Lucky numbers

In addition, the casinos offer a bet on “ties”. Tie is the case when The Player and The Banker hands have the same total. This bet gains 9 times the amount bet, otherwise the bet is lost. The probability of a tie is 9.5156%, therefore the expectation of the bet is -4.884%. It is obvious that the probability hence the expectation of a tie depends on the subset of not played cards. Based on computer simulations, there was selected random subset of different sizes from a complete 416 pack of cards (8 decks). The result is quite disappointing from the money making point of view. The advantages from the knowledge of the used cards are limited to the extreme end of the pack they are not large. The card counting strategies, which are very practical, are marginal and precarious because they are easily eliminated simply by shuffling the deck with 26 remaining cards.

To begin the Baccarat game, 8 decks of cards are shuffled and near the end a joker is placed face up. The cards are placed into wooden dealing box which is called a shoe. The first card is opened and its value is known. The Baccarat table has 12 seats which are occupied by shills and customers. A shill is an employee of the house who bets money and pretends to be a player in order to stimulate a play or attract customers. There are two principal bets, called “Player” and “Banker”. Any player makes one of this bets before each round. Two of the players are singled out - one is The Banker and the other is The Player. The seats are numbered counterclockwise from 1 to 12. The Banker is usually player number 1, unless he refuses to be. If that is the case, the opportunity passes counterclockwise around the table until somebody accepts. The Player is generally chosen to be that player who has the largest bet on the Player (other than The Banker). The Banker retains the shoe and deals as long as the bet “Banker” doesn’t lose. When the bet “Players” wins, the shoe moves to the player on the right. Now, this player becomes The Banker. If we have a Tie, the players are allowed to change bets in any way they want to. After that the same banker deals another coup. To begin a coup, The Player and The Banker are dealt each two cards. After each of them receives two cards, the croupier faces their cards. If both two-card total equals 8 or 9 (natural 8 or 9), all bets are settled at once. If neither The Banker nor The Player has a natural, then they stand or draw according to the following set of rules:

If The Player has:
0-5 he draws a card
6-7 he stands
8-9 he turns card over

The Banker has two choices- (1) to draw when the player draws and (2) not to draw when the player draws.
If The Banker has in case (1):
0 draws when The Player draws none, 0-9
1 draws when The Player draws none, 0-9
2 draws when The Player draws none, 0-9
3 draws when The Player draws none, 0-7, 9
4 draws when The Player draws none, 2-7
5 draws when The Player draws none, 4-7
6 draws when The Player draws 6, 7
7 he stands
8 he turns cards over
9 he rurns cards over

If the Banker has in case (2)
3 doesn’t draw when The Player draws 8
4 doesn’t draw when The Player draws 0, 1, 8, 9
5 doesn’t draw when The Player draws 0-3, 8, 9
6 doesn’t draw when The Player draws none, 0-5, 8, 9
7 he stands
8 he turns cards over
9 he turns cards over

The higher hand wins. If the hands are equal, there is a Tie and no money changes hands. At that moment players are free to change their bets in a way they want to. The played coup is complete when the joker is reached, the shoe ends and the cards are reshuffled. However, the casino is allowed to reshuffle the cards any time during the coups.

Two main bets can be made against the house. A player can bet on either The Player or The Banker. The Winning bets on The Player are paid at even money. Winning bets on the Banker are paid 0.95 of the amount of the bet. The rest is the tax which is imposed on the even money pay-off. For complete 8 decks the probability of a tie is 0.0952, and the probability The Banker to win is 0.4586. The fundamental idea of this calculation is to consider all possible distinct 6 card sequence. The outcome for each sequence is computed and the related probability of that sequence is computed and accumulated in the appropriate register. The house advantage over The Player is 1.2351 %. The house advantage over The Banker is 0.4586 x 5%- 1.2351 % or 1.0579 %, where 2.2930 % is the house tax on the winnings of The Banker. If the ties are not counted as trials, then the figures for house advantage should be multiplied by 1/0.904844, which gives an advantage of the house per bet that isn’t a tie - over The Banker of 1.1692 % and over The Player of 1.3650%. In this situation the tax on the banker is 2.5341%.