Expected number of tails before the third head I have a question that I am currently modeling on coin tosses. Suppose, a number of coin tosses are done with $p = 1/4$ such that I have a string of tails, then one head, then a string of tails, then another head, and a string of tails and then a third head and so on. Something like TTTTHTTTTHTTTH. I have to find the expected number of tails that occur before the third head. If I consider all the subsequences of TTTTH as random variables $X1, X2, X3$ with the same mean and distribution, then could I do something like $n*E(Xi) = n/p$  where $n = 3 $ to get the expected number of tails before the first head? Is this a good way to approach?

If I were you, I would consider the negative binomial distribution, which illustrates, collectively, the number of failures before the third head. 
You already have your probability of success, $p=\frac 14$ and $r=3$ and so, knowing that your distribution is parameterized, you go on to solve for the expected number of failures.
