1981 AMC 12 Problem 26
Attempt Problem 26 of the 1981 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1981 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
26.
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.)
Answer: D
Small hint:
For Carol to win in a given round, Alice and Bob must fail before Carol succeeds
Big hint:
If all three fail, the process restarts with the same probabilities
Solution:
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.
Problem 26 in other years
- 1950 AMC 12
- 1951 AMC 12
- 1952 AMC 12
- 1953 AMC 12
- 1954 AMC 12
- 1955 AMC 12
- 1956 AMC 12
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- 1958 AMC 12
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