casino math

Casino Mathematics

(How Casinos Make Money)

Casino Mathematics

(How Casinos Make Money)• Probabilities• Law of Large Numbers• Expected Value / House Advantage• Standard Deviation • Confidence Limits• The Normal Curve• Game Volatility / Volatility Index• Table Win/Loss Probabilities • SM Jackpot Probabilities• Exit Strategies • Gambler’s Ruin• Rebates and Discounts (Marketing)

• Probabilities• Law of Large Numbers• Expected Value / House Advantage• Standard Deviation • Confidence Limits• The Normal Curve• Game Volatility / Volatility Index• Table Win/Loss Probabilities • SM Jackpot Probabilities• Exit Strategies • Gambler’s Ruin• Rebates and Discounts (Marketing)

A casino is a mathematics palace set up to separate players from their money. Every bet made in a casino has been calibrated within a fraction of its life to maximize profit while still giving the players the illusion that they have a chance.

A casino is a mathematics palace set up to separate players from their money. Every bet made in a casino has been calibrated within a fraction of its life to maximize profit while still giving the players the illusion that they have a chance.

Nicholas Pillegi, Casino (1995)Nicholas Pillegi, Casino (1995)

ADVANCES IN CASINO GAMINGADVANCES IN CASINO GAMING

Table Game Slot Machine

Player Rating Systems Slot TrackingDynamic Reporting Systems Virtual ReelsChipper Champs TokenizationWinning Result Displays Multi-line GamesCS, LIR and Super Bucks Jackpots Feature GamesE.S.S. Mystery Jackpots“Madness 21” – Interactivity JackpotsAutomatic Shufflers Bonusing

MessagingAuto-pagingNote Acceptors

Table Game Slot Machine

Player Rating Systems Slot TrackingDynamic Reporting Systems Virtual ReelsChipper Champs TokenizationWinning Result Displays Multi-line GamesCS, LIR and Super Bucks Jackpots Feature GamesE.S.S. Mystery Jackpots“Madness 21” – Interactivity JackpotsAutomatic Shufflers Bonusing

MessagingAuto-pagingNote Acceptors

Comparative Experiences and “Value” Derived

Table Game Player Slot Player

Product Gaming entertainment Gaming entertainment

Cost $90 over 2 hours (e.g.) $90 over 2 hours (e.g.)

Benefits $3 in points $9 in points

$4 - $8 bounce back

$5 snack voucher

May double or treble stake May experience significant win

(in credits)

Chance to win jackpots including

cars

Involvement in sporadic

promotions often not limited

to members

Involvement in regular EGM

promotions limited to members

Total Benefits $3 and limited $20 and extensive

Other benefits associated with Slot play include ease of play, low limits, low bankroll, automaticrecognition and experience that even when you lose you have grown your point balance.

Probability is the very guide of life. Probability is the very guide of life.

Cicero (106-48BC)Cicero (106-48BC)

Probability is like a wave. Because of the house advantage, over time the player dips lower and lower until he stops crossing the mid-point and ultimately loses all his money, unless he quits first.

Probability is like a wave. Because of the house advantage, over time the player dips lower and lower until he stops crossing the mid-point and ultimately loses all his money, unless he quits first.

Jeff Marcum in Temples of Chance, (1992) Jeff Marcum in Temples of Chance, (1992)

Basic Probability Rules:

1. The probability of an event will always be between 0 and 1.

2. The probability of an event occurring plus the probability of an eventnot occurring equals I. (Complement Rule)

3. For mutually exclusive events, the probability of at least one of theseevents occurring equals the sum of their individual probabilities.(Addition Rule)

4.a. For independent events, the probability of all of them occurring equals the product of their individual probabilities. (Multiplication Rule – dependent events)

4.b. For non-independent events, the probability of all of them occurring equals the product of their conditional probabilities, where the conditional probability of one event is affected by the event(s) that came before it. (Multiplication Rule – dependent events)

How Do Casinos Make Money? How Do Casinos Make Money?

