Now suppose that we have just tossed four heads in a row, so that if the next coin toss were also to come up heads, it would complete a run of five successive heads. Since the probability of a run of five successive heads is only 1⁄32 (one in thirty-two), a person subject to the gambler's fallacy might believe that this next flip was less likely to be heads than to be tails. However, this is not correct, and is a manifestation of the gambler's fallacy; the event of 5 heads in a row and the event of "first 4 heads, then a tails" are equally likely, each having probability 1⁄32.
Assuming that the win/loss outcomes of each bet are independent and identically distributed random variables, the stopping time has finite expected value.[citation needed] This justifies the following argument, explaining why the betting system fails: Since expectation is linear, the expected value of a series of bets is just the sum of the expected value of each bet.[dubious – discuss] Since in such games of chance the bets are independent, the expectation of each bet does not depend on whether you previously won or lost. 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.
In other words using a system that is not profitable will not turn profitable simply by applying a Martingale strategy.
Moreover Martingale as a strategy in of itself simply relies on having infinite wealth:
Since a gambler with infinite wealth will, almost surely, eventually flip heads, the martingale betting strategy was seen as a sure thing by those who advocated it. Of course, none of the gamblers in fact possessed infinite wealth, and the exponential growth of the bets would eventually bankrupt "unlucky" gamblers who chose to use the martingale. It is therefore a good example of a Taleb distribution – the gambler usually wins a small net reward, thus appearing to have a sound strategy. However, the gambler's expected value does indeed remain zero (or less than zero) because the small probability that he will suffer a catastrophic loss exactly balances with his expected gain. (In a casino, the expected value is negative, due to the house's edge.) The likelihood of catastrophic loss may not even be very small. The bet size rises exponentially. This, combined with the fact that strings of consecutive losses actually occur more often than common intuition suggests, can bankrupt a gambler quickly.
Sources:
http://en.wikipedia.org/wiki/Gambler's_fallacy
http://en.wikipedia.org/wiki/Martingale_(betting_system)
