Mark and Alice agree to play the following game: they will flip a fair coin, with the winner receiving even money, and will play this game over and over until one side is wiped out. Mark has a finite bankroll of $n$ units. Alice has an infinite bankroll and can never be ruined.
Given enough time, will Mark get ruined? When do we expect Mark will be ruined?
EDIT: I adapted this from the finite version, where Alice has only $m$ units. There it's easy to see that a wipeout sequence has probability at least $\frac{1}{2}^{m+n}$. The law of large numbers assures us that as the number of flips approaches infinity we will get close to the expectation of this wipeout sequence. However, this argument only works because we can get a finite bound on the total bankroll (namely, $m+n$). In the case where Alice's bankroll is infinite things are less clear because there's no finite bound on Mark's bankroll either, since he can win coins from Alice. My intuition is that Mark will get ruined eventually, but I don't know how to prove it, nor do I have any idea when we'd expect that ruin to occur.