Martingale - SpeedyLook encyclopedia

Various martingales

Many martingales are only the dream of their author, some are in fact inapplicable, some allow indeed to cheat a little. The money plays are in general inequitable: whatever the played blow, the probability of profit of the casino (or the State in the case of a lottery) is more important than that of the player. In this type of play, it is not possible to reverse the chances, only to minimize the probability of ruin of the player.

The traditional martingale

It consists in playing a simple chance with the Roulette (black or red, even or odd) in order to gain, for example, a unit in a series of blows by doubling its setting if one loses, and that until one gains. Example: the player put 1 unit on the red, if the red leaves, it stops playing and it gained 1 unit (2 units of profit minus the unit of setting), if the black leaves, it doubles its setting by betting 2 units on the red and so on until he gains.

Having a chance on two to gain, it can think that it will end up gaining; when it gains, it is inevitably refunded of all that it played, once its starting setting.

This martingale seems to be sure in practice. To note that on the theoretical level, to be sure to gain, it would be necessary to have the possibility of playing an unlimited number of times. What presents major disadvantages:

This martingale in fact is limited by the settings which the player can make because it is necessary to double the setting with each blow as long as one loses: 2 times the starting setting, then 4,8,16…. if it loses 10 times of continuation, it must be able to advance 1024 times its setting initial for the 11th part! One thus needs much money to gain little.
the casters comprise a " 0" who is neither red nor black. The risk to lose at the time of each blow is thus greater than 1/2.
Moreover, to paralyze this strategy, the casinos propose tables of play by section of setting: from 1 to 100 euros, 2 to 200, 5 to 500, etc Impossible thus to use this method on a great number of blows, which increases the risk all to lose.

The large martingale

It is similar to the traditional martingale, except that the player is not satisfied to double his setting with each loss, he adds also a unit.

Example:

the player put a unit; if it gains, it leaves the play with 2 units - 1 unit which it played = 1 unit

If it loses first once, it plays 3 units; if it gains, it carries 6 units - 3 (that it has just played in the 2nd part) - 1 (which it played in the 1st part) = 2 units

If it loses second once, it plays 7 units; if it gains, it carries 14 units - 7 (that it has just played in the 3rd part) - 3 (which it played in the 2nd part) - 1 (that it played in the 1st part) = 3 units

etc

This martingale is as not very sure as the traditional martingale (the player with the impression which it can nothing lose, but it is true only if it succeeded with miser just, before leaving the table of play!), on the other hand it makes it possible to increase the profits. Attention with the fact that it presents the same disadvantages as the traditional martingale, but especially it is limited even by the settings than the player can make: it is enough that it loses three times, to have to play 15 times its setting with the next blow (2047 times for the 11th part).

Piquemouche

It is another alternative of the traditional martingale. The player starts again with a unit when it gains, but when it loses, it increases its setting by a unit, it doubles it only after three consecutive losses. It does not require to increase as of the beginning the settings in the event of successive losses, it is surer, but the profits are weak (null if one does not gain as of the first part) or requires 2 profits.

Example:

the player put a unit; if it gains, it leaves the play with 2 units - 1 unit which it played = 1 unit

If it loses first once, it plays 1 unit; if it gains, it carries 2 units - 1 (that it played in the 2nd part) - 1 (which it played in the 1st part) = 0 unit

If it loses second once, it plays 1 units; if it gains, it carries 2 units - 1 (that it played in the 3rd part) - 1 (which it played in the 2nd part) - 1 (that it played in the 1st part) = -1 unit

One thus needs a second profit to be gaining.

Continuation of setting so always losing 1 - 1 - 1 - 2 - 2 - 2 - 4 - 4 - 4 - 8…

etc

Whittacker

The player plays a whittacker when it put the sum of his two preceding settings as long as it loses, and starts again with a unit when it gains.

The pyramid of Alembert

The name is a reference to Jean the Round of Alembert, mathematician of the XVIIIe century. The principle consists in increasing the setting by a unit after a loss and decreasing the setting of a unit after a profit.

Used when it is thought that a profit decreases the chance to still gain, whereas a loss increases the chance to gain thereafter (magazine by famous the " principle Shadok ")

Against of Alembert

This martingale takes again the principle of that of Alembert but the settings are done in the other direction: it is thus necessary here to decrease the setting by a unit when one loses and to increase the setting by a unit when one gains.

Conversely with the preceding one, it is used when it is thought that the last chance is representative of the future chance (for example vis-a-vis a line of machines with under).

One can see the Vie like a large illustration of this martingale.

The paroli

This martingale consists to double setting with each profit (thus to garage what one gained), then, starting from a number of profit defined in advance, to stop and start again with the starting setting. One speaks about paroli of 1, if one stops after having gained twice his setting, paroli of 2 if one gained four times his setting, paroli of 3 if one stops after having gained eight times his setting, etc

The American martingale

More complex than the preceding ones, this martingale requires a good memory, or what to note its settings. The player starts by increasing his settings by a unit as long as it gains. As soon as it loses, it retains the setting which it has just lost, and given the sum of the last and the first setting. When it gains, it retains the setting which it gained, and raye the first setting of its list. Then it given the sum of the last and the first setting of its list, by not taking account of that which it striped.

The Dutch martingale

The player implements this martingale when it loses. He retains all the settings which he lost. It setting setting more weak among those that he lost (if he lost some several), by adding 1. Then it given the following setting, in the ascending order.

Martingales and mathematics

A martingale is intended to optimize the mathematical Espérance of a strategy of play.

