Blackjack: A Beatable Game David Parker Advisor: Dr. Wyels California Lutheran University ‘05

Why is Blackjack Beatable? Only game in a casino where the probabilities change from game to game. If a player can take full advantage of favorable probabilities, they might be able to win more money then the dealer over a period of time.

Rules of Blackjack Player(s) vs. Dealer Object: Closest to 21 without going over Card Values Face Cards = 10 Aces = 1 or 11 (Player’s choice) 2,3,4,5,6,7,8,9,10 = Numerical value of card drawn.

Rules of Blackjack Player Dealer

Basic Strategy S = Stand H = Hit D = Double Down P = Split Pair Dealer Card Up Dealer Card Up Player Player

How to Count Cards Dr. Edward Thorp (1962) High cards are good for the player. Card Counting Cards 2,3,4,5,6 are worth +1 Cards 10,J,Q,K,A are worth -1 Cards 7,8,9 are neutral and are worth 0 Player keeps a running total of cards played in their head. Once the deck is reshuffled the count is reset to zero.

The Truecount Julian H. Braun (1964) A high count becomes more beneficial to the player as the number of cards played increases. A truecount of +8 after 8 cards have been played: A truecount of +8 after 44 cards have been played:

Truecount (Cont.) Player still keeps track of count. Player keeps track of total number of cards played. Complete Count = Count divided by the number of decks have not been completely exhausted. Truecount = Floor (Complete Count).

Maple Simulation Dealer Card Up Player Cards Final Player Cards Outcome Count Probability of winning at count Number of Cards Played Truecount Probability of Winning at Truecount

6 Deck Shoe

Betting Strategies Thorp – Bet Count Braun – Bet Truecount Hi-Low When the truecount is in the player’s favor (>2), bet 20 chips, otherwise bet 1 chip. MIT Team Pick a betting unit. When there is a favorable truecount (>2), bet the [truecount x (betting unit)]. Otherwise bet half the betting unit.

Maple Simulation Dealer Card Up Player Cards Final Player Cards Outcome Count Probability of winning at count Number of Cards Played Truecount Probability of Winning at Truecount Betting Consistently Thorp Braun Hi-Low MIT Blackjack Team Amount Bet Amount Won/Lost Total amount Won/Lost

Maple Simulation (Cont.) Study was conducted with the same rules as if we were playing at a 5 dollar minimum Las Vegas blackjack table. 6 deck shoe. Single player vs. dealer. Trials of 500 hands 500 hands takes between 7.5 – 10 human hours to play.

Normal Distributions 10,000 trials of 500 hands -10.41 0.55 -5.87 -7.59 6.09

Max Amount Won 10,000 Trials of 500 Hands 95% Confidence Intervals

Conclusions 6.09

Conclusions Hi-Low strategy wins the most money. Chances of getting caught are high. High Standard Deviation. Need to buy 860 Chips.

0.55

Conclusions Hi-Low strategy wins the most money. Chances of getting caught are high. High Standard Deviation. 860 Chips to Play. MIT Strategy is the only other strategy in which the player wins money Proven to work. Good Standard Deviation. 366 Chips to Play.

Conclusions Not many chips (0.55) earned for number of hours spent playing (7-10 hours). Dealers are taught the betting strategies to spot card counters. Casinos take measures to improve their odds. Not allowing the player to double down with certain hands. Dealer has to hit on 17. Reshuffling with cards left in the shoe.

However…. Player has a 0.13% edge on the dealer! 0.0013*500 = 0.65 Better than all 6-deck strategies with the exception of the Hi-Low Method. Recommendation: learn basic strategy and find a 1-deck game that reshuffles after every hand!

Further Studies Rules Variations Play at numerous tables. Player is allowed to re-split aces. Blackjack pays 6-5 instead of 2-1. Play at numerous tables. Increase the number of players. Various other card counting strategies. Write an NSF grant to obtain funding to test findings in a Casino setting.

References Baldwin, Roger, Wilbert Cantey, Herbert Maisel, and James McDermott. "The Optimum Strategy to Blackjack." Journal of the American Statistical Association 51.275 (1956): 429-439. Manson, A.R., A.J. Barr, and J.H. Goodnight. "Optimum Zero-Memory Strategy and Exact Probabilities for 4-deck Blackjack." The American Statistician May 1975: 84-88. Mezrich, Ben. Bringing Down the House. 1st ed. New York: Free Press, 2003. Millman, Martin. "A Statistical Analysis of Casino Blackjack." The American Mathematical Monthly Aug - Sep 1983: 431-436. Tamhane, Ajit, and Dorothy Dunlop. Statistics and Data Analysis. Upper Saddle River: Prentice Hall, 2000. Thorp, Edward. "A Favorable Strategy for twenty-one." Proc Natl Acad Sci Jan 1961: 110–112. Thorp, Edward. Beat the Dealer. 2nd ed. New York: Random House, 1966. Thorp, Edward. The Mathematics of Gambling. 1st ed. New York: Gambling Times, 1985. Larsen, Richard, and Morris Marx. An Introduction to Mathematical Statistics and its Applications. 2nd ed. Eaglewood Cliffs: Prentice Hall, 2000.

Special Thanks! Dr. Cindy Wyels – California Lutheran University. Dr. Karrolyne Fogel – California Lutheran University. Dr. David Kim – Manhattan College. Larry Coaly – California Lutheran University. Bryan Parker – University of California Los Angeles.