Mathematics of the Big Four Casino Table Games: Blackjack, Baccarat, Craps, & Roulette
In this first chapter probability theory is placed on a firm foundation. Then a variety of interesting and entertaining examples are introduced and investigated. These include French and American roulette; dice games such as craps, hazard, chuck-a-luck, crown and anchor, and sic-bo; and card games such as blackjack, macao, baccarat, and solitaire. Beyond the realms of card and casino games, topics ranging from the Martingale betting system and the Fibonacci betting system, setting winnings and losses bounds, the “optimum” strategy for blackjack, the gamblers’ fallacy, inherited diseases, monopoly, weather forecasting, chaos theory, horseracing odds, elections, your intuition, and paradoxes are examined. While we should expect that in each and every casino game the house has a statistical advantage, in this chapter we shed light on which games offer players a good chance.
italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Cuckoo
is a popular card game, which originated in France during the 15th century and then spread throughout Europe, where it is currently well-known under distinct names and with different variants. Cuckoo is an imperfect information game-of-chance, which makes the research regarding its optimal strategies determination interesting. The rules are simple: each player receives a covered card from the dealer; starting from the player at the dealer's left, each player looks at its own card and decides whether to exchange it with the player to their left, or keep it; the dealer plays at last and, if it decides to exchange card, it draws a random one from the remaining deck; the player(s) with the lowest valued card lose(s) the round. We formulate the gameplay mathematically and provide an analysis of the optimal decision policies. Different card decks can be used for this game, e.g., the standard 52-card deck or the Italian 40-card deck. We generalize the decision model for an arbitrary number decks' cards, suites, and players. Lastly, through numerical simulations, we compare the determined optimal decision strategy against different benchmarks, showing that the strategy outperforms the random and naive policies and approaches the performance of the ideal oracle.
A winning strategy is developed for the nine to one side bet on a Banker natural nine. Let n be the number of cards that remain for play. Let t be the number of nines that remain for play. If p(n, t) is the probability of a natural nine when n and t are given, then p(n, t) is greater than 0.1 frequently enough to make the counting of n and t the basis for a practical winning strategy. The Kelly criterion (play to maximize the expected value of the log of capital) is used to determine bet sizes for favorable situations. Similar strategies are developed for the side bets on Banker natural eight, Player natural nine, and Player natural eight. The relationships between the four side bets are analyzed. It is shown that the main (roughly 1:1) bets on Banker and Player occasionally favor the player. There are theoretical favorable strategies but none exists which is currently practical.
To get around an old California law that prohibits the game of "21," California card casinos introduced a variation of standard Blackjack, called LA Blackjack, in which the objective is to get as close as possible to 22 without going over.
The standard game of Blackjack, or "21", pits the player against the dealer ("the house"). Money lost by the player represents an equivalent gain by the house, and vice versa. In contrast, LA Blackjack pits the player against other players. Each player gets a turn to be the "banker" and collect money from the other players at the table, while the house collects a percentage of the total money bet. Mathematicians developed optimal plays for standard Blackjack many years ago, while the optimal plays for LA Blackjack are largely unexplored.
We analyze this game, develop a basic strategy for the player, explore the possibility of employing bet variance to raise the player's expected return, and conclude that a player cannot develop a profitable strategy.
This article discusses the card game blackjack as played in the casinos of Las Vegas. The basic rules for the game are described in detail. The player's strategic problems are analyzed with the objective of finding the strategy maximizing his mathematical expectation.A mathematical expression is derived giving a general solution to the player's problem of standing pat with a given hand versus drawing additional cards. No general solutions are possible for the other major strategic problems, however, and a detailed examination of individual situations is required. The formulas and methods for the case analysis are stated, but computational details are omitted. Similarly, the formula for the player's mathematical expectation is stated, but its numerical evaluation is not described. Detailed discussion is given to the problems arising in the combinatorial type of computations required by blackjack.The “optimum strategy” determined by the above analysis differs substantially from the published strategies of card experts and the usual style of play in the casinos.
