What's the probability of tossing two heads if the game started with a head? A player is tossing an unbiased coin until two heads or two tails occur in a row. What's the probability of heads winning the game if the game started with a head?
I looked at Two tails in a row - what's the probability that the game started with a head? and came up with this solution, but I'd like someone to confirm that it's correct or show some alternative/more elegant solution:
First toss was a head. Now there is an infinite number of ways how the game can end with heads winning:


*

*$H$ — probability is $0.5$ 

*$THH$ - probability is $0.5^3$ 

*$THTHH$ - probability is $0.5^5$ 

*$THTHTHH$ - probability is $0.5^7$

*$\ldots$


So, total probability of heads winning the game is:
$$P(H) = \sum_{i=0}^{\infty}\frac{1}{2^{2i+1}} = \frac{0.5}{1 - 0.25} = \frac{2}{3}$$
I also checked that probability of tails winning is $\frac{1}{3}$ (calculated using pretty much the same method), so that's at least some sanity.

If $p$ is the probability you're looking for, then $p= 0.5 + 0.5(1-p)$. Can you see why?
