# Expected Mean using Total Law of Expectation

So I have an idea of conditioning on a new set variable say y that is 0 if the first toss is tails and 1 if heads although I am not sure how to structure this answer. I believe the law of total expectation is the most useful formula to solve this although any direction would be greatly appreciated.

After the first toss, you'll always have a dependency on the last toss. If the last toss is $$H$$ or $$T$$, let the expected number of tosses forward be $$E_h, E_t$$ respectively. We'll write two recursive equations:
For $$E_h$$, we the next toss is Heads (with prob. $$p$$) it's finished. If not, we have an expected number of tosses while the last toss is $$T$$:
$$E_h=p\times 1+(1-p)\times E_t$$
Similarly, for $$E_t$$, we have:
$$E_t=p\times E_h+(1-p)\times E_t$$
Finally, $$E[N]=1 + p\times E_h+(1-p)\times E_t=1 + E_t$$