Retailer: Revenue = Quantity x Price

Retailer: Revenue = Quantity x Price

Casino:

1. Theoretical Win = Handle x House Advantage

Handle = Ave. Bet x Hands per Hour x Hours Playedor

Handle = Ave. Bet x No. of Hands Played

2. Actual Win = End. Capital + Cash Sales + Excess Chips + Chip Yield – Table Refill – Beg. Capital

where:

How Do Casinos Make Money?

EXPECTED WIN

DEAL PLAYER BANK CUSTOMERS HOUSE Handle X H.A.

1 1,000 1,000 2,000 25.4

2 1,000 1,000 1,950 50 25.4

3 1,000 1,000 2,000 25.4

4 1,000 1,000 1,950 50 25.4

5 1,000 1,000 2,000 25.4

6 1,000 1,000 1,950 50 25.4

7 1,000 1,000 2,000 25.4

8 1,000 1,000 1,950 50 25.4

9 1,000 1,000 2,000 25.4

10 1,000 1,000 1,950 50 25.4250 250

FLOW OF CHIPS

Assumption: Baccarat H.A. = 1.27% (excluding ties)

Game Location Rate Advantage

Game Location Rate Advantage BJ CGA 63 1.30 %BJ IR 76 1.30 AR CGA 36 2.70AR IR 33 2.70MB CGA 58 1.26MB IR 53 1.26BA IR 41 1.26TWO UP CGA 45 3.13CRAPS CGA 50 1.50BIG WHEEL CGA 42 7.69B&S CGA 46 2.78KENO CGA 1 23.00

Comp value = comp % x (theoretical win – tax – staff cost) Comp % = 50% standardTheoretical win = ave. bet x hands/hour x time played x edgeTax % = 20%Staff cost = $10 / hour of recorded play

Decision House

ADELAIDE CASINO System Standards

100 55 0.550 5 0.050

1,000 525 0.525 25 0.025

10,000 5,100 0.510 100 0.010

100,000 50,500 0.505 500 0.005

1,000,000 501,000 0.501 1,000 0.001

FAIR COIN PROBABILITIESHEAD = 50% TAIL = 50%

No. of Tosses

No. of Heads

Percent Heads

Deviation - No. of Heads

Deviation - Percent Heads

Law of Large Numbers

Theoretical Actual TotalMonth Win Win Bets Theoretical Actual

Jan 988,131 3,886,299 85,924,400 0.011500 0.045229 Feb 609,621 1,325,950 53,010,483 0.011500 0.037516 Mar 672,802 1,601,550 58,504,515 0.011500 0.034511 Apr 1,317,763 1,810,200 114,588,067 0.011500 0.027639 May 1,116,195 2,452,000 97,060,412 0.011500 0.027075 Jun 1,758,309 2,677,100 152,896,417 0.011500 0.024472 Jul 3,261,150 4,678,600 283,578,245 0.011500 0.021798

Aug 1,166,140 2,067,000 101,403,467 0.011500 0.021647 Sep 1,270,092 876,500 110,442,778 0.011500 0.020215 Oct 1,389,633 (190,700) 120,837,683 0.011500 0.017980 Nov 1,571,716 5,173,900 136,670,943 0.011500 0.020046 Dec 1,264,212 2,119,250 109,931,492 0.011500 0.019986

Cumulative Win Rates

CF-BacolodBaccarat Win Rate (Theoretical vs. Actual)

January to December, 2005

Note: 9 tables

-20.00%

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

NO. OF DEALSNO. OF DEALS

Law of Large NumbersWin Percent

Law of Large NumbersWin Percent

100100 1,0001,000 2,0002,000 10,00010,000 100,000100,000

Assumption: P 1.00 bet per hand

Upper Limit (95%)

Theoretical Win

Lower Limit (95%)

Theoretical Win

Upper Limit (95%)

Lower Limit (95%)

5,000

5,000

10,000 10,000 15,000 15,000 20,000 20,000 25,000 25,000 30,000 30,000 35,000 35,000 40,000 40,000 45,000 45,000 50,000 50,000 55,000 55,000


(.20)(.20)

