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.

Does it really matter if you get a 5 or a 6? It doesn't really,  because you just end up back to where you started. At the end of this whole game, you can only lose 1 or win 2.
So there is a $\frac12$ chance you have a net profit of $-1$, and a $\frac12$ chance you have a net profit of $2$. 
$\frac12*-1+\frac12*2=\frac12$
In case you still think that the reroll affects the expected win rate, let's go through that process.
There is a 2/3 chances you get 1-4, and so we multiply the above fraction with 2/3. The other 1/3 chance means we do this all over again from the beginning. You end up with an infinite expression that looks like:
$$x=\frac23*\frac12+\frac13(\frac23*\frac12+\frac13(\frac23*\frac12+\frac13(...)))\\x=\frac23*\frac12+\frac13(x)\\6x=2*1+2x\\4x=2\\x=1/2$$
