Expected value problem with two dice

Solution 1:

When you roll $k$ dice, the expected value of their product is $(3.5)^k$ (since the expected value of each die is $3.5$, and the dice are independent).

Let $P$ denote the product of the red dice. We compute $E[P]$ by conditioning on the value of $B$: $$ E[P]=\sum_{k=1}^6 E[P\mid B=k]\cdot P(B=k)=\sum_{k=1}^6 (3.5)^k\cdot (1/6) $$ This is a finite geometric series, so you can evaluate this easily.

In this case, visualizing the probability space is not especially helpful to the problem. However, for completeness, each element in the probability space looks like one of the ordered tuples below. Each $*$ in a tuple represents a number between $1$ and $6$. $$ (1,*),\;(2,*,*),\;(3,*,*),\;(4,*,*,*,*),\;(5,*,*,*,*,*),\;(6,*,*,*,*,*,*) $$ The probability of a tuple of length $k$ is $(1/6)^k$.