Blackjack

Blackjack, also known as Twenty-one or Vingt-et-un (French: "twenty-one"), is the most widely played casino banking game in the world.[1] Blackjack is a comparing card game between a player and dealer and played with one or more French decks of 52 cards.

The player is dealt an initial two card hand with the option of drawing cards to bring the total value to 21 or less without exceeding it, so that the dealer will lose by having a lesser hand than the player or by exceeding 21. Many rule variations of blackjack exist. Since the 1960s, blackjack has been a high profile target of advantage players, particularly card counters, who track the profile of cards yet to be dealt, and adapt their wager and playing strategy accordingly.

Other casino games inspired by blackjack include Spanish 21 and pontoon. The recreational British card game of black jack is a shedding-type game and unrelated to the subject of this article.

Contents

History

Blackjack's precursor was twenty-one, a game of unknown origin. The first written reference is found in a book by the Spanish author Miguel de Cervantes, who is most famous for writing Don Quixote. Cervantes was a gambler, and the main characters of his tale Rinconete y Cortadillo, from Novelas Ejemplares, are a couple of cheats working in Seville. They are proficient at cheating at ventiuna (Spanish for twenty-one), and state that the object of the game is to reach 21 points without busting and that the ace values 1 or 11. The game is played with the Spanish baraja deck, which lacks eights, nines and tens. This short story was written between 1601 and 1602, implying that ventiuna was played in Castilia since the beginning of the 17th Century or earlier. Later references to this game are found in France and Spain.[2]

When twenty-one was introduced in the United States, gambling houses offered bonus payouts to stimulate players' interest. One such bonus was a ten to one payout if the player's hand consisted of the ace of spades and a black jack (either the jack of clubs or the jack of spades). This hand was called a "blackjack" and the name stuck to the game, even though the ten to one bonus was soon withdrawn. In the modern game, a blackjack refers to any hand of an ace plus a ten or face card, regardless of suits or colours.

Rules of play at casinos

At a casino blackjack table, the dealer faces between five to seven playing positions from behind a semicircular table. At the beginning of each round, up to three players place their bets in the "betting box" at each position in play. The player whose bet is at the front of the betting box is deemed to have control over the position, and the dealer will consult the controlling player for playing decisions regarding the hand; the other players of that box are said to "play behind". Any player is usually allowed to control or bet in as many boxes as desired at a single table, but it is prohibited to play on more than one table at a time or to place multiple bets in a single box.

Each box is dealt an initial hand of two cards visible to the people playing on it, and often to any other players. The dealer's hand receives its first card face up, and in "hole card" games receives its second card face down immediately (the hole card), which the dealer peeks at but does not reveal unless it makes the dealer's hand a blackjack. Hole card games are sometimes played on tables with a small mirror or electronic sensor which are used to peek securely at the hole card. In European casinos, "no hole card" games are prevalent; the dealer's second card is neither drawn nor consulted until the players have all played their hands.

Cards are dealt either from one or two hand-held decks, from a dealer's shoe, or from a shuffling machine. Single cards are dealt to each of wagered-on position clockwise from the dealer's leftmost position, followed by a single card to the dealer, followed by an additional card to each of the positions in play. The players' initial cards may be dealt face-up, or face-down (more common in single-deck games).

The players' object is to win money by creating card totals which will turn out to be higher than the dealer's hand, but without exceeding 21 ("busting"/"breaking"). On their turn, players must choose whether to "hit" (take a card), "stand" (end their turn), "double" (double wager, take a single card and finish), "split" (if the two cards have the same value, separate them to make two hands) or "surrender" (give up a half-bet and retire from the game). Number-cards count as their natural value; the jack, queen, and king (also known as "face cards" or "pictures") count as 10; aces are valued as either 1 or 11 according to the player's best interest. If the hand value exceeds 21 points, it busts, and all bets on it are immediately forfeit. After all boxes have finished playing, the dealer's hand is resolved by drawing cards until the hand busts or achieves a value of 17 or higher (a dealer total of 17 including an ace, or "soft 17", must be drawn to in some games and must stand in others). The dealer never doubles, splits nor surrenders. If the dealer busts, all remaining player hands win. If the dealer does not bust, each remaining bet wins if its hand is higher than the dealer's, and loses if it is lower. In the case of a tied score, known as "push" or "standoff", bets are normally returned without adjustment; however, a blackjack beats any hand which is not a blackjack, even with value 21. Blackjack vs. blackjack is a push. Wins are paid out at 1:1, or equal to the wager, except for winning blackjacks, which are traditionally paid at 3:2, or one and a half times the wager. Many casinos today pay blackjacks at less than 3:2 at some tables.[3]

Blackjack games almost always provide a side bet called insurance, which may be played when dealer's upcard is an ace. At least one further side bet is usually provided.

