Math gambling question. Why is the net loss $0$? Suppose you are playing a gambling game. There is a $50/50$ chance of losing $1$ dollar, and winning $1$ dollar. Your starting money is $1$ dollar, and you keep playing until you either lose all your money, or you finish $1000$ rounds.
Using a computer, with a sample size of $100,000$ players, the average player ends with about $1$ dollar. My gut tells me that the average person should lose money, since hitting $0$ dollars locks you out from the game. Why does kicking players with $0$ dollars not affect the average significantly?
An observation is, when looking at an individual's money at the end, there are many people with $0$ dollars, but also a few big winners.

On each round of the game, one's expected loss is zero, whether or not
one has been eliminated. If one is active, one's expected loss on a round
is $\frac12(1+(-1))=0$. If one is eliminated, one always loses zero.
By linearity of expectation, one's expected loss in $10^4$ rounds
is $10^4\times 0$.
This problem is essentially random walk with one absorbing barrier.
