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?

$\dfrac{3}{8}$

$\dfrac{4}{9}$

$\dfrac{5}{9}$

$\dfrac{9}{16}$

$\dfrac{5}{8}$

Solution:

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.

2021 AMC 10A Fall Problem 9

Solution:

Problem 9 in Other Years