Boardwalker, Not So Progressive 21…(yawn)
- Barbara J Bennett
- Randomness
Bennett, Barbara J., Randomness. Harvard University Press, Cambridge, MA, 1998.
Boardwalker, Not So Progressive 21…(yawn). Blackjack Forum XIII #3, September 1993, p. 42.
Bollman, Mark, Basic Gambling Mathematics: The Numbers Behind The Neon. CRC
Press/Taylor & Francis, Boca Raton, FL, 2014.
The Bone Man, New Craps Fire Bet Emerging in Las Vegas
- Mark Bollman
Bollman, Mark, Mathematics of Casino Carnival Games. CRC Press/Taylor & Francis, Boca
Raton, FL, 2020.
The Bone Man, New Craps Fire Bet Emerging in Las Vegas. Online at
http://www.nextshooter.com/firebet. Accessed 23 June 2012.
Gaming Company Makes Roulette More Colorful
- John Brokopp
Brokopp, John, Gaming Company Makes Roulette More Colorful. Online at
http://brokopp.casinocitytimes.com/article/gaming-company-makes-roulette-more-colorful-58988. Accessed 21 May 2012.
Atlantic City Update
- Chris Buro
Buro, Chris, Atlantic City Update. Blackjack Forum XVI #4, Winter 1996, pp. 49-52.
Why Casino Craps Can't Be Beaten. Las Vegas Advisor
- Bryce Carlson
Carlson, Bryce, Why Casino Craps Can't Be Beaten. Las Vegas Advisor, 26 October 2020.
Online at https://www.lasvegasadvisor.com/gambling-with-an-edge/why-casino-craps-cant-bebeaten/. Accessed 4 November 2020.
Cash, Les, More Indian Reservation Problems
- Mike Caro
Caro, Mike. Games You Can Beat and Games You Can't. Online at
http://www.gamblingtimes.com/writers/mcaro/mcaro_winter2001.html. Accessed 15 July 2015.
Cash Card Games, LLC. Online at http://www.cashcardgames.com. Accesse 12 June 2020.
Cash, Les, More Indian Reservation Problems. Blackjack Forum XVII #4, Winter 1997, pp.
30-32.
Hard Hardway Bet, in Finding the Edge
- Donald E Catlin
- Really
Catlin, Donald E., A Really Hard Hardway Bet, in Finding the Edge, Olaf Vancura, Judy A.
Cornelius, and William R. Eadington, editors. Institute for the Study of Gambling and
Commercial Gaming, Reno, NV, 2000, pp. 297-302.
Beating The Blackjack Slot Machines
- R Cohen
- Carl
Cohen, R. Carl, Beating The Blackjack Slot Machines, R. Carl Cohen, Publisher, Philadelphia,
1985.
- Anthony Curtis
- Las Vegas Advisor
Curtis, Anthony, Las Vegas Advisor. Blackjack Forum XI #1, March 1991, pp. 22-24, 29.
Curtis, Anthony, Las Vegas Advisor. Blackjack Forum XI #4, December 1991, pp. 32-34.
The Gambling Conference, or, How I Finally Found Positive Expectation in Atlantic City
- Anthony Curtis
- Las Vegas Advisor
Curtis, Anthony, Las Vegas Advisor. Blackjack Forum XII #4, December 1992, pp. 27-28, 33.
Curtis, Anthony, The Gambling Conference, or, How I Finally Found Positive Expectation in
Atlantic City. Blackjack Forum V #1, March 1985, pp. 9-18.
Devlin, Keith, The Unfinished Game
- Mark Dace
Dace, Mark, Around the World: Peru. Blackjack Forum XXI #2, Sumemr 2001, p. 70.
Devlin, Keith, The Unfinished Game. Basic Books, New York, 2008.
