2021 AMC 10B Spring Problem 18

Question

A fair

6

-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?

\dfrac{1}{120}

\dfrac{1}{32}

\dfrac{1}{20}

\dfrac{3}{20}

\dfrac{1}{6}

Answer: C

Concepts:dice (probability)basic probability

Difficulty rating: 1220

Small hint:

Ignore repeated rolls until a new face type appears

Big hint:

The first three distinct outcomes must be the three even numbers before any odd number appears

Video solution:

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Written solution:

The probability that the first number is even is

\frac 36.

The probability that the second distinct number is even is

\frac 25.

The probability that the third distinct number is even is

\frac 14.

The combined probability is

\dfrac{3\cdot 2\cdot 1}{6\cdot 5\cdot 4} = \dfrac 1{20}.

Thus, the answer is C .