2012 AMC 10A Problem 9

Question

A pair of six-sided fair 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 each of

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}

List the even-odd pairs that sum to

7

Each listed value appears on two faces of its die

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.

Problem 9 in other years