Player decisions

After receiving an initial two cards, the player has up to four standard options: "hit," "stand," "double down," or "split". Each option has a corresponding hand signal. Some games give the player a fifth option, "surrender".

signal: (handheld) Scrape cards against table. (face up) Tap the table or wave hand toward body
signal: (handheld) Slide cards under chips. (face up) Wave hand horizontally.
signal: Place additional chips beside the original bet, and point with one finger.
signal: Place additional chips next to the original bet outside the betting box. Point with two fingers spread into a V formation.

Hand signals are used to assist the "eye in the sky", a person or video camera located above the table and sometimes concealed behind one-way glass. The eye in the sky usually makes a video recording of the table, which helps in resolving disputes and identifying dealer mistakes, and is also used to protect the casino against dealers who steal chips or players who cheat. The recording can further be used to identify advantage players whose activities, while legal, make them undesirable customers. In the event of a disagreement between a player's hand signals and their words, the hand signal takes precedence.

Each hand may normally "hit" as many times as desired so long as the total is not above hard-20. On reaching 21 (including soft 21), the hand is normally required to stand; busting is an irrevocable loss and the players' wagers are immediately forfeited to the house. After a bust or a stand, play proceeds to the next hand clockwise around the table. When the last hand has finished being played, the dealer reveals the hole card, and stands or draws further cards according to the rules of the game for dealer drawing. When the outcome of the dealer's hand is established, any hands with bets remaining on the table are resolved (usually in counter-clockwise order): bets on losing hands are forfeited, the bet on a push is left on the table, and winners are paid out.

Insurance

If the dealer's upcard is an ace, the player is offered the option of taking "insurance" before the dealer checks the hole card.

Insurance is a side bet that the dealer has blackjack and is treated independently of the main wager. It pays 2:1 and is available when the dealer's exposed card is an ace. The idea is that the dealer's second card has a fairly high probability (nearly one-third) to be ten-valued, giving the dealer blackjack and disappointment for the player. It is attractive (although not necessarily wise) for the player to insure against the possibility of a dealer blackjack by making a maximum "insurance" bet, in which case the "insurance proceeds" will make up for the concomitant loss on the original bet. The player may add up to half the value of their original bet to the insurance and these extra chips are placed on a portion of the table usually marked "Insurance Pays 2 to 1".

Players with a blackjack may also take insurance, and in taking maximum insurance they commit themselves to winning an amount exactly equal to their main wager, regardless of the dealer's outcome. Fully insuring a blackjack against blackjack is thus referred to as "taking even money", and paid out immediately, before the dealer's hand is resolved; the players need not produce to place more chips for the insurance wager.

Insurance bets are expected to lose money in the long run, because the dealer is likely to have blackjack less than one-third of the time. However the insurance outcome is strongly anti-correlated with that of the main wager, and if the player's priority is to reduce variation, it is reasonable to pay for this.

Furthermore, the insurance bet is susceptible to advantage play. It is advantageous to make an insurance bet whenever the hole card has more than a chance of one in three of being a ten. Advantage play techniques can sometimes identify such situations. In a multi-hand, face-up, single deck game, it is possible to establish whether insurance is a good bet simply by observing the other cards on the table after the deal; even if there are just 2 player hands exposed, and neither of their two initial cards is a ten, then 16 in 47 of the remaining cards are tens, which is larger than 1 in 3, so insurance is a good bet. This is an elementary example of the family of advantage play techniques known as card counting.

Bets to insure against blackjack are slightly less likely to be advantageous than insurance bets in general, since the ten in the player's blackjack makes it less likely that the dealer has blackjack too.[4]

Rule variations and their consequences for the house edge

The rules of casino blackjack are generally determined by law or regulation, which establishes certain rule variations allowed at the discretion of the casino. The rules of any particular game are generally posted on or near the table, failing which there is an expectation that casino staff will provide them on request. Over 100 variations of blackjack have been documented.[5]

As with all casino games, blackjack incorporates a "house edge", a statistical advantage for the casino which is built into the game. The advantage of the dealer's position in blackjack relative to the player comes from the fact that if the player busts, the player loses, regardless of whether the dealer subsequently busts. Nonetheless, blackjack players using basic strategy will lose less than 1% of their total wagered amount with strictly average luck; this is very favorable to the player compared to other casino games. The loss rate of players who deviate from basic strategy through ignorance is in general expected to be greater.

