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?

$\dfrac{1}{120}$

$\dfrac{1}{32}$

$\dfrac{1}{20}$

$\dfrac{3}{20}$

$\dfrac{1}{6}$

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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 .

2021 AMC 10B Spring Problem 18

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Problem 18 in Other Years