2008 AMC 10B Problem 16

Question

Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, the sum is

0.

)

\dfrac{3}{8}

\dfrac{1}{2}

\dfrac{43}{72}

\dfrac{5}{8}

\dfrac{2}{3}

Answer: A

Concepts:conditional probability dice (probability)parity

Difficulty rating: 1490

Small hint:

If at least one die is rolled, the sum is odd exactly half the time

Big hint:

Subtract the probability of rolling no dice (both coins tails), then take half of the rest

Solution:

Whenever at least one die is rolled, by symmetry the sum is odd with probability

\tfrac12.

No die is rolled only when both coins are tails, with probability

\tfrac14;

that sum

0

is even. So the answer is

\left(1-\tfrac14\right)\cdot\tfrac12=\tfrac38.

Thus, the correct answer is A.