Each game has a rule about whether the dealer must hit or stand on soft 17, which is generally printed on the table surface. The variation where the dealer must hit soft 17 is abbreviated "H17" in blackjack literature, with "S17" used for the stand-on-soft-17 variation. Substituting an "H17" rule with an "S17" rule in a game benefits the player, decreasing the house edge by about 0.2%.
All things being equal, using fewer decks decreases the house edge. This mainly reflects an increased likelihood of player blackjack, since if the players draws a ten on their first card, the subsequent probability of drawing an ace is higher with fewer decks. It also reflects a decreased likelihood of blackjack-blackjack push in a game with fewer decks.
Casinos generally compensate by tightening other rules in games with fewer decks, in order to preserve the house edge. When offering single deck blackjack games, casinos are more likely to disallow doubling on soft hands or after splitting, to restrict resplitting, and to pay the player less than 3:2 for a winning blackjack.
The following table illustrates the mathematical effect on the house edge of the number of decks, by considering games with various deck counts under the following ruleset: double after split allowed, resplit to four hands allowed, no hitting split aces, no surrender, double on any two cards, original bets only lost on dealer blackjack, dealer hits soft 17, and cut-card used. The increase in house edge per unit increase in the number of decks is most dramatic when comparing the single deck game to the two-deck game, and becomes progressively smaller as more decks are added.
Number of Decks House Advantage
Single deck 0.17%
Double deck 0.46%
Four decks 0.60%
Six decks 0.64%
Eight decks 0.66%
Late/early surrender

Surrender, for those games that allow it, is usually not permitted against a dealer blackjack; if the dealer's first card is an ace or ten, the hole card is checked to make sure there is no blackjack before surrender is offered. This rule protocol is consequently known as "late" surrender. The alternative, "early" surrender, gives player the option to surrender before the dealer checks for blackjack, or in a no-hole-card game. Early surrender was originally the norm in Atlantic City casinos but is much more favourable to the player than late surrender, and has largely disappeared from the United States. Nonetheless early surrender games can still be found in several countries. Most medium-strength hands should be surrendered against a dealer Ace if the hole card has not been checked.

For late surrender, however, while it is tempting opt for surrender on any hand which will probably lose, the correct strategy is to only surrender on the very worst hands, because having even a one in four chance of winning the full bet is better than losing half the bet and pushing the other half, as entailed by surrendering.

Resplitting
If the cards of a post-split hand have the same value, most games allow the player to split again, or "resplit". The player places a further wager and the dealer separates the new pair dealing a further card to each as before. Some games allow unlimited resplitting, while others may limit it to a certain number of hands, such as four hands (for example, "resplit to 4").
Hit/resplit split aces
After splitting aces, the common rule is that only one card will be dealt to each ace; the player cannot split, double, or take another hit on either hand. Rule variants include allowing resplitting aces or allowing the player to hit split aces. Games allowing aces to be resplit are not uncommon, but those allowing the player to hit split aces are extremely rare. Allowing the player to hit hands resulting from split aces reduces the house edge by about 0.13%; allowing resplitting of aces reduces house edge by about 0.03%. Note that a ten-value card dealt on a split ace (or vice versa) is a "soft 21" and not a "natural".
No double after split
After a split, most games allow doubling down on the new two-card hands. Disallowing doubling after a split increases the house edge by about 0.12%.
Double on 9/10/11 or 10/11 only
Under the "Reno rule", double down is only permitted on hard totals of 9, 10, or 11 (under a similar European rule, only 10 or 11). Basic strategy would otherwise call for some doubling down with hard 9 and soft 13-18, and advanced players can identify situations where doubling on soft 19-20 and hard 8,7 and even 6 is advantageous. The Reno rule prevents the player from taking advantage of double down in these situations and thereby increases the player's expected loss. The Reno rule increases the house edge by around one in 1000, and its European version by around two in 1000.
No hole card and OBO

In most non-U.S. casinos, a 'no hole card' game is played, meaning that the dealer does not draw nor consult his or her second card until after all players have finished making decisions. With no hole card, it is almost never correct basic strategy to double or split against a dealer ten or ace, since a dealer blackjack will result in the loss of the split and double bets; the only exception is with a pair of A's against a dealer 10, where it is still correct to split. In all other cases, a stand, hit or surrender is called for. For instance, holding 11 against a dealer 10, the correct strategy is to double in a hole card game (where the player knows the dealer's second card is not an ace), but to hit in a no hole card game. The no hole card rule adds approximately 0.11% to the house edge.

