Mathematics in Martingale strategy
Martingale system – one of the most popular systems of betting management at casino. Adhering to this strategy, a player can stay in the black, even if often losing. But it has one drawback - high risk of losing the total budget in pursuit of a fine win (as well as the negative mathematical expectation at a distance). At a prolonged unsuccessful band, you can lose the entire deposit. Below we will explain the mathematical basis of the system, as well as the usefulness (and the probability in general) of its practical application.
The essence of the classic Martingale system
Martingale strategy is known from the second half of the 18 century. Its core consists in the following: the bet amount should be doubled after a loss and lowered to initial value after a win.
Example of using this scheme when playing roulette:
- Play European roulette.
- The minimum bet is €1. Play on red.
- The first bet is lost, then put €2 on red.
- In the case of a third consecutive loss, the rate will rise to four dollars, eight, sixteen, thirty-two and so on.
- If the bet won, the next bet will be €1 (initial value).
- The system allows to get small winnings, and the loss may be huge, though rare.
Disadvantages of Martingale system
When using the classic version of the system, you may encounter several problems:
- Lack of funds for a new bet. In a series of 10 losses, the eleventh time should be 1024 minimum bets!
- Next bet might be more than the table limit.
- Psychologically it is difficult to continually doubling the bet. At a minimum rate of €1, we should play €2, €4, €8 , €16, €32, €64, €128, €256, €512, €1024 and so forth.
- There is no such thing as infinite sequences, budgets and unlimited limits.
Calculating the mathematical expectation
Suppose we toss a coin. The probability of tails or heads is 0.5 (50% - half of 1). On martingale system, we always play on tails, and if heads are dropped, we double our bet. We have a roughly unlimited budget in a series of "N" bets on the tails.
In this case, the probability of losing the entire budget will be 0.5 at Nth degree, and of winning is: 1 - (0.5 at Nth degree). Move on to the figures: the initial bet is €1, having a budget of N=10 bets, which are doubled every time — a total of €1023.
The probability of losing the entire budget:
- The results of these ten tosses could be completely different: all coins' tails, all coin's heads, 5 coins tails, and 5 coins heads, etc. The total number of combinations is 1024.
- All outcomes have the same probability – 1/1024. Note that of all the possible outcomes, the big loss will happen at 10 consecutive heads.
- Hence, the probability of losing the entire budget is equal to 1/1024.
The probability of winning a bet:
- The probability of winning after these 10 tosses, i.e. getting any other combination of outcomes (except for ten consecutive heads), is 1 - 1/1024 = 1023/1024.
- Therefore, the ratio of the probability of the loss of the total budget and of winning is equal to 1/1024, divided by (1 - 1/1024) = 1/1023.
Zero mathematical expectation:
In our example, the amount of potential winnings when using the Martingale system will be €1. And the whole capital is exposed to risk with €1023. If you toss a coin an infinite number of times, on average, each 1024th time you're going to lose all the winnings of 1023 tosses. In this game the mathematical expectation will be equal to 0. Stats will deviate from the mean value but getting an enough (very large) number of tosses, the result will seek precisely the above data.
A theoretically working system of betting:
If you slightly change the strategy (after losing placing another bet equal to the total amount of loss plus X), at a certain value (more than a third of the amount of the first two bets), the mathematical expectation will be higher than zero. I.e. the system will work theoretically, provided the unrestricted sequence of betting and budget. But at an infinite number of bets and budget, the doubling should work using a classic version of the Martingale strategy. But only theoretically. In the real world there is no such thing as unlimited budgets and limitless opportunities to increase rates.
Martingale system is a working system, but only theoretically. Even in the absence of zero and other ways of loss, you get a zero mathematical expectation. Moreover, especially against the players who use the Martingale system, the casinos set limits. If the minimum bet is €1, the maximum will be €100; Want to play bigger? Here it is: the maximum bet is €1000, the minimum will increase by 10 times (up to €10). In other words, often you will not be able to increase the initial bet by more than 100 times.
Martingale as a scamming system
The Martingale system is very often considered by scammers as an ideal opportunity of guaranteed winnings at online casino. There are many websites and videos, where it is told about the ease of solid earnings: $200 per hour and more. Remember, small frequent winnings will be equal to one big loss at a distance. But due to zero in Roulette or dealer's Blackjack, the system will be negative for player and positive for casino.