Two coins decide how many dice: odd total

Abdelrahman tosses two fair coins and then throws one fair six-sided die for every head that appeared. If no dice are thrown the total is $0$. What is the probability that the total shown on the dice is odd?

$3/8$. Condition on the number of heads: with $0$ heads (prob $\tfrac14$) the total is $0$, even; with $1$ head (prob $\tfrac12$) one die is odd half the time; with $2$ heads (prob $\tfrac14$) the sum of two dice is odd half the time. So $\tfrac12\cdot\tfrac12+\tfrac14\cdot\tfrac12=\tfrac38$.