I have run VP simulations and if you leave the simulator on long enough it always hits the expected return of the game.
You could do this with BJ as well. It would take much less time to see the expected return in BJ compared to VP.
Yep, on a fair game this always happens. Mathematically it is the same that if you do a random walk starting from zero then you will eventually come back to zero at some point in the future.
But what do you think of the following actual bankroll graph?
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