# 2021 Fall AMC 10A Problems/Problem 9

## Problem

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?

$\textbf{(A)}\ \frac{3}{8} \qquad\textbf{(B)}\ \frac{4}{9} \qquad\textbf{(C)}\ \frac{5}{9} \qquad\textbf{(D)}\ \frac{9}{16} \qquad\textbf{(E)}\ \frac{5}{8}$

## Solution 1

Since an even number is $3$ times more likely to appear than an odd number, the probability of an even number appearing is $\frac{3}{4}$. Since the problem states that the sum of the two die must be even, the numbers must both be even or both be odd. We either have EE or OO, so we have $$\frac{3}{4}\cdot \frac{3}{4} + \frac{1}{4} \cdot \frac{1}{4} = \frac {1}{16} + \frac {9}{16} = \frac{10}{16} = \boxed{\textbf{(E)}\ \frac{5}{8}}.$$

~Arcticturn ~Aidensharp

## Solution 2 (Complementary Counting)

As explained in the above solution, the probability of an even number appearing is $\frac{3}{4}$, while the probability of an odd number appearing is $\frac{1}{4}$. Then the probability of getting an odd and an even (to make an odd number) is $\frac{3}{4} \cdot \frac{1}{4} \cdot 2 = \frac{3}{8}.$ Then the probability of getting an even number is $1 - \frac{3}{8} = \boxed{\textbf{(E)}\ \frac{5}{8}}.$

~littlefox_amc

## Video Solution (HOW TO THINK CREATIVELY!!!)

https://youtu.be/xZo7pKxrnGA


~Education, the Study of Everything

## Video Solution by WhyMath

https://youtu.be/oOKx2Wqp_ig ~savannahsolver