Rolling and rerolling dice

Question

I roll $10$ initial dice. Whenever I roll a $6$, I roll $5$ more dice.

What's the expected number of dice I'm going to throw?

I was thinking about it when trying to create a die system, but I can't solve the problem.

Answer 1

If for every six you roll you roll five more dice, then you can compute the expected number for rolling one dice $E_1$ as follows: there is a $\frac 56$ chance that the first throw is the last, and a $\frac 16$ chance that you will be rolling five dice next time. Let $E_n$ be the expected number of dice you roll if you start by rolling $n$, then you have $$E_1=1+\frac 16E_5$$

Now you could equally do this by rolling the first of the five extra dice until you finish that trial, and then the second etc - the result of rolling the first is independent of the result of rolling the second. This is just a way of saying $E_5=5E_1$ so that $$E_1=1+\frac 56E_1$$And that is easy to solve. Then you need to find $E_{10}$.

Answer 2

This can be modeled as a branching process. Chapter $3$ of this book discusses branching processes. Useful for you would be equation $(3.1.16)$ on page $90$ and Theorem $3.3$ and Theorem $3.5$ on page $92$.