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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:

  1. What is the probability that I stop after exactly 11 throws? (or $k$ throws in general?)
  2. What is the probability that I roll the die at least 9 times? (or $k$ times in general?)
  3. 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:

Exercise 44 question

and the solution:

Exercise 44 answer

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    @wj32 you are right it is not exactly this, but perhaps you can adapt some of the ideas, perhaps not.2012-10-27

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