OK, interesting take which I had not realized: the problem is also different because there is a limited amount of cash, I now realize. However, the idea that “

OK, interesting take which I had not realized: the problem is also different because there is a limited amount of cash, I now realize. However, the idea that “the mean distribution (the uniform, in this case)” needs to be near the N-th step one is what is statistically wrong, that is why I turned to the coin example. But yes, thanks for pointing out the differences!