The "original bets only" rule variation appearing in certain no hole card games states that if the player's hand loses to a dealer blackjack, only the mandatory initial bet ("original") is forfeited, and all optional bets, meaning doubles and splits, are pushed. "Original bets only" is also known by the acronym OBO; it has the same effect on basic strategy and house edge as reverting to a hole card game.[6]

Altered payout for a winning blackjack
In many casinos, a blackjack pays only 6:5 or even 1:1 instead of the usual 3:2. This is usually at tables with the lowest table minimums and single-deck games. Among common rule variations in the U.S., these altered payouts for blackjack are the most damaging to the player, causing the greatest increase in house edge. Since blackjack occurs in approximately 4.8% of hands, the 1:1 game increases the house edge by 2.3%, while the 6:5 game adds 1.4% to the house edge. Video blackjack machines generally pay 1:1 payout for a blackjack. The 6:5 rule is most commonly employed on table blackjack at single deck games, where they help the house to compensate for low house edge intrinsic in using one deck only.[3]
Dealer wins ties
The rule that bets on tied hands are lost rather than pushed is catastrophic to the player. Though rarely used in standard blackjack, it is sometimes seen in "blackjack-like" games such as in some charity casinos.

Blackjack strategy

Basic strategy

Each blackjack game has a basic strategy, which is the way of playing a hand with any total value against any dealer upcard which will lose least money to the house in the long term with strictly average luck.

An example of basic strategy is shown in the table below for a common ruleset:

Player hand Dealer's face-up card
2 3 4 5 6 7 8 9 10 A
Hard totals (excluding pairs)
17-20 S S S S S S S S S S
16 S S S S S H H SU SU SU
15 S S S S S H H H SU H
13–14 S S S S S H H H H H
12 H H S S S H H H H H
11 Dh Dh Dh Dh Dh Dh Dh Dh Dh H
10 Dh Dh Dh Dh Dh Dh Dh Dh H H
9 H Dh Dh Dh Dh H H H H H
5–8 H H H H H H H H H H
Soft totals
2 3 4 5 6 7 8 9 10 A
A,8–A,9 S S S S S S S S S S
A,7 S Ds Ds Ds Ds S S H H H
A,6 H Dh Dh Dh Dh H H H H H
A,4–A,5 H H Dh Dh Dh H H H H H
A,2–A,3 H H H Dh Dh H H H H H
Pairs
2 3 4 5 6 7 8 9 10 A
A,A SP SP SP SP SP SP SP SP SP SP
10,10 S S S S S S S S S S
9,9 SP SP SP SP SP S SP SP S S
8,8 SP SP SP SP SP SP SP SP SP SP
7,7 SP SP SP SP SP SP H H H H
6,6 SP SP SP SP SP H H H H H
5,5 Dh Dh Dh Dh Dh Dh Dh Dh H H
4,4 H H H SP SP H H H H H
2,2–3,3 SP SP SP SP SP SP H H H H

Key:

S = Stand
H = Hit
Dh = Double (if not allowed, then hit)
Ds = Double (if not allowed, then stand)
SP = Split
SU = Surrender (if not allowed, then hit.)

The bulk of basic strategy is common to all blackjack games, with most rule variations calling for changes in only a few situations. For example, if the above game used the hit soft 17 rule, common in Las Vegas Strip casinos, only 5 cells of the table would need to be changed: double on 11 vs. A, double on A/7 vs. 2, double on A/8 vs. 6 and surrender 15 or 17 vs. A.

Estimates of the house edge for blackjack games quoted by casinos and gaming regulators are generally based on the assumption that the players follow basic strategy and do not systematically change their bet size. Most blackjack games have a house edge of between 0.5% and 1%, placing blackjack among the cheapest casino table games. Casino promotions such as complimentary matchplay vouchers or 2:1 blackjack payouts allow the player to acquire an advantage without deviating from basic strategy.

