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.

You pay one if either you roll $1$ or $2$, or you roll $5$ or $6$ for $k$ times and then you roll $1$ or $2$ again. Therefore the probability of paying one is:
$$
\Pr(W=-1)=\frac 13+\frac 13\left(\frac 13+(\frac{1}3)^2+...\right)=\frac 13\times \frac{1}{1-\frac 13}=\frac 12
$$
Similarly probability of winning two is equal to $\frac 12$ and you can find your expected value which is $\frac 12$.
The idea is that the probability of not winning anything is to roll $5$ or $6$ for infinite amount of times whose probability is zero and because of symmetry of the problem, the probability of each other cases are $\frac 12$.
