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Mathematically, the probability is equal to one that gains will eventually equal losses and a gambler will return to his starting point; however, the expected number of times he has to play is infinite, and so is the expected amount of capital he will need! A similar argument shows that the popular doubling strategy (start with $1, if you lose, bet $2, then $4 etc., until you win) does not work; see St. Petersburg paradox. Situations like these are investigated in the mathematical theory of random walks. This and similar strategies either trade many small wins for a few huge losses (as in this case) or vice versa. With an infinite amount of working capital, one would come out ahead using this strategy; as it stands, one is better off betting a constant amount if only because it makes it easier to estimate how much one stands to lose in an hour or day of play. Should the stakes of the punters exceed the amount for, the time being in the bank, the banker is not responsible for the amount of such excess. In the event of his losing, the croupier pays the punters in order of rotation, so far as the funds in the bank will extend; beyond this, they have no claim. The banker, may, however, in such a case, instead of resting on his right, declare the stakes accepted, forthwith putting up the needful funds to meet them. In such event the bank thenceforth becomes unlimited, and the banker must hold all stakes (to whatever amount) offered on any subsequent hand, or give up the bank. A random event is more likely to occur because it has not happened for a period of time; The breaking of the bank does not deprive the banker of the right to continue, provided that he has funds with which to replenish it, up to the agreed minimum. There are three general categories of casino games:
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-------------------------------------------------------------------------------- A player going bank may either do so on a single hand, in the ordinary course, or a cheval, i.e. on two hands separately, one-half of the stake being played upon each hand. A player going bank and losing may, again go bank; and if he again loses, may go bank a third time, but not further. The traditional form of punto banco baccarat is played at an oval table, similar to the chemin de fer version. The table is staffed by a croupier, who directs the play of the game, and two dealers who collect and pay bets as well as tallying commissions due. Six or eight decks of cards are used, normally shuffled only by the croupier and dealers. Like chemin de fer, the shoe is passed around from player to player, who acts as the dealer of the cards and as "banker," but he or she does not actually bank the game. Indeed, the "banker" may bet on the player hand if he or she wishes, or may pass the shoe along to another player - the role of the "banker" is merely ceremonial. The person who bet the highest amount on the player hand is given the player-hand cards, though he or she simply turns the cards over, annoucing their total. The croupier instructs the "banker" on if or when to deal third cards, and then announces the winning hand. Consider players with 90 and 10 pennies respectively, repeating the game 100 times. The player with 90 pennies is expected to win 90 out of 100 complete games, winning 10 pennies each game. However, he is also expected to lose 10 games, each time forfeiting all 90 of his pennies. So after the series of 100 games, the richer player is expected to win 90*10=900 pennies, and lose 10*90=900 pennies. Despite the fact that after any single game, one player ends up with all the pennies, the expected result over many games is for both players to break even. Rh = suRrender if allowed, otherwise hit Playing Blackjack as a Business, Lawrence Revere, 1971 Non-examples
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