Disclaimer: I know all about the house edge, negative expectation, expected value, gambler's fallacy, etc.
In the roulette forum I frequent, there are various debates about the inevitability of loss of a roulette player. There are people who insist that "the math" says that that any roulette system will lose and they can prove it. They base their reasoning on spin independence and house edge.
I try to explain that since there are probabilities involved, one can not be certain, let alone prove, that a specific system will lose after say 10,000 spins.
Yes, the expected value is negative and the probability of winning may diminish, but there can be no proof, in the mathematical sense, that a player will definitely lose playing a specific system. Because there will always be a chance that he will win. After all we can not exclude the case that he is lucky and wins more bets that he "should".
- Am I correct that there can be no mathematical proof that a specific roulette player will lose money by playing roulette for, say, 10,000 spins?
- How can I explain mathematically the fact that it can not be proven that a roulette player will lose?
PS: It is interesting that, although by "lose" I meant "overall monetary loss" (which would require many losing bets), most replies interpreted "lose" as "lose at least one bet". Which is even more probable, yet it can not be proved it will happen.