I have an interesting roulette problem that I initially thought was easy but now I'm second guessing my self. The problem is as follows:
A friend of yours thinks that he has devised a purely mathematical way of beating the standard European roulette wheel, which has 37 pockets. His plan is to come to the roulette table with 100 units with which to bet. On each spin, he places a one-unit, single-number bet. (Recall that winning such a bet returns 36 units to the winner.) He will make exactly 100 bets, no matter how much he wins or loses. He claims that this strategy results in a greater than 50% chance that he will come out ahead at the end of 100 spins, so using this strategy repeatedly will make him a winner.
a) Does he have a greater than 50% chance of coming away with more money than he started with?
b) If the answer to the previous question is yes, then does that make his strategy a winning one, meaning that it will lead to a long-term increase in bankroll?
So for part A I think he does have a greater than 50% chance of coming away ahead. He needs to win at least 3 times in order to walk away with more than he started with. And the probability of winning at least three times is 50.939% (I did 1 - the probability of winning 0, once, or twice with p=(1/37).)
So based on that it seems like he does have a greater than 50% chance of walking away ahead...
This goes against my intuition, especially for part b. If everyone could do this strategy and come out ahead then they would. But surely it can't be this simple... I don't see how this would lead to a long term increase in bank roll. Any thoughts?