00

1010

2020

3030

40 40

5050

6060

IN M

ILL

ION

PE

SO

SIN

MIL

LIO

N P

ES

OS

Assumption: P1.00 bet per handAssumption: P1.00 bet per hand

Law of Large NumbersWin Amount

Law of Large NumbersWin Amount

Expected Value or House Advantage

where, Net Pay = net payoff

P = probability of net pay

HA =EV

wagerX 100

EV = Σ (Net Payi x P)

Formulas:

Type of Bet Probability True Odds Payoff Odds

1 Number 1/38 37 to 1 35 to 12 Numbers 2/38 36 to 2 17 to 13 Numbers 3/38 35 to 3 11 to 14 Numbers 4/38 34 to 4 8 to 15 Numbers 5/38 33 to 5 6 to 16 Numbers 6/38 32 to 6 5 to 1Dozens/Columns (12 Numbers) 12/38 26 to 12 2 to 1Red/Black/Odd/Even/High/Low 18/38 20 to 18 1 to 1 (18 Numbers)

DOUBLE ZERO ROULETTE

4

1 1

2

8

1

1

BACCARAT BACCARAT . Probabilities .

Including Draw Excluding DrawBANKER 0.4585974 0.5068248PLAYER 0.4462466 0.4931752DRAW 0.0951560 –

. Probabilities . Including Draw Excluding DrawBANKER 0.4585974 0.5068248PLAYER 0.4462466 0.4931752DRAW 0.0951560 –

SUPER 6 . Probabilities .

Including Draw Excluding Draw

BANKER 0.404518 0.447222

PLAYER 0.446123 0.493220

DRAW 0.095488 –

SUPER 6 0.053871 0.059558

Table Occupancy and Productivity

Definition of terms :

Table Occupancy – the ratio of the number of playing customers to the total number of table betting slots

Table Utilization – the ratio of the utilized time of the table to the total time the table is open

Comparative Analysis Casinos A & B

Casino A - 28 tables at 1 customer per table Casino B - 4 tables at 7 customers per table

Assumptions: 1. Both casinos have 28

customers.2. Table occupancy:

4. Table utilization for both casinos is 100%.

3. Standard no. of deals per hour:

Casino A - 60Casino B - 30

Bet per hand P 300 P 300Hands per hour per table x 60 x 210

18,000 63,000

House Advantage x 1.15% x 1.15%

Win per hour per table 207 724.5Hours per shift x 8 x 8

Win per shift per table 1,656 5,796No. of tables x 28 x 4Gross profit/win 46,368 23,184

Operating Expenses: Salaries - Casino Shift Mgr. - - Pit Managers (2:1) 3,524 1,762 Pit Supervisors (7:1) 9,317 1,331 Dealers (42:6) 42,840 6,120 Other OPEX-Opns. (20% GP) 9,274 4,637Total Operating Expenses 64,955 13,850Net Profit before taxes (18,587) 9,334

Contribution Margin 40.09% 40.26%

Casino A (28) Casino B (4)

Bet per hand P 500 P 500Hands per hour per table x 60 x 210

30,000 105,000


Win per hour per table 345 1,207.50 Hours per shift x 8 x 8

Win per shift per table 2,760 9,660No. of tables x 28 x 4Gross profit/win 77,280 38,640

Operating Expenses: Salaries - Casino Shift Mgr. - - Pit Managers (2:1) 3,524 1,762 Pit Supervisors (7:1) 9,317 1,331 Dealers (42:6) 42,840 6,120 Other OPEX-Opns. (20% GP) 15,456 7,728Total Operating Expenses 71,137 16,941Net Profit before taxes 6,143 21,699



Bet per hand P 1,000 P 1,000Hands per hour per table x 60 x 210

60,000 210,000


Win per hour per table 690 2,415Hours per shift x 8 x 8

Win per shift per table 5,520 19,320No. of tables x 28 x 4

Gross profit/win 154,560 77,280

Operating Expenses: Salaries - Casino Shift Mgr. - - Pit Managers (2:1) 3,524 1,762 Pit Supervisors (7:1) 9,317 1,331 Dealers (42:6) 42,840 6,120 Other OPEX-Opns. (20% GP) 30,912 15,456

