Question

A pair of six-sided dice are labeled so that one die has only even numbers (two each of $2, 4,$ and $6$ ), and the other die has only odd numbers (two of each $1, 3,$ and $5$ ). The pair of dice is rolled. What is the probability that the sum of the numbers on the tops of the two dice is $7?$

$\dfrac{1}{6}$

$\dfrac{1}{5}$

$\dfrac{1}{4}$

$\dfrac{1}{3}$

$\dfrac{1}{2}$

Solution:

The pairs of numbers that sum to $7$ are $(2, 5), (4, 3), \text{ and } (6, 1).$

There is a $\dfrac{1}{3} \cdot \dfrac{1}{3} = \dfrac{1}{9}$ chance that we get any of these pairs.

There are $3$ pairs, which means that the total probability that the rolls sum to $7$ is $3 \cdot \dfrac{1}{9} = \dfrac{1}{3}.$

Thus, D is the correct answer.

2012 AMC 10A Problem 9

Solution:

Problem 9 in Other Years