Approaching the following problem:
Gambles are independent, and each one results in the player being equally likely to win or lose 1 unit. Let $W$ denote the net winnings of a gambler whose strategy is to stop gambling immediately after his first win. Find $E[W]$
Realizing that the expectation is not just $E[ W] = \sum\limits_{x} i \left(\frac{ 1}{ 2}\right)$ I am unsure of how to approach the problem.
Letting $W_L$ be the value of $W$ accumulated by loses and $W_W$ be the value of $W$ accumulated by wins, I am inclined to believe we are looking at $E[ W_L] = \sum\limits_{i = 0}^\infty -i \left(\frac{ 1}{ 2}\right)^i$ and the $E[ W_W] = 1$.
However, 1) I do not know if this is the correct approach and 2) should this indeed be the correct approach, I do not understand [conceptually ] how to merge these two summations?