Expected value for games where you can replay? Lets just say there's a game where you roll one fair die.  If you roll a 1 or 2, you pay 1.  If you roll a 3 or 4, you win 2.  If you roll a 5 or 6 you roll again until you get a 1, 2, 3, or 4.
How much are you expected to win?  I can't figure out how to think about this.
Thanks.

Note that your expected gain given that you first rolled a $5$ or $6$ is the same as your expected gain initially… you just get to start over.
Using linearity of expectation, then, you can write your expected gain as
$$
E[G]=\sum_{i=1}^{6}E[G\;\vert\;X_1=i]\cdot P[X_1=1]=-\frac{1}{3}+\frac{2}{3}+\frac{1}{3}E[G]=\frac{1}{3}+\frac{1}{3}E[G],
$$
and then solve to yield $E[G]=1/2$.  