Dice-The Soul of Honesty. Casino & Sports 22, 1982, pp. 26-55
Hard Pass Craps Wager, United States Patent # US 8
- Stacey Friedman
Friedman, Stacey, Hard Pass Craps Wager, United States Patent # US 8,118,309 B1, 21
Gambler's Book Club staff, $100,000 Club
February 2012. Online at https://patents.google.com/patent/US8118309. Accessed 4 May 2020.
Gambler's Book Club staff, $100,000 Club. Systems & Methods 5, 1975, pp. 59-60.
Around the World: Russia. Blackjack Forum XXI #1
- Garry
Garry, Around the World: Russia. Blackjack Forum XXI #1, Spring 2001, pp. 67-70.
Group of eight glass astragali (knucklebones)
- Tina Griffin
- Letter To Ashford
- Ashford Kneitel
- Llc Gaming
Griffin, Tina, Letter to Ashford Kneitel, Ashford Gaming LLC, 24 April 2014. Online at
https://fortress.wa.gov/wsgc/etransfer/OnlineServices/activities/ViewGameRule.cshtml?gid=9.
Accessed 2 March 2019.
Group of eight glass astragali (knucklebones). Online at
https://commons.wikimedia.org/wiki/File:Group_of_eight_glass_astragali_(knucklebones)_MET
_DP121111.jpg. Accessed 29 August 2020.
No dice: Casino invents version of craps played with cards
- Joe Hadsall
Hadsall, Joe, No dice: Casino invents version of craps played with cards. The Joplin Globe,
Joplin, MO, 16 April 2010.
Institute for the Study of Gambling and Commercial Gaming
- Robert C Hannum
- Anthony N Cabot
Hannum, Robert C., and Anthony N. Cabot, Practical Casino Math, 2nd edition. Institute for the
Study of Gambling and Commercial Gaming, Reno, NV, 2005.
United States Patent Application Publication
- Samuel C Hunter
Hunter, Samuel C., Craps game with progressive jackpot. United States Patent Application
Publication, # US 2004/0130094 A1, 8 July 2004. Online at
https://patents.google.com/patent/US20040130094. Accessed 7 May 2020.
A Variation of Blackjack Created for Card Rooms
- Kim Jacobs
- California No Bust
- Blackjack
Jacobs, Kim, California No Bust Blackjack: A Variation of Blackjack Created for Card Rooms.
Cardroom Insider, 7 October 2015. Online at
https://cardroominsider.wordpress.com/2015/10/07/california-blackjack-blackjack-variationcreated-for-card-rooms/. Accessed 24 May 2020.
Baccarat Combinatorial Analysis The Easy Way
- Eliot Jacobson
Jacobson, Eliot, Advanced Advantage Play. Blue Point Books, Santa Barbara, CA, 2015.
Jacobson, Eliot, Baccarat Combinatorial Analysis The Easy Way. Online at
https://www.888casino.com/blog/baccarat-tips/baccarat-combinatorial-analysis-the-easy-way.
Accessed 15 May 2018.
Card Counting the Natural 9 and Natural 8 Baccarat Side Bets
- Eliot Jacobson
Jacobson, Eliot, Card Counting the Natural 9 and Natural 8 Baccarat Side Bets. 11 August
2013. Online at https://www.888casino.com/blog/side-bets/card-counting-the-natural-9-andnatural-8-baccarat-side-bets. Accessed 5 August 2020.
Lightning Launch Roulette-Scientific Games
Lightning Launch Roulette-Scientific Games. Casino Journal, 3 May 2018. Online at
https://www.casinojournal.com/articles/92082-lightning-launch-roulette-scientific-games.
Accessed 11 June 2018.
United States Patent Application Publication
- Jerrery Lowery
- Roy
Lowery, Jerrery Roy, Craps game with jackpot. United States Patent Application Publication, #
US 2008/0128990 A1, 5 June 2008. Online at
https://patents.google.com/patent/US20080128990. Accessed 7 May 2020.
The Essentials of Casino Game Design
- Dan Lubin
Lubin, Dan, The Essentials of Casino Game Design. Huntington Press, Las Vegas, 2016.