Total Operating Expenses 86,593 24,669

Net Profit before taxes 67,967 52,611



CONFIDENCE LIMITSand

WIN/(LOSS) PROBABILITIES

CONFIDENCE LIMITSand

WIN/(LOSS) PROBABILITIES

1. About two-thirds of the time, the actual win will be one standard deviation of the theoretical win. (68.26%)

2. About 95% of the time, the actual win will be two standard deviations of the theoretical win. (95.44%)

3. The actual win will never be more than three standard deviations from the theoretical win. (99.74%)

VOLATILITY PRINCIPLE

0-1-2-3 +1 +2 +3-3.7 +3.7

.3413

.4772

.4987

.4999

.3413.3413.1359 .1359 .0215.0215.0012 .0012

.6826.9544

.9974

.9998

The Normal Curve

0-1-2-3 +1 +2 +3-3.7 +3.7

The Normal Curve

575 T 2.06 M-910 T 3.55 M-2.40 M 5.03 M-3.88 M

.3413 .3413.1359 .1359.0215 .0215

Assumption: 1,000 hands at P50,000 per hand

Confidence LimitsWin Percent

Confidence LimitsWin Percent

100 1,000 1,500 3,000 7,500 50,000 100,000100 1,000 1,500 3,000 7,500 50,000 100,000

-30%

-25%

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

25%

30%

NO. OF DEALS

-400,000

-200,000

0

200,000

400,000

600,000

800,000

1,000,000

1,200,000

1,400,000

5,0005,000 10,00010,000 15,00015,000 20,00020,000 25,00025,000 30,00030,000 35,00035,000 40,00040,000 45,00045,000 50,00050,000 55,00055,000


Confidence LimitsWin Amount

Confidence LimitsWin Amount

Formula:

Varwager= [(Net Payi – EV)2 x Pi]

SDwager= Varwager

where, EV = expected value Net Pay = net payoff P = probability of net pay

Standard DeviationStandard Deviation

3 Ways of Examining Fluctuations or Volatility

3 Ways of Examining Fluctuations or Volatility

Win Percent EVwin % = EVpu

SDpu

SDwin % = ----------

√ n

Win Amount Evwin = unit wager x n x EVpu

SDwin = unit wager x √ n x SDpu

Win Units EVunits = n x EVpu SDunits = √ n x SDpu

To determine Confidence Limits:

EV ± ( Z x SDwin )

where Z = standard normal value depending on the confidence level

Actual Play at 95% Confidence LimitActual Play at 95% Confidence Limit

-15,000,000

-10,000,000

-5,000,000

0

5,000,000

10,000,000

15,000,000

1 2 3 4 5 6 7 8 9 1 11

6363 124124 186186 235235 297297 361361 429429 491491 554554 614614

No. OF DEALSNo. OF DEALSAverage Bet = P250,000Average Bet = P250,000

-15,000,000

-10,000,000

-5,000,000

0

5,000,000

10,000,000

15,000,000

20,000,000

1 2 3 4 5 6 7 8 9 1 11

Actual Play at 97% Confidence LimitActual Play at 97% Confidence Limit

6363 124124 186186 235235 297297 361361 429429 491491 554554 614614

No. OF DEALSNo. OF DEALS

Average Bet = P250,000Average Bet = P250,000

Questions to ask if actual win falls outside normal confidence levels:

1. Is the data correct?

2. Is there an extraordinary event that caused the

deviation?

