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?

The idea is good. You are letting $X_1$ be the number of tails before the first head, and $X_2$ the number of tails between the first head and the second head, and so on. 
We want $E(X_1+\cdots+X_n)$, which by the linearity of expectation is $E(X_1)+\cdots+E(X_n)$. 
We have $E(X_i)=\frac{1}{p}-1$, where $p$ is the probability of
 head.
Thus the expected number of tails before the $n$-th head is $n\left(\frac{1}{p}-1\right)$.
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