2021 AMC 10A Fall Problem 9

Question

When a certain unfair die is rolled, an even number is

3

times as likely to appear as an odd number. The die is rolled twice. What is the probability that the sum of the numbers rolled is even?

\dfrac{3}{8}

\dfrac{4}{9}

\dfrac{5}{9}

\dfrac{9}{16}

\dfrac{5}{8}

Let the probability of odd be

p,

so even is

3p

An even sum means two odds or two evens

Let

p

be the probability that an odd number is rolled. Then

3p

is the probability an even number is rolled. We know that

p + 3p = 1 \Rightarrow p = \dfrac{1}{4}.

The only way for the sum to be even is if both rolls have the same parity. This happens with a probability of

\dfrac{1}{4}^2 + \dfrac{3}{4}^2 = \dfrac{10}{16} = \dfrac{5}{8}.

Thus, E is the correct answer.

Problem 9 in other years