The martingale relies on the false reasoning that, when a series of even chances have resulted in similar outcomes, the odds are in favor of a different outcome next time.  If, for example, a red number has come up five successive times, then money poker is placed on black for the sixth spin.  If red comes up again- thus making six successive reds- the bet is doubled and put on black.  But if the first bet on black wins, the player waits for a further series of five manqué numbers, five passé, five pair, five impair, five rouge, or five noir numbers before betting again.  A succession of five similar outcomes may not be enough for him; on the other hand he may think two or three enough.  It’s just a question of personal psychology .
The Reverse martingale is played in a similar way i.e., bets are always doubled up after a loss-but after each successive win stakes and winnings are left in place-the hope being for a run of eight successive wins, which would bring a profit of 127 times the original stake.  Since all Martingale bets are on even chances the bettor’s profits, if any, will be small.  But losses may well be big-for no matter how long a preceding “run” may be, the probability of his poker winning any one bet remains ½.  Also, casino gamblers intent on doubling up bets after each successive loss will find that sooner or later they are checked by the bank’s maximum limit.
Although this system was first used in the 17th century, it became popularized and more widely known as “the martingale” in 1850 when the British novelist William Thackeray published Pendennis. In this novel one of the characters, Sir Francis Clavering, suffers enormous financial loss through his blind faith in the martingale.  (Actually, the system’s name is thought to derive from the name of a particular kind of “checking” rein intended to prevent a horse from rearing its head when excited.  Similarly, the system is supposed to “check” disaster by recouping losses.)
The martingale and the innumerable other variations of the doubling-up systems are called collectively the d’Alembert systems rather unfairly, since this 18th century French mathematician was the first person to point out the fallacy in what was then known as the “Law of Equilibrium.”  “This law,” d’Alembert wrote in his Traité de dynamique, “supposes a balancing of events in an interminable series, not in a brief array of events limited by the mind and time of man” a brief array like five or six spins of a wheel.  Yet somehow he has been saddled with the responsibility of applying the law Equilibrium to doubling-up systems gambling.
The Biarritz system is more for players who find the even chance systems too slow and too tame players who would rather try fro the 35 to 1 rake-off on an en plein  win.  The Biarritz system is based on the assumption that any given number of spins of the wheel over 111 (that is, three times 37) should give each en plein number a chance of appearing at least once.  To follow it, the gambler must watch at least 111 consecutive spins of the wheel without betting.  He must note every number that comes up and count how many times each occurs.  He then chooses a number that has occurred fewer than 10 times and backs it en plein, on the assumption that if it has come up only nine (or fewer) times in 111 spins of the wheel, it’s likely to appear again during the next 34 spins.

He now permits himself up to 34 successive bets bingomania and if the number doesn’t come up in that sunbingo series he loses.  (If zero comes up he also loses, for then all except the even-chance bets are seized by the bank.)  Should the number come up on the 34th spin, 35 units of money are won-one more than originally staked.  But if the number came up on the first spin the profit would be 35 units of money.

Some of the followers of the Biarritz would have the bettor go on for 111, not  34, spins in an effort to backgammon the lucky number ; and not only go on, but actually increase the stake every 10 losing bets after the 30th.  So if another 111 spins went by before the number came up, he would have staked 530 units of money for a return of 195 units of money. All this assuming most rashly-that the bank would accept a stake of 500 times the minimum and just as rashly that zero wouldn’t show up once in 111 spins!

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