An amusing observation in connection with the first proof is that if you start with m dollars and can choose at each stage what the amount is that you will win/lose if the coin is heads/tails (the two amounts being equal of course), subject to the condition that you are not allowed to bet an amount that would take you over n or below zero, then your probability of getting to n before you get to 0 is still m/n, as long as you bet a positive integer number each time. In other words, if you try to do better by developing a strategy that involves betting different amounts at each stage, you won't. (But at least you won't do worse either.)
On the other hand, if you are playing roulette and your probability of winning goes very slightly down because of the 37, then your best hope is to bet the maximal amount every time, so as to minimize the chance that a 37 ever occurs during the process. (That's not a proof, but the conclusion is sound: if you take very small steps then the slight bias towards the bank means you will almost certainly lose.)
