2005 AMC 10B Problem 9

One fair die has faces

1,

1,

2,

2,

3,

3

and another has faces

4,

4,

5,

5,

6,

6.

The dice are rolled and the numbers on the top faces are added. What is the probability that the sum will be odd?

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Solution

The first die is odd (a

1

3

) with probability

\dfrac23

and even with probability

\dfrac13.

The second die is odd (a

5

) with probability

\dfrac13

and even with probability

\dfrac23.

The sum is odd when the two parities differ:

\dfrac13 \cdot \dfrac13 + \dfrac23 \cdot \dfrac23 = \dfrac19 + \dfrac49 = \dfrac59.

Thus, D is the correct answer.