Law of Dubins and Savage

Mathematically, Lester Dubins and Leonard Savage showed in 1956 that the best way of playing in a play where the probabilities are unfavourable to the player always consists with miser what makes it possible the most quickly to approach set aim. Intuitively this result seems obvious: so with each part one is likely more to lose than to gain, to as much minimize the number of played parts. This result also means, that unless having an infinite starting setting, there do not exist strategies making it possible to reverse the probabilities in your favor in a play which is unfavourable for you.

It should be noted that even in the case of a fair game, the player who has at the same time the possibility and the will of miser more gives more chances to ruin its adversary and thus to prevent it from continuing to play: thus, to the price of a larger potential loss, it gives itself also more chance of profits. As in any martingale, that does not modify however the hope of the two players (i.e more " small joueur" with less of chance to gain but, as paradoxical as that appears, it can gain more!).

Probabilities

There exist however certain games of chance which are not systematically unfavourable to the player. One can quote for example the case of William Jaggers which gained a large sum with Monte Carlo at the 19th century by systematically studying the frequencies of exit of the numbers to the Roulette. It could thus determine certain numbers which had a probability of exit which was favorable for him. Today the casinos are protected from this kind from practices by maintaining their material carefully, so that dispersions are extremely weak. This means that the probabilities of exit of a given number are as well as possible very slightly favorable to the player. It would thus be necessary to bet an immense number (often during several months) of times of the small sums to probably hope for a profit very far from remunerating the authorized efforts.

The black jack is a play which has gaining strategies: several techniques of play, which generally require to memorize the charts, make it possible to reverse the chances in favor of the player. The mathematician Edward Thorp thus published in 1962 a book Beat the Dealer which was at the time a true best-seller. But all these methods ask for long weeks of drive and are easily detectable by the croupier (the abrupt changes of amount of the settings are characteristic). The casino has then any leisure to draw aside from its establishment the players in question. The black jack however remains the least unfavourable play with the player: the advantage of the casino is only of 0,66% vis-a-vis a good player, it is of 2,7% with the caster and up to 10% for the machines with under.

The Backgammon although being a set of dice allows to develop strategies gaining on a great number of parts. Indeed the arbitration enters various displacements of pawns are connected with a quasi mathematical movement of style wargame and being able to be represented by graph S probabilists. The play can be summarized in a sequential process of Markov. As strange as that can appear, this play can apply in Assurance in the management of the Risque S in a general way. The constant arbitrations that the players must carry out can be represented in a matrix of Léontiev. Such tools can " perdre" in front of an even inexperienced player if this one profits from jets of favorable dice but it is undeniable that the higher the number of parts is, the more the formula of Stirling and the law of the great numbers of Bernoulli apply and make it possible an intelligent machine to gain any tournament beyond 50 parts.

Advanced methods for the lotto

It should be noted that there exist rather advanced methods. One of them rests on the least played combinations. In the plays where the profit depends on the number of gaining players (Lotto…), to play the least played combinations will optimize the profits. Thus certain people sell combinations which statistically would be very seldom used by the other players. One can guess all the same that certain numbers are more often played: many players notching their birth date, or another date, numbers 1,9, and 19 correspondents at the year are very often played. It is the same of the first 12 numbers corresponding to the months.

On the basis of this reasoning, one can still conclude that a player who would have succeeded in thus determining statistically the least played combinations, in order to optimize his hope of profit will not be in fact certainly not the only player to have obtained by the analysis these famous combinations, and all these players are thus likely finally to be very disappointed by their profits if it proved that this equiprobable combination left to pulling! In other words, the numbers in theory the least played in fact are surjoués by combinations, best would be perhaps to carry out a scientist mixes under-played numbers and numbers surjoués to obtain the ideal combinations, which can in addition be observed in last pullings when there was not gaining. Another conclusion with all that is perhaps that best is still to play of the random combinations which are likely finally less to be also selected by the players who incorporate an human factor and harmonious in the choice of their numbers.

Other less analytical players are tempted to bet when special kitties are concerned, because the hope of profit is then optimal even higher than the setting.

Miraculous methods

A certain number of reviews or Internet sites claims to inform you about the “form” of the numbers, i.e. their probability of leaving in next pullings. Here for example a pulling of 50 balls of lotto: 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. One realizes that ball 9 left 4 times whereas ball 8 never left. Following erudite calculations, the authors of these “methods” will say to you whereas figure 9 is in form and that it thus will leave in next pullings or on the contrary that the Loi of the great numbers implies that the 8 has a stronger probability to leave to fill its delay.

It is of course there about one error in extreme cases of the Escroquerie characterized. The balls of lotto do not have fun to count the number of times where they left the machine, more especially as it would be necessary that they are sufficiently vain not to take into account pullings of tests or calibration of the machines. If each ball has on average a chance on 49 to leave, this Probabilité is reached only for one infinitely great number of pullings. The fact that ball 9 left 4 times moreover than ball 8 thus does not have any importance since the probabilities do not guarantee that each ball will leave the same number of times, but simply that the difference of the number of exits of two balls will be very small compared to the full number of pullings: nothing says that the ball eight finally will make up for its lost time. For example, so at the end of ten thousand pullings ball 9 left 206 times and ball 8 left 202 times, one will obtain a frequency of 1,01/49 and 0,99/49. To the millionth pulling if ball 9 left 20410 times and ball 8 left 20406 times one will obtain 1,0001/49 and 0,9999/49 respectively. The frequencies approach more and more the theoretical probability of 1/49, however ball 9 preserves its advance of four exits on ball 8.

Others rest on the bet of a systematic skew: pullings are not exactly equiprobable, after for example negligible differences in weight of the balls. Even if the calculation of the mathematical Espérance of this type of martingale is much more complex, the good sense indicates that if the author of the receipt finds more profitable to sell it than to use it for his account, it is probably that its effectiveness is about null.

See too

External bonds

Martingales and other illusions, article drawn from the newspaper Science

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