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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
NO. OF DEALSNO. OF DEALS
(.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
House Advantage x 1.15% x 1.15%
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
Contribution Margin 7.95% 56.16%
Casino A (28) Casino B (4)
Bet per hand P 1,000 P 1,000Hands per hour per table x 60 x 210
60,000 210,000
House Advantage x 1.15% x 1.15%
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
Contribution Margin 43.97% 68.08%
Casino A (28) Casino B (4)
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
NO. OF DEALSNO. OF DEALS
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