3. Is there a mechanical or personnel error?

4. Are players cheating you?

5. Are your employees stealing from your casino?

Determining Win/(Loss) Probabilities using as example a “Freeze-Out” Game

In February 1990, a Japanese whale won $6 M from the Trump Plaza in Atlantic City and $19 M at the Diamond Beach Casino in Darwin, Australia. Upon returning to the Trump Plaza in May, the casino accepted the whale’s challenge to play Baccarat at $200,000 per hand on the condition that he do so until he was either ahead $12 M or behind $12 M. The game lasted 70 hours with a total of 5,600 hands dealt. Compute:

1. the probability that the casino willa. lose > 12 Mb. win > 12 M

2. the probability that the casino willa. lose > 24 M h. win 12 to 18 Mb. lose 24 to18 M i. win 18 to 24 Mc. lose 18 to12 M j. win 24 to 30 Md. lose 12 to 6 M k. win 30 to 36 Me. lose 6 to 0 M l. win 36 to 42 Mf. win 0 to 6 M m. win > 42 Mg. win 6 to 12 M

GAMBLER’S RUIN CONCEPT

GAMBLER’S RUIN CONCEPTThis is the simple proposition that all

other things being equal, the gambler with more money in a bust out game, i.e., where play continues until you win or lose all your money, is more likely to prevail over the gambler with less money. Of course, casinos not only have more money than almost all players, they also possess a house advantage to assure that all things are not equal…

Gambler’s Ruin Formula:

P (success) =

P (ruin) =

p

q

p

q

n

a

1

1

p

q

p

q

p

q

n

na

1

Double or Nothing Formula:

P (doubling before ruin) =

qpp

aa

a

P (ruin before doubling) =

qpq

aa

a

2. Dead Chip Program 2. Dead Chip Program

REBATES AND DISCOUNTS

Effective H.A. = H.A. − )1( R

R

x Plose

where:H.A. = the normal house advantageR = rebate percentagePlose = probability of losing the wager

1. Rebate on (Player’s) Theoretical Loss

Comp Value = average bet x decisions per hour x hours played x HA x comp rate

3. Rebate on Actual Loss3. Rebate on Actual Loss

REBATES AND DISCOUNTS

where:N = number of handsHA = house advantageLE = theoretical loss equivalencySD = wager standard deviationz = (N x HA) / ( x SD)UNLLI (z) = unit normal linear loss integral for z

Equivalent Rebate = ])([)( SDxNxzUNLLIHAxN

LExHAxN

N

PAYOUT HITS PAYOUT HITS PAYOUT HITS0 566966 0 879816 0 7268172 296827 2 300 2 1289245 20624 5 300 5 1425610 50 10 300 10 984025 50 25 400 25 240040 50 40 400 40 80050 40 50 500 50 500100 30 100 800 100 320150 25 150 600 150 240180 20 200 400 200 180200 15 300 400 250 240300 12 500 300 300 100400 10 1000 200 500 50500 8 2000 10 750 321000 5 3000 6 1250 241200 3 5000 3 1500 12100000 1 10000 1 2000 1

Cycle 884,736 Cycle 884,736 Cycle 884,736Hits 317,770 Hits 4,920 Hits 157,919

Payback % 94.50% Payback % 94.84% Payback % 94.66%Hit Freq. 35.92% Hit Freq. 0.56% Hit Freq. 17.85%

V.I. 175.57 V.I. 43.64 V.I. 21.04

PAYOUT DISTRIBUTIONS FOR 3 SLOT MACHINES

SLOT MACHINE 1 SLOT MACHINE 2 SLOT MACHINE 3

1 94.947% 36.80% 325,570

PAYOUT HITS

0 559,166

2 295,827

5 28,256

10 400 Handle Pulls Lower % Upper %

25 300 1,000 60.19 129.70

40 240 10,000 83.96 105.94

50 180 100,000 91.47 98.42

100 150 1,000,000 93.85 96.05

150 100 10,000,000 94.60 95.29

200 50

250 20

300 15

500 12

750 10

1,250 6

1,500 3

2,000 1

TOTAL 884,736

PAR SHEET SUMMARY DATA

COIN #PERCENT PAYBACK

HIT FREQ TOTAL HITS TOTAL PAYS

840,034

Reel Strip Number 1A HOLD % 5.053

MODEL # : HYPO - A PAYTABLE XYZ - A

Reels: 3 Stops: 96 Reel Combos: 884,736

VOLATILITY INDEX = 10.990

90% CONFIDENCE VALUES

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