Expected winnings of betting game The game rules are: bet an amount, flip a fair coin, if heads I win the bet and get the amount, if tails I lose the bet and pay the amount. I quit after getting my first head. If I lose, I double the bet and play again. What are my expected earnings (start with a $1 bet)? 
I visualized a few rounds and saw that I can play until I get net \$1 back. So I always can regain my losses. So I think expected earnings is $1$? I'm not sure how to justify this answer.

HINT: You have a Geometric distribution here. This has a pmf as $f(x) = (1-p)^x\cdot p$ where $p$ is the probability of getting heads. Can you find the expected value of this and apply it to the betting game?
Edit: the geometric distribution has two schools of thought. the pmf that I gave you comes from counting the number of failures before one success. The other idea, which I think you are using, is counting the number of trials before one success. the pmf for this is $f(x) = (1-p)^{x-1}p$
