Blackjack

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   We first present the probabilities attached to card dealing and initial predictions. In making this calculus, circumstantial information such as fraudulent dealing is not taken into account (as in all situations corresponding to card games). All probabilities are calculated for cases using one or two decks of cards. Let us look at the probabilities for a favorable initial hand (the first two cards dealt) to be achieved. The total number of possible combinations for each of the two cards is C(52, 2) = 1326, for the 1-deck game and C(104, 2)=5356 for the 2-deck game.

Probability of obtaining a natural blackjack is  P = 8/663 = 1.20663%  in the case of a 1-deck game and  P = 16/1339 = 1.19492%  in the case of a 2-deck game.

 Probability of obtaining a blackjack from the first two cards is  P = 32/663 = 4.82654%  in the case of a 1-deck game and  P = 64/1339= 4.77968%  in the case of a 2-deck game.

 Similarly, we can calculate the following probabilities:

 Probability of obtaining 20 points from the first two cards is  P = 68/663 = 10.25641%  in the case of a 1-deck game and  P = 140/1339 = 10.45556%  in the case of a 2-deck game.

 Probability of obtaining 19 points from the first two cards is  P = 40/663 = 6.03318%  in the case of a 1-deck game and  P = 80/1339 = 5.97460%  in the case of a 2-deck game.

 Probability of obtaining 18 points from the first two cards is  P = 43/663 = 6.48567%  in the case of a 1-deck game and  P = 87/1339 = 6.4973%  in the case of a 2-deck game.

 Probability of getting 17 points from the first two cards is  P = 16/221 = 7.23981%  in the case of a 1-deck game and  P = 96/1339 = 7.16952%  in the case of a 2-deck game.

 A good initial hand (which you can stay with) could be a blackjack or a hand of 20, 19 or 18 points. The probability of obtaining such a hand is calculated by totaling the corresponding probabilities calculated above:  P = 32/663 + 68/663 + 40/663 + 43/663 = 183/663, in the case of a 1-deck game and  P = 64/1339 + 140/1339 + 80/1339 + 87/1339 = 371/1339, in the case of a 2-deck game.

Probability of obtaining a good initial hand is  P = 183/663 = 27.60180%  in the case of a 1-deck game and  P = 371/1339 = 27.70724%  in the case of a 2-deck game.

 The probabilities of events predicted during the game are calculated on the basis of the played cards (the cards showing) from a certain moment. This requires counting certain favorable cards showing for the dealer and for the other players, as well as in your own hand. Any blackjack strategy is based on counting the cards played. Unlike a baccarat game, where a maximum of three cards are played for each player, at blackjack many cards could be played at a certain moment, especially when many players are at the table. Thus, both following and memorizing certain cards require some ability and prior training on the playerís part.

 The formula of probability for obtaining a certain favorable value is similar to that for baccarat and depends on the number of decks of cards used. If we denote by x a favorable value, by nx the number of cards showing with the value x (from your hand, the hands of the other players and the face up card in the dealerís hand) and by nv the total number of cards showing, then the probability of the next card from the deck (the one you receive if you ask for an additional card) having the value x is:

 

This formula holds for the case of a 1-deck game. In the case of a 2-deck game, the probability is:

 Generally speaking, if playing with m decks, the probability of obtaining a card with the value x is:

 

  Example of application of the formula:  Assume play with one deck, you are the only player at table, you hold Q, 2, 4, A (total value 17) and the face up card of the dealer is a 4. Let us calculate the probability of achieving 21 points (receiving a 4).

 We have  nx = 2, nv = 5, so:

.

For the probability of achieving 20 points (receiving a 3), we have  nx = 0, nv = 5, so:

.

For the probability of achieving 19 points (receiving a 2), we have  nx = 1, nv = 5, so:

.

If we want to calculate the probability of achieving 19, 20 or 21 points, all we must do is total the three probabilities just calculated. We obtain  P = 9/47 = 19.14893%. 

Unlike in baccarat, where fewer cards are played, the number of players is constant (two), and the number of gaming situations is very limited, in blackjack, the number of possible playing configurations is in the thousands and, as a practical matter, cannot be entirely covered by tables of values.

 

 Sources

A big part of the gaming situations that require a decision, where the total value held is 15, 16, 17, 18, 19 or 20 points, is comprised in tables in the section titled Blackjack of the book PROBABILITY GUIDE TO GAMBLING: The Mathematics of Dice, Slots, Roulette, Baccarat, Blackjack, Poker, Lottery and Sport Bets. You will also find there other issues of probability-based blackjack strategy . See the Books section for details.

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