A martingale is intended to maximize the probability of winning but does not change the expected gains, which remains at the detriment of the player.

#### Dubins and Savage Law

Mathematically, Lester Dubins and Leonard Savage showed in 1956 that the best way to play in a game where the odds are unfavorable to the player is to always bet which allows to approach the fastest the goal. Intuitively this result seems obvious: if each party is more likely to lose than to win, so minimize the number of games played. This result also means that unless you have an infinite initial bet, there are no strategies to reverse the odds in your favor in a game that your chance is unfavorable.

Even in the case of a fair game, the player who has both the ability and willingness to bet most will have more chances to ruin his opponent and thus prevent him from continuing to play; this way at a price of a greater potential loss, it also gives more chance of winning. As in any martingale, this does not change the expectation of the players (that is to say the most “small player” has less chance of winning but, paradoxical as it may seem, he can earn more!) .

#### Probabilities

However, there are some games of chance that are not systematically unfavorable to the player. We can cite for example the case of William Jaggers who won a large sum in Monte Carlo in the nineteenth century by studying systematically the output frequencies of the roulette numbers. He was able to identify some issues that had an output probability in his favor. Today thw casinos protect against such practices by carefully maintaining their equipment, so that the dispersions are extremely low. This means that the output probabilities of a given number are at best marginally favorable to the player. So he should bet a huge number of times (often several months) small amounts to expect a gain probably far to remunerate his efforts.

Blackjack is a game that has winning strategies: several playing techniques, which generally require memorizing the cards, allow to reverse the odds in favor of the player. The mathematician Edward Thorp published also in 1962 a book, * Beat the Dealer*, which was at the time a real bestseller. But all these methods require long weeks of training and are easily detectable by the dealer (sudden changes in the amount of bets is typical). The casino then has the liberty to banish from its establishment the players in question. Blackjack remains yet the least unfavorable game to the player: the casino advantage is only 0.66% against a good player, and it is 2.7% in roulette and up to 10% for slot machines.

Backgammon, although a dice game, helps develop winning strategies on a large number of parties. Indeed arbitration between the different movements of pawns is like an almost mathematical movement wargame style and can be represented by probabilistic graphs. The game can be summed up in a Markov sequential process. Strange as it sounds, this game can be applied in insurance risk management in general. The constant trade-offs to be made by players can be represented in a matrix of Leontiev. Such tools can “lose” even before one inexperienced player if he benefits from favorable dice rolls, but it is undeniable that the greater the number of parts increases, Stirling formula and the law of large numbers of Bernoulli s’ apply and allow an intelligent machine to win almost any tournament beyond 50 games.

### The advanced methods for lotto

There are quite advanced methods. One of them is based on the least played combinations. In games where winning depends on the number of winning players (Loto …), playing the least played combinations maximize gains. We can still guess that some numbers are played more often: many players checking their birth date, or another date, numbers 1, 9, and 19 corresponding to the year are often played. It is the same for the first 12 numbers corresponding to the month. But it is not enough to take the numbers above 31 and combine them randomly; the psychology of the players is much more complex. For example 41-42-43-44-45 not part of the least played combinations. One can even say that it is among the most played. This method loses interest as in the various lotteries the proportion of players who request a ticket with numbers drawn randomly by a computer ( “flash draw”) increases. If human bias is easily measurable and quantifiable, one can consider that the computer is not (or at least not to a level that would boost its earnings).

### The miraculous methods

A number of magazines and websites claim to provide information on the “form” of numbers, that is to say, their probability to be drawn in the next draws. Here for example a draw of 50 lotto balls 39, 38, 42, 29, 18, 48, 40, 36, 9, 24, 49, 33, 47, 9, 45, 7, 11, 49, 16, 28 , 27, 25, 16, 27, 22, 48, 5, 24, 16, 6, 4, 14, 17, 44, 46, 9, 37, 22, 39, 12, 33, 9, 21, 44, 11 , 33, 19, 20, 37, 18. We see that the ball 9 is output 4 times while the ball 8 was never drawn. Following scientific calculations, the authors of these “methods” will tell you when the number 9 is fit and he will therefore be drawn in the next draw, or on the contrary the law of large numbers implies that 8 will be drawn to make up his delay.

This is, of course, a logical error that, when is knowingly spread, it is a scam. The lotto balls are not interested to count the number of times they were drawn of the machine, the more it would means to be sufficiently coquettish not take into account the drawn tests or calibration of machines. If each ball has a chance on average of 49 to go out, this probability is attained only for an infinitely large number of draws. The fact that the ball 9 is drawn 4 times more than the ball 8 has no importance, since the odds do not guarantee that each ball will come out the same number of times, only the difference in the number of two outputs balls will be very small relative to the total number of prints: nothing says that the 8 ball will eventually make up his delay. For example, if after ten thousand runs the ball 9 is drawn 206 times and the ball 8 202 times, we get a frequency of 1,01/49 and 0,99/49. The millionth draw, if the ball 9 is output 20410 times and the ball 8 20406 times, we obtain 1,0001/49 and 0,9999/49 respectively. The frequencies are approaching more and more of the theoretical probability of 1/49, yet the ball 9 retains his lead of four draws on the ball 8.

Others are based on the gamble of a systematic bias: the draws are not exactly equally probable, following such infinitesimal differences in weight of balls. Although the calculation of the mathematical expectation of this martingale is much more complex, common sense says that if the author of the recipe is more profitable to sell than using it for its own account, it is likely that its effectiveness is almost zero.

Translated from Wikipedia

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