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 $0$.)
I thought that since odd+even =odd, we then have to get (odd,even) and (even,odd) pairs from $\{0,1,2,3,4,5,6\}$. Thus, since there are a total of $7*7$ possibilities for pairs and $4*3*2$ pairs that add to an odd number I get $\dfrac{4*3*2}{7*7} = \dfrac{24}{49}$ but the right answer is $\dfrac{3}{8}$. What did I do wrong?
Best Answer
There is a $1/4$ chance of no heads, and therefore no dice. The sum of no dice is $0$, which is even.
There is a $1/2$ chance of one head, and therefore one die. For one die, there is a $1/2$ chance of landing on an odd total $1,3,5$.
There is a $1/4$ chance of getting two heads, and therefore two dice. There are an equal number of possibilities giving even and odd, so there is again a $1/2$ chance of landing on an odd total.
So in total, we have total probability $$ \frac{1}{4} \cdot 0 + \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{4} \cdot \frac{1}{2} = \frac{3}{8},$$ as you indicated was the correct answer. $\diamondsuit$