Here's the situation: I have a (fair) die, which I roll successively until I get 5 consecutive ones, in which case I stop rolling. My questions are:
- What is the probability that I stop after exactly 11 throws? (or $k$ throws in general?)
- What is the probability that I roll the die at least 9 times? (or $k$ times in general?)
- More generally, is there a name for the kind of probability distribution involved?
For the first question, I'm thinking of the following string: $ABCDEZ11111$ where $ABCDE$ can be anything other than $11111$, and $Z$ can be anything other than $1$. The probability is therefore $\left(1-\left(\frac{1}{6}\right)^5\right)\left(\frac{5}{6}\right)\left(\frac{1}{6}\right)^5$ by this reasoning (or is it?).
I have no idea how to approach the second question. Is it valid to use the inclusion-exclusion principle there?
Edit: After a bit of searching, I found this in Enumerative Combinatorics Volume 1, exercise 44 in Chapter 4:
and the solution: