Martingale is originally referred to a class of betting strategies popular during 18th century France. The simplest strategies was designed for a game in which the gambler wins his stake if a coin comes up heads and loses it if the coin comes up tails. This strategy enabled the gambler to double his bet after every loss, enabling the first win to recover all previous losses plus win a profit equal to the original stake.A gambler with infinite wealth with probability 1 eventually flips heads, the martingale betting strategy was seen as a sure thing by those who practiced it. However, none of these practitioners in fact possessed infinite wealth (indeed, why would one bet if he possesses infinite wealth?), and the exponential growth of the bets would eventually bankrupt those foolish enough to use the martingale over a long losing streak.

Analysis

Assume someone applies the martingale betting system at an American roulette table, with 0 and 00 values; a bet on either red or black will win 18 times out of each 38. If the player's initial bankroll is $160 and the betting unit is $10, the player will make a win in approximately 96% of sessions, gaining an average of $4.30 from each winning session. In the remaining 4% of sessions, the player will "bust", exhausting his bankroll, for a loss of $160. It follows then that the average session losses of a gambler employing this strategy are $2.27. With a larger bankroll, the odds of making a win before running out of cash increase; however, the average winnings grow more slowly than the average losses, therefore the game remains a losing proposition.
Modern casinos normally have table minimums and maximums in order to prevent players from doubling their bets more than five or six times, thus rendering the martingale system obsolete.

Explanation

The betting strategy seems to work. We think of long streaks as impossible, and they are the only thing that could actually bankrupt the betting party. However, they're not actually impossible, just unlikely. One can easily demonstrate that they are possible enough to keep the balance at the casino.

Let's say that a very large number of people are each gambling $1 on the flip of a coin in their hand. About half will win, and about half will lose, and thus the casino will pull in as much as it puts out. For the next flip, it branches off and half bet $1, half bet $2, in line with the strategy. Remember that previous flips won't affect future flips, so half of the people betting $1 will win and half will lose. Half of the people betting $2 will win and half will lose. The casino will take in as much as it loses, again.

This simply continues to branch out. Half betting $4 win on the next round, half lose. Even out at round 25, when gamblers are betting $33,554,432 the casino will not become off-balance for winnings and losings, because half of the gamblers betting that much will win and half will lose.

Let's look at a different angle, suppose the game being played results in 51% losing and 49% winning. This branches in the same way, except the casino will always take in 2% of the total money gambled. Group them by how much they bet, and we find that in each of those categories the casino makes 2%, so in total they will make 2%. Interestingly, the casino would make less money per round if everyone continued to bet $1 each time, instead of increasing the betting pool -- but each individual would play for a greater number of rounds before quitting or going bankrupt. The casino still takes in the same 2% of the total money gambled.