Assume the following well known result: If a coin is flipped independently many times and its probability of heads $(H)$ is $p \in(0,1)$ and probability of tails $(T)$ is $(1-p)$, then the expected number of coin flips till the first time a heads is observed is $1 / p$.
What is the expected number of coin flips till the sequence $H T$, i.e., tails immediately following a heads, is observed for the first time?
- $\frac{1}{1-(1-p)^{2}}\left(2+\frac{1+p^{2}}{1-p}+p\right)$
- $\frac{2}{p(1-p)}$
- $2+\frac{1}{p(1-p)}$
- $\frac{1}{1-(1-p)^{2}}(4+1 / p)$
- $\frac{1}{p}+\frac{1}{1-p}$