Quote
The fundamental reason why all martingale-type betting systems fail is that no amount of information about the results of past bets can be used to predict the results of a future bet with accuracy better than chance. In mathematical terminology, this corresponds to the assumption that the win-loss outcomes of each bet are independent and identically distributed random variables, an assumption which is valid in many realistic situations. It follows from this assumption that the expected value of a series of bets is equal to the sum, over all bets that could potentially occur in the series, of the expected value of a potential bet times the probability that the player will make that bet. In most casino games, the expected value of any individual bet is negative, so the sum of lots of negative numbers is also always going to be negative.
The martingale strategy fails even with unbounded stopping time, as long as there is a limit on earnings or on the bets (which is also true in practice).[1] It is only with unbounded wealth, bets and time that it could be argued that the martingale becomes a winning strategy.
https://en.m.wikipedia.org/wiki/Martingale_(betting_system) The martingale strategy fails even with unbounded stopping time, as long as there is a limit on earnings or on the bets (which is also true in practice).[1] It is only with unbounded wealth, bets and time that it could be argued that the martingale becomes a winning strategy.
