What is the pmf of rolling a die until obtaining three consecutive $6$s? I am trying to find the pmf of rolling a die until 3 consecutive 6s turn up. I am able to find the expected value using a tree diagram, but the pmf is not obvious to me.
Let A be the event of not rolling 6, and let B be the event of rolling a 6.
The geometric distribution does not work, because we could have any number of As and Bs (ex. ABBAAABBA...) until we reach BBB. But using a binomial doesn't make sense to me either, because we don't care how many As or Bs we have, we just care about the final 3 consecutive Bs.

The probabilities can be calculated recursively by $p(1)=p(2)=0$, $p(3)={1\over 216}$ and for $n>3$, $$p(n)={5\over 6}\,p(n-1)+{5\over 36}\,p(n-2)+{5\over 216}\,p(n-3).\tag1$$
I suspect that an explicit formula for $p(n)$ will be too complicated to be useful, but you may get useful information via the probability generating function $$G(s)={s^3\over 216-180s-30s^2-5s^3}.\tag 2$$  For example, differentiating $G$ and setting $s=1$ shows us that the expected number of throws until we see three 6s in a row is $258$. 