Lucky 13s Blackjack. Online at http://lucky13s.com.au/. Accessed 6 May 2014.
Lucky 13s Blackjack Rack Card. Online at http://lucky13s.com.au/ETG-Brochure.pdf. Accessed
6 May 2014.
Marina Bay Sands Casino, Singapore Table Games
- Mason Malmuth
- Blackjack Essays
Malmuth, Mason, Blackjack Essays. Two Plus Two Publishing, Las Vegas, 2000.
Marina Bay Sands Casino, Singapore Table Games. Online at
https://www.sandscasino.com/singapore/games/table-games.html. Accessed 21 June 2020.
Nevada OKs craps-style table game. Gaming Today
- Philip Nehrt
Nehrt, Philip, Winning Gambling Strategies. AuthorHouse, Bloomington, IN, 2012.
Nelson, Rex, California Aces. WIN Magazine 14 #1, June 1992, pp. 36-37, 57.
Nevada OKs craps-style table game. Gaming Today, 22 April 2009. Online at
https://www.gamingtoday.com/casino_games/table_games/article/21067-Nevada_OKs_craps_style_table_game. Accessed 28 Auugust 2020.
New betting option livens up craps, Gaming Today, 14 October 2008. Online at
https://www.gamingtoday.com/casino_games/table_games/article/18459-New_betting_option_livens_up_craps. Accessed 16 March 2018.
Ohio Lottery, The Lucky One
- The Norseman
- Ol
Norseman, The Ol', Blackjack in Denmark. Blackjack Forum XII #2, June 1992, p. 34.
Ohio Lottery, The Lucky One. Online at https://www.ohiolottery.com/Games/The-Lucky-One#1.
Accessed 17 April 2018.
Just How Much of an Edge Does 'Crap-less' Craps Offer?
- Dennis Paulsen
Paulsen, Dennis, Just How Much of an Edge Does 'Crap-less' Craps Offer?. Casino & Sports
16, 1981, pp. 42-43.
Computer Analysis of Vegas World's Innovative 'No Lose Free Roll
- Dennis Paulsen
- Theoretical
Paulsen, Dennis, A Theoretical & Computer Analysis of Vegas World's Innovative 'No Lose
Free Roll' Craps. Casino & Sports 26, pp. 71-78.
- Pentagon Craps
- Win Magazine
Pentagon Craps. WIN Magazine, 13, #12, May 1992, pp. 8-9.
Pontoon Pandemonium (MBS) Game Rules, Version 4, 4 November 2016. Online at
https://www.cra.gov.sg/docs/default-source/game-rule-documents/mbs/pontoon-pandemoniummbs-game-rules-version-4.pdf. Accessed 21 June 2018.
Prime Time (Roulette side bet). Online at
https://gaming.nv.gov/Modules/ShowDocument.aspx?documentid=8997. Accessed 12 Ocotber
2018.
Progressive Roulette, United States Patent # US 6,776,714 B2. Online at
http://www.google.com/patents/US6776714. Accessed 17 July 2015.
Is Baccarat Card Counting Worth the Effort? Gamblingsites.net, 15
- Joey Richardson
Richardson, Joey, Is Baccarat Card Counting Worth the Effort? Gamblingsites.net, 15 August
2018. Online at https://www.gamblingsites.net/blog/is-baccarat-card-counting-worth-the-effort/.
Accessed 31 July 2020.
Riverboat Roulette" to be rebranded as "Rainbow Bet Roulette
- Rebecca Richfield
Richfield, Rebecca, How the Free Ride Really Died. Blackjack Forum XVI #4, Winter 1996, pp.
30-32.
"Riverboat Roulette" to be rebranded as "Rainbow Bet Roulette". Online at
http://doubleluckgaming.com/riverboat-roulette-to-be-rebranded-as-rainbow-bet-roulette/.
Accessed 1 May 2015.
