Po-Shen Loh。你也可以参加完整限时模拟考试、查看全部 1981 AMC 12 解答，或核对答案。
所有题目均经美国数学协会（MAA）官方合法授权使用。
26.
爱丽丝、鲍勃和卡萝尔轮流反复掷一个骰子。爱丽丝先掷；爱丽丝之后总是轮到鲍勃；鲍勃之后总是轮到卡萝尔；卡萝尔之后总是轮到爱丽丝。求卡萝尔第一个掷出六点的概率。（每次掷出六点的概率为 ，且与其他任何一次投掷的结果相互独立。）
Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is independent of the outcome of any other toss.)
小提示：
要让卡萝尔在某一轮获胜，爱丽丝和鲍勃必须先失败，然后卡萝尔成功
For Carol to win in a given round, Alice and Bob must fail before Carol succeeds
大提示：
如果三人都失败，则过程以相同的概率重新开始
If all three fail, the process restarts with the same probabilities
解答：
卡萝尔在第一轮获胜的概率为 。完整一轮都没有出现六点的概率为 。因此所求的等比级数为
所以正确答案是 D。
Carol wins in the first round with probability A complete round with no six has probability Therefore the desired geometric series is
Therefore, the correct answer is D.
其他年份的第 26 题
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- 1951 AMC 12
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- 1954 AMC 12
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- 1982 AMC 12
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- 1989 AMC 12
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- 1991 AMC 12
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- 1993 AMC 12
- 1994 AMC 12
- 1995 AMC 12
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- 1998 AMC 12
- 1999 AMC 12