Composition-dependent strategy

Basic strategy is based upon a player's point total and the dealer's visible card. Players may be able to improve on this decision by considering the precise composition of their hand, not just the point total. For example, players should ordinarily stand when holding 12 against a dealer 4. However, in a single deck game, players should hit if their 12 consists of a 10 and a 2. The presence of a 10 in the players' hand has two consequences:[7]

However, even when basic and composition-dependent strategy lead to different actions, the difference in expected reward is small, and it becomes even smaller with more decks. Using a composition-dependent strategy rather than basic strategy in a single deck game reduces the house edge by 4 in 10,000, which falls to 3 in 100,000 for a six-deck game.[8]

Advantage play

Main article: Advantage player

Blackjack has been a high profile target for advantage players since the 1960s. Advantage play is the attempt to win more using "honest" skills such as memory, computation and observation. These techniques, while generally legal, can be powerful enough to give the player a long-term edge in the game, making them an undesirable customer for the casino and leading to ejection or blacklisting if they are detected. The main techniques of advantage play in blackjack are as follows:

Card counting

Main article: Card counting#Blackjack

During the course of a blackjack shoe, the dealer progressively exposes cards which are dealt to his or her own and the players' hands. Careful accounting of the exposed cards allows a player to make inferences about the cards which remain to be dealt, and use these inferences in one of two ways:

A typical card counting system assigns a point score to each rank of card (e.g., 1 point for 2–6, 0 points for 7–9 and −1 point for 10–A). Whenever a card is exposed, a counter adds the score of that card to a running total, the 'count'; the count is used to make betting and playing decisions according to a table which they have learned. The count starts at 0 for a freshly shuffled deck for "balanced" counting systems. Unbalanced counts are often started at a value which depends on the number of decks used in the game.

In comparison to the house edge of 0.5-1% which can be achieved by playing basic strategy in most blackjack games,[9] card counting can give the player an advantage of up to 2% over the house.[10]

Card counting is most rewarding near the end of a complete shoe when as few as possible cards remain. Single deck games are therefore particularly susceptible to card counting. As a result, casinos are more likely to insist that players do not reveal their cards to one another in single deck games. In games with more decks of cards, casinos limit penetration by ending the shoe and initiating a reshuffle when one or more decks remain undealt, or by using a shuffling machine to reintroduce the exhausted cards every time a deck has been played.

Card counting mentally is legal and is not considered cheating.[11] However, it usually needs to be done discreetly; if a player is detected while counting, the casino may inform them that they are no longer welcome to play blackjack there, or they may be completely banned from the property.[12]

The use of any external devices to assist with counting cards is illegal in all US states that license blackjack card games.[13]

Shuffle tracking

Techniques other than card-counting can swing the advantage of casino blackjack toward the player. All such techniques are based on the value of the cards to the player and the casino, as originally conceived by Edward O. Thorp.[14] One technique, mainly applicable in multi-deck games, involves tracking groups of cards (aka slugs, clumps, packs) during the play of the shoe, following them through the shuffle and then playing and betting accordingly when those cards come into play from the new shoe. Shuffle tracking requires excellent eyesight and powers of visual estimation, but is more difficult to detect since the player's actions are largely unrelated to the composition of the cards in the shoe.[15]

Arnold Snyder's articles in Blackjack Forum magazine brought shuffle tracking to the general public. His book, The Shuffle Tracker's Cookbook, mathematically analyzed the player edge available from shuffle tracking based on the actual size of the tracked slug. Jerry L. Patterson also developed and published a shuffle-tracking method for tracking favorable clumps of cards and cutting them into play and tracking unfavorable clumps of cards and cutting them out of play.[16][17][18]

Identifying concealed cards

The player can also gain an advantage by identifying cards from distinctive wear markings on their backs, or by hole carding (observing during the dealing process the front of a card dealt face down). These methods are generally legal although their status in particular jurisdictions may vary.

Side bets

Many blackjack tables offer a side bet on various outcomes including[19]:

The side wager is typically placed in a designated area next to the box for the main wager. A player wishing to wager on a side bet is usually required to place a wager on blackjack. Some games require that the blackjack wager should equal or exceed any side bet wager. A non-controlling player of a blackjack hand is usually permitted to place a side bet regardless of whether the controlling player does so.

The house edge for side games is generally higher than for the blackjack game itself. Nonetheless side games can be susceptible to card counting, often requiring bespoke counting systems. Most side games do not offer sufficient win rate to justify the effort of advantage play; exceptions are "Lucky ladies" and "Over/Under".

In team play it is common for team members to be dedicated toward counting only a sidebet using a specialized count.

Blackjack tournaments

Blackjack can be played in tournament form, where players start with equal numbers of chips and the aim is to finish among the top chip-holders. Depending on the number of competitors, tournaments may be held over several rounds, with one or two players qualifying from each table after a set number of deals to meet the qualifiers from the other tables in the next round. Alternatively the Elimination Blackjack format drops the lowest-stacked player from the table at pre-determined points in the tournament, for instance after every four deals. Good strategy for blackjack tournaments can be very different from non-tournament strategy, and has the added dimension of choosing the amount to be wagered. As in poker tournaments, players pay the casino an initial entry fee to participate in a tournament, and re-buys are sometimes permitted.

