Let's say I have a bag with numbered slips of paper from 0 to 10. If I draw a 0 then I stop adding to the sum and return the sum. Else I add the number I draw to the current sum and place the slip back in the bag. I keep doing this until I draw the 0 slip. (The initial sum is 0).
What would be the expected value of the sum?
So this is what I've been trying to do.
I let $S$ = the total sum, $X_i $ = the ith slip drawn.
$\operatorname{E}[S] = \operatorname{E}[\operatorname{E}[S\mid X_1]] $
$\operatorname{E}[S] = \sum\limits_{i=1}^{10} \operatorname{E}[S \mid X_1 = i] * \operatorname{P}(X_1 = i) $
This is the part where I'm a bit confused.
Am I correct to say that $\operatorname{E}[S\mid X_1 = 1] $ is equal to $1 + \operatorname{E}[S] $ and so on?