How to Play Card-Based Roulette at Riverwind Casino
- Riverwind Casino
Riverwind Casino, How to Play Card-Based Roulette at Riverwind Casino. Online at
https://www.youtube.com/watch?v=MTO3-yncZ_4. Accessed 28 February 2018.
- Stanley Roberts
- Roberts' Rules
Roberts, Stanley, Roberts' Rules. WIN Magazine 11, #11, June 1990, pp. 10-11.
Roulette Dinner, Casino & Sports 2, p. 30, 1978. Roulette Evolution
- Stanley Roberts
- Roberts' Rules
- Magazine
Roberts, Stanley, Roberts' Rules. WIN Magazine 11, #12, July 1990, pp. 10-11, 68.
Roberts, Stanley, Roberts' Rules. WIN Magazine 12, #1, August 1990, pp. 10-11, 62.
Roulette Dinner, Casino & Sports 2, p. 30, 1978.
Roulette Evolution. Online at http://www.igt.com/us-en/games/gamepage.aspx?type_id=5075&showtab=1. Accessed 12 September 2015.
Inside The World Of Advantage-Play Craps
- Frank Scoblete
- Am A Dice
- Controller
Scoblete, Frank, I Am A Dice Controller: Inside The World Of Advantage-Play Craps. Triumph
Books, Chicago, 2015.
Baccarat-The Number of Combinations of Each Possible Hand
- Michael Shackleford
Shackleford, Michael, Baccarat-The Number of Combinations of Each Possible Hand. Online
at https://wizardofodds.com/games/baccarat/appendix/1/. Accessed 1 August 2020.
Dealer probabilities in blackjack
- Michael Shackleford
Shackleford, Michael, Craps Side Bets. Online at
http://wizardofodds.com/games/craps/appendix/5/#firebet. Accessed 23 June 2012.
Shackleford, Michael, Dealer probabilities in blackjack. Online at
http://wizardofodds.com/games/blackjack/appendix/2b/. Accessed 6 April 2015.
Gambling in the United Kingdom
- Michael Shackleford
Shackleford, Michael, Gambling in the United Kingdom. Online at
https://wizardofodds.com/blog/gambling-united-kingdom/. Accessed 16 February 2018.
Raving Table Games Show
- Michael Shackleford
Shackleford, Michael, Raving Table Games Show 2014. Online at
http://wizardofodds.com/blog/raving-2014/. Accessed 6 April 2015.
Roul 8: Roulette Side Bet
- Michael Shackleford
Shackleford, Michael, Roul 8: Roulette Side Bet. Online at
https://wizardofodds.com/games/roulette/side-bets/roul-8/, 26 July 2017. Accessed 12 June
2018.
The Rise and Fall of Bob Stupak and Las Vegas' Stratosphere Tower
- R J Smart
- John Smith
- No Limit
Smart, R.J., Playing Roulette as a Business. Carol Publishing Group, Secaucus, NJ, 1996.
Smith, John, No Limit: The Rise and Fall of Bob Stupak and Las Vegas' Stratosphere Tower.
Huntington Press, Las Vegas, 1997.
The Big Book of Blackjack
- Arnold Snyder
Snyder, Arnold, The Big Book of Blackjack. Cardoza Publishing, New York, 2006.
Snyder, Arnold, Inside the Over/Under. Blackjack Forum IX, #3, September 1989, p. 5.
Surprise Party at the Klondike Hotel
- Arnold Snyder
Snyder, Arnold, Surprise Party at the Klondike Hotel. Blackjack Forum XVI #3, Fall 1996, pp.
9-17.
There's Gold in Them Thar Hills!
- Arnold Snyder
Snyder, Arnold, There's Gold in Them Thar Hills!. Blackjack Forum XX, #4, Winter 2000, pp.
7-14.
The Unbalanced Over/Under
- Arnold Snyder
Snyder, Arnold, The Unbalanced Over/Under. Blackjack Forum XI, #1, March 1991, pp. 7-11.