I can only find a brute-force solution to the following problem:
A fair coin is flipped until it comes up "heads" for the first time. Every time the coin is flipped, a fair six-sided die is thrown. Let $N$ be the number of coin-flips and $S$ be the sum of all the die-throw results. What is the probability of the event $N\leq 4 \; \cap S\leq 7$? (or equivalently, what is the value of $F_{N,S}(4,7)$ ?)
Is there a neat way (either combinatoric or probability-theoretic) to find the solution?
I know that $N\sim Geom(1/2)$, so that all we need to find is the conditional probabilities of $S\leq 7$ on the different values of $N$, but this still leaves a lot of dirty counting to be done.
Trying to present $S$ as a linear combination of simpler random variables also didn't seem to provide any shortcuts.. (But maybe I'm missing a clever way to do it)
Thanks