Since a gambler will almost surely eventually flip heads, the martingale betting strategy is certain to make money for the gambler provided they have infinite wealth and there is no limit on money earned in a single bet.
Additionally, as the likelihood of a string of consecutive losses is higher than common intuition suggests, martingale strategies can bankrupt a gambler quickly.
The martingale strategy has also been applied to roulette, as the probability of hitting either red or black is close to 50%.
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).
[3] It is only with unbounded wealth, bets and time that it could be argued that the martingale becomes a winning strategy.
Let q be the probability of losing (e.g. for American double-zero roulette, it is 20/38 for a bet on black or red).
With a win on any given spin, the gambler will net 1 unit over the total amount wagered to that point.
[4] However, bold play is not always the optimal strategy for having the biggest possible chance to increase an initial capital to some desired higher amount.
[5] The previous analysis calculates expected value, but we can ask another question: what is the chance that one can play a casino game using the martingale strategy, and avoid the losing streak long enough to double one's bankroll?
As before, this depends on the likelihood of losing 6 roulette spins in a row assuming we are betting red/black or even/odd.
Many gamblers believe that the chances of losing 6 in a row are remote, and that with a patient adherence to the strategy they will slowly increase their bankroll.
In fact, while the chance of losing 6 times in a row in 6 plays is a relatively low 1.8% on a single-zero wheel, the probability of losing 6 times in a row (i.e. encountering a streak of 6 losses) at some point during a string of 200 plays is approximately 84%.
These unintuitively risky probabilities raise the bankroll requirement for "safe" long-term martingale betting to infeasibly high numbers.
The anti-martingale approach, also known as the reverse martingale, instead increases bets after wins, while reducing them after a loss.