Blackjack or twenty one is one of the most widely played games in American homes and clubs and traditionally rivals poker for popularity in the armed forces. In las vegas, Reno, and other parts of the wide open spaces. Blackjack ranks with poker, roulette and craps as one of the four standard gambling games. Of these four, however, blackjack is by far the most neglected in the scientific literature of gambling and offers a relatively unexplored area for mathematical and statistical analysis.
In should be made clear at the outset that this paper deals exclusively with the "house" game of blackjack and not the "private" game. In the house game a representative of the gambling casino is permanent dealer, and his strategy is completely fixed by known house rules. The fixed and known nature of the dealer's strategy is vital in reducing the mathematical and computational problems in analyzing blackjack to manageable proportions.
Each gambling casino has a set of blackjack rules which agree with those of other casinos on the main points but which usually differs on details. Therefore, in selecting a variation of the game of blackjack for analysis, the best that could be done was to consider rules which are common but not universal. A presentation of these rules follows.
1. Number of players - A dealer and from one to six players.
2. The pack - An ordinary 52 cards deck
3. Betting - The player makes their bets before any cards are dealt. The house establishes a minimum and a maximum bet.
4. The deal - The player and the dealer each receive two cards. Each player gets both cards face down. The dealer receives one card face up and one card face down. Cards received face down in the deal or draw are commonly known as "hole cards".
5. The numerical value of the cards - The numerical value of an ace is 1 or 11 as the player chooses, the numerical value of a face card is 10, and the numerical value of all other cards is simply their face value. The numerical value or total of a hand is the sum of the numerical values of the cards in the hand.
6. Object of the player - To obtain a total which is greater than the dealer's but does not exceed 21.
7. Naturals - An ace and a face card or ten dealt on the first two cards to either player or dealer constitutes a "natural" or "blackjack". If a player has a natural and the dealer does not, the player receives 1.5 times his original bet from the dealer. If a player does not have a natural and the dealer does, the player loses his original bet. If both player and dealer have naturals, no money changes hands.
8. The draw - A player is not required to increase the number of cards in his hand and may look at his hole cards and elect to "Stand". Otherwise, he may require that the dealer give him additional cards, face up, one at a time. If the player goes over 21 ("busts"), he immediately turns up his hole cards and pays his bet to the dealer. After each player has drawn his cards, starting with the player at the dealer's left and proceeding in a clockwise fashion, the dealer turns up his hole cards. If his total is 16 or less, he must draw a card and continue to draw cards until his total is 17 or more, at which point he must stand. If the dealer has an ace, and counting it as 11 would bring his total to 17 or more without exceeding 21, he must count the ace as 11 and stand.
9. The settlement - If the player does not go over 21 ("bust") and the dealer does, the player wins an amount equal to his original bet. If neither player nor dealer busts, the person with the higher total wins and amount equal to the player's original bet. If neither player nor dealer busts and both have the same total, no money changes hands.
10. Splitting Pairs - In the following rules a pair is defined as two cards which are identical except for suit, such as two jacks, two aces, or two tens. If the player's hole cards form a pair, he may choose to turn them face up and treat them as the initial cards in two separate "twin" hands.
This strategy is known as "splitting pairs". The original bet goes on one of the split cards, and an equal amount is bet on the other card. The player automatically receives a second card face down on each of the split cards and may continue drawing cards face up to both twin hands as long as he desires. An excepction to this rule is made in the case of split aces where the player may draw only one more card to each ace. Furthermore, if a face card or ten falls on one of the split aces, the hands is not counted as a natural but as ordinary 21. (Similarly, the player splitting a pair of face cards or tens who draws an ace holds an ordinary 21) Finally, If a player splits a pair and receives a third card of the same type, he is not permitted to undertake further splitting.
11. Doubling down - after looking at his hole cards a player may elect to double his bet and draw one and only one more card. This strategy is known as "doubling down". A player who elects to double down turns up his hole cards and receives his third card face down. A player splitting any pair except aces, after receives an additional card on each of the split cards, may elect to double down on one or both of his twin hands.