Blackjack: A Beatable Game

Amber GuoAdapted from:David Parker

Advisor: Dr. WyelsCalifornia 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

2 3 4 5 6 7 8 9 10 A

5 H H H H H H H H H H 6 H H H H H H H H H H 7 H H H H H H H H H H 8 H H H H H H H H H H 9 H D D D D D H H H H

10 D D D D D D D D H H 11 D D D D D D D D D H 12 H H S S S S H H H H 13 S S S S S S H H H H 14 S S S S S S H H H H 15 S S S S S S H H H H 16 S S S S S S H H S H 17 S S S S S S S S S S 18 S S S S S S S S S S 19 S S S S S S S S S S 20 S S S S S S S S S S 21 S S S S S S S S S S

2 3 4 5 6 7 8 9 10 A

2-2 P P P P P P H H H H 3-3 P P P P P P H H H H 4-4 H H H H H H H H H H 5-5 D D D D D D D D H H 6-6 H P P P P P H H H H 7-7 P P P P P P H H H H 8-8 P P P P P P P P P P 9-9 P P P P P P P P S S

10-10 S S S S S S S S S S A-A P P P P P P P P P P A2 H H H D D H H H H H A3 H H H D D H H H H H A4 H H D D D H H H H H A5 H H D D D H H H H H A6 H D D D D H H H H H A7 S D D D D S S H H S A8 S S S S S S S S S S A9 S S S S S S S S S S

A10 S S S S S S S S S S

Basic Strategy

Pla

yer

Pla

yer

Dealer Card UpDealer Card Up

S = Stand

H = Hit

D = Double Down

P = Split Pair

Example x2

Player Dealer

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.

True Count 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.

True count = Floor (Complete Count).

True Count (Cont.) Higher the true count, the greater the

advantage to the player 6 deck shoe

Shuffled running count = 0, dealer advantage of 0.5%

+1 count even with the dealer Each +1 after +1 count, additional 0.5% player

advantage

True Count in Blackjack Strategy True count < or = +1, bet base amount True count +2 or +3, bet 2 times base unit True count +4 or +5, bet 3 times base unit True count +6 or +7, bet 4 times base unit True count > or = +8, bet 5 times base 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

1 Deck Shoe500 trials of 20,000 hands42

.21% 43

.75%

44.5

1%

45.6

7% 47.1

8%

47.9

4%

47.6

4%

47.8

6%

48.6

7%

49.3

4%

49.5

3%

52.2

3%

50.6

8%

49.9

9%

47.9

7%

47.8

1%

47.5

0%

47.2

8%

46.8

3%

45.9

1%

45.5

1%

49.1

7%

35%

40%

45%

50%

55%

≤-5 -4 -3 -2 -1 0 1 2 3 4 ≥5

True Count

Win

nin

g P

ecen

tag

e

Player Dealer

Count vs. Truecount (Player's Edge)

y = -2E-06x3 - 0.0002x2

+ 0.0082x - 0.0132

y = -1E-05x3 - 5E-05x2 + 0.0036x - 0.0172

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-10 -8 -6 -4 -2 0 2 4 6 8 10

Edge

Cou

nt

Count Truecount

6 Deck Shoe

Betting Strategies Bet Count 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 Distributions10,000 trials of 500 hands

-400 -300 -200 -100 0 100 200 300 400

Number of Chips WonNot Counting Thorp Braun MIT Blackjack Team Hi-Low

-10.41

-5.870.55

-7.59 6.09

Max Wins and Losses10,000 Trials of 500 Hands

-859.5

-240.5

-746.5

-366

Not Counting Braun

Hi - Low

MIT Team

Thorp

-96

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

Ch

ips

Max Amount Won10,000 Trials of 500 Hands

Not Counting

Braun

Thorp

HI - Low

MIT Team

0 20 40 60 80 100 120

Chips

95% Confidence Intervals

Conclusions

Normal Distributions

-400 -300 -200 -100 0 100 200 300 400

Number of Chips WonNot Counting Thorp Braun MIT Blackjack Team Hi-Low

6.09

Conclusions Hi-Low strategy wins the most money.

Chances of getting caught are high. High Standard Deviation. Need to buy 860 Chips.

Normal Distributions

-400 -300 -200 -100 0 100 200 300 400

Number of Chips WonNot Counting Thorp Braun MIT Blackjack Team Hi-Low

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.

Double-Deck Basic Strategy

We will be playing with Hits, Stands, Doubles, and Splits. Double after Splits are allowed. Surrendering is not allowed.