conditional expectation? I'm trying to solve an expected value problem where a biased coin is flipped until a run of five heads is achieved. I need to compute the $E(X)$ where $X$ is the number of tails expected before the run of five heads.
Would this require conditional expectation, since $E(X)$ is dependent on $P(Y)$ which is the probability of a run of five heads? 
I know how to calculate the expected value of flips, but I'm pretty lost on counting the number of tails. 
If $E(Y)$ is value $n$, then would I solve like so?
$P(X = k \mid X E {n})$
$E(X) = P(X)E(Y)$

Let $p$ be the chance of a head on a single toss of the coin.
Every time you try to throw 5 heads in a row, you either succeed or fail
 by getting a tail "too soon". 
$$\underline{HHT}\ \ \underline{T}\ \ \underline{HT}\cdots\underline{HHT}\ \ \underbrace{\underline{HHHHH}}_{\mbox{success!}} $$
The number of tails $X$ observed is the same 
as the number of failures before the first success, so $X$ has a geometric 
distribution with $\mathbb{P}(\mbox{success})=p^5$. 
Therefore $\mathbb{E}(X)={1\over p^5}-1.$
