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2021 AMC 10B Problem 18

Problem 18 of 25IntermediateCounting & Probability

A fair 66-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?

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Solution

The probability that the first number is even is 36.\frac 36. The probability that the second distinct number is even is 25.\frac 25. The probability that the third distinct number is even is 14.\frac 14. The combined probability is 3⋅2⋅16⋅5⋅4=120.\dfrac{3\cdot 2\cdot 1}{6\cdot 5\cdot 4} = \dfrac 1{20}. Thus, the answer is C .

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Concepts: dice (probability) · basic probability

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.