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Split: Double the wager and have each card be the first card in a new hand. This option is available only when both cards have the same value.  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. Casino games generally provide a predictable long-term advantage to the casino, or "house", while offering the player the possibility of a large short-term payout. Casino games often include an illusion of control, in which the player must make choices. However, in most cases it is not mathematically possible for a player to eliminate his or her inherent long-term disadvantage (the house advantage) in a casino game. Million Dollar Blackjack, Ken Uston, 1981 If the dealer's upcard is an Ace, the player is offered the option of taking Insurance before the dealer checks his 'hole card'. There are many scenarios where the gambler's fallacy might superficially seem to apply, where it in fact does not. Hit: Take another card. 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. Encyclopedia of Casino Twenty-One Gaming machines I also like the straightforward but generous 100% Match Casino Bonus up to $100 Free. Especially if you take in consideration that Europa online casino offers an additional 10% or 15% bonus if you don't make your deposit with credit card but with one of the alternative payment methods, like NETeller, Citadel or Moneybookers! An example: coin-tossing The stakes are made afresh, and the game proceeds. If the banker has been the winner, he deals again. If otherwise, the cards are passed to the player next in order, who thereupon becomes banker in his turn. 
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