Video blackjack

Some casinos, as well as general betting outlets, provide blackjack among a selection of casino-style games at electronic consoles. Video blackjack game rules are generally more favorable to the house; e.g., paying out only even money for winning blackjacks. Video and online blackjack games deal each coup from a fresh shoe, rendering card counting much less effective.

Variants of the game

Blackjack is a member of a large family of traditional card games played recreationally all around the world. Most of these games have not been adapted for casino play. Furthermore, the casino game development industry is very active in producing blackjack variants, most of which are ultimately not adopted for widespread use in casinos. The following are the prominent twenty-one themed comparing card games which have been adapted or invented for use in casinos and have become established in the gambling industry.

Examples of the many local traditional and recreational blackjack-like games include French/German Blackjack, called Vingt-et-un (French: Twenty-one) or "Siebzehn und Vier" (German: Seventeen and Four). The French/German game does not allow splitting. An ace can only count as eleven, but two aces count as a blackjack. It is mostly played in private circles and barracks. A British variation is called "Pontoon", the name being probably a corruption of "Vingt-et-un".

Blackjack Hall of Fame

Main article: Blackjack Hall of Fame

In 2002, professional gamblers around the world were invited to nominate great blackjack players for admission into the Blackjack Hall of Fame. Seven members were inducted in 2002, with new people inducted every year after. The Hall of Fame is at the Barona Casino in San Diego. Members include Edward O. Thorp, author of the 1960s book Beat the Dealer which proved that the game could be beaten with a combination of basic strategy and card counting; Ken Uston, who popularized the concept of team play; Arnold Snyder, author and editor of the Blackjack Forum trade journal; Stanford Wong, author and popularizer of the "Wonging" technique of only playing at a positive count, and several others.

See also

Blackjack literature

Regulation in the United Kingdom

References

  1. ^ Scarne's New Complete Guide to Gambling, p. 342
  2. ^ Fontbona, Marc (2008). Historia del Juego en España. De la Hispania romana a nuestros días. Barcelona: Flor del Viento Ediciones. ISBN 978-84-96495-30-2. http://www.loteriasyapuestas.es/index.php/mod.pags/mem.libros/libro.18/idpag.600008/relcategoria.271032. 
  3. ^ a b Taking a hit: New blackjack odds further tilt advantage toward the house by Jeff Haney, Las Vegas Sun, November 13, 2003.
  4. ^ Blackjack Insurance Exceptions
  5. ^ QFIT.com 100+ Blackjack variations
  6. ^ Modern Blackjack page 39
  7. ^ "The Wizard of Odds". Fine points of basic strategy in single-deck blackjack. http://wizardofodds.com/blackjack/appendix3c.html. Retrieved December 8, 2006. 
  8. ^ "The Wizard of Odds". Total Dependent and Composition Dependent Basic Strategy in Blackjack. http://wizardofodds.com/blackjack/appendix15.html. Retrieved December 19, 2006. 
  9. ^ Rules & House Edge Table
  10. ^ Theory of Blackjack, p. 5
  11. ^ Theory of Blackjack, pp 6–7
  12. ^ Thorp, Edward O. (1966) Beat The Dealer, Vintage Books, a Division of Random House, ISBN 978-0-394-70310-7, pp 132-136
  13. ^ US State Gambling Laws
  14. ^ The Mathematics of Gambling
  15. ^ Shuffle Tracking Counts
  16. ^ The Gambling Times Guide to Blackjack; Gambling Times Incorporated, Hollywood, CA; © 1984; Page 110; ISBN 0-89746-015-4 Shuffle-Tracking An Easy Way to Start
  17. ^ Break the Dealer; by Jerry L. Patterson and Eddie Olsen; Perigee Books; A Division of Penguin Putnam; © 1986; ISBN 0-399-51233-0 Shuffle-Tracking; Chapter 6, Page 83]
  18. ^ Blackjack: A Winner's Handbook; by Jerry L. Patterson; Perigee Books; A Division of Penguin Putnam; © 1990; ISBN 0-399-51598-4 Shuffle-Tracking; Chapter 4, Page 51]
  19. ^ Blackjack side bets — analyzed by The Wizard of Odds

External links

Blackjack calculators