Martingale probability theory
The Martingale betting systems, also known as doubling-up or progression systems, are the most popular with casino players. The basic principle of this system was invented about 300 years ago and was extremely popular in 18th century France. The classic Martingale betting system is based on the martingale probability theory introduced by French mathematician Paul Pierre Lévy. The system is still used by roulette players for repeating even-money outside roulette bets.
Classic Martingale System
The idea of the classic Martingale system is to double up the next bet after each loss and to reduce it to the initial amount after each win. Eventually the win covers all the net losses of the previous bets.
For example, if you bet 5 units and lose the bet, the next bet is doubled up to 10 units. Then if you lose your 10-unit bet, you must bet 20 units on the next spin of the roulette wheel. After next loss the bet is increased to 40 units. At last you win the bet which means that the next roulette bet is reduced to the starting amount and you bet 5 units again. As a result, you have a gain of 5 units: 40 - (5+10+20)=5.
NOTE: The main problem about Martingale betting system is that the player can reach the table betting limit very quickly as a result of continuous losses. Unfortunately, long losing streaks are not rare in roulette games, and that is why you can exceed the maximum betting limit in a few games.
Most roulette tables have a limit of $500. As a result of seven losses in a row you would reach the betting limit with an initial bet of, for example, $5.
In order not to lose large sums of money and avoid going over the maximum allowed bet, you are recommended to start the betting cycle with small amounts, such as a $1 bet.
Grand Martingale System
The Grand or also Great Martingale is a variation of the classic Martingale system which involves an even greater growth of bets. Here after each loss the player must double the bet and add one more unit to the doubled amount. As a result, the sequence of bets will be like this (if you start betting with 5 units):
If there is a series of constant losses, the player can reach the table limit much quicker than when using classic system. That is why the Grand Martingale system is good for roulette tables with at least a $1,000 limit. This is so the bet can grow and not go bust as a result of a losing streak:
If the series of losses is finally broken by win, the net gain is always the initial bet plus one unit, but not only the initial bet like when using the classic Martingale. Of course, it is possible to use other variations of this system, for example, add 2 or more units to the doubled bet amount after each loss. But such a roulette strategy involves greater risk of losing money.
Reverse Martingale System
If you prefer to take advantage of winning streaks rather than losing ones, you can use the Reverse Martingale system. This means, that you double the next roulette bet after each win rather than loss and go back to the initial bet when you lose. This is the formula of Reverse Martingale (if you start betting with 5-unit bet):
It is considered to be less risky than the traditional Martingale betting system. However, the first loss after a sequence of successive wins will wipe out the potential gains from the whole streak.