Casino MathematicsCasino Mathematics

Or why they keep building casinos

Or why they keep building casinos

The Odds Are StackedThe Odds Are Stacked

The first principle of casino gaming is “the house shall always win”

The house stacks the odds in its favor The casino counts on the fact that, in the long

run, players will lose more money than they make. Even very shrewd players struggle just to break even.

How can casinos insure they make money off gamblers regardless of the skill of the player?

The first principle of casino gaming is “the house shall always win”

The house stacks the odds in its favor The casino counts on the fact that, in the long

run, players will lose more money than they make. Even very shrewd players struggle just to break even.

How can casinos insure they make money off gamblers regardless of the skill of the player?

“There is no such thing as luck.It is all mathematics”

- Nico Zographos, Legendary Baccarat Player

“There is no such thing as luck.It is all mathematics”

- Nico Zographos, Legendary Baccarat Player

The entire basis of the casino industry’s wealth does not lie in its glitter or extravagance.

The real driving force behind the success of casino gaming is basic probability and algebra.

The entire basis of the casino industry’s wealth does not lie in its glitter or extravagance.

The real driving force behind the success of casino gaming is basic probability and algebra.

What is probability?What is probability?

The likelihood that a certain event occurs.

For particular casino games, there are specific probabilities for different outcomes.

To take a familiar example, the probability of flipping two coins and both coming up heads is 1/4 or 25%.

Let’s look at an example of probability in the casino…

The likelihood that a certain event occurs.

For particular casino games, there are specific probabilities for different outcomes.

To take a familiar example, the probability of flipping two coins and both coming up heads is 1/4 or 25%.

Let’s look at an example of probability in the casino…

Probability & RouletteProbability & Roulette

Roulette (“little wheel”) is a game popularized in 19th century France. A ball is dropped in the counterclockwise direction onto a wheel spinning clockwise. On the wheel are 38 numbered slots 1-36 and 0 and 00. The players place bets on a grid that contains all the numbers on the wheel. The ball stops rolling and lands in one of the slots. The croupier pays off the winning bets and rakes in the losing bets.

Roulette (“little wheel”) is a game popularized in 19th century France. A ball is dropped in the counterclockwise direction onto a wheel spinning clockwise. On the wheel are 38 numbered slots 1-36 and 0 and 00. The players place bets on a grid that contains all the numbers on the wheel. The ball stops rolling and lands in one of the slots. The croupier pays off the winning bets and rakes in the losing bets.

Probability & Roulette (cont.)Probability & Roulette (cont.)

The probability P of an event E where each outcome is equally likely and mutually exclusive is given by

The probability P of an event E where each outcome is equally likely and mutually exclusive is given by

€

P(E) = # of outcomes favorable to E

total number of outcomes

Probability & Roulette (cont.)Probability & Roulette (cont.)

There are 38 slots on the roulette wheel and thus there are 38 possible outcomes each of which is equally likely. Thus, the probability of getting any of the 38 numbers is 1/38

There are 38 slots on the roulette wheel and thus there are 38 possible outcomes each of which is equally likely. Thus, the probability of getting any of the 38 numbers is 1/38

Let’s take a look at how the laws of probability apply to a certain roulette wager - the corner bet

Let’s take a look at how the laws of probability apply to a certain roulette wager - the corner bet

What is a corner bet?What is a corner bet?

A corner bet is a wager in which a player places a chip at the intersection of four adjacent squares.

The player is betting on any of the four numbers to hit.

In the picture on the left, the player has money on 4, 5, 7, and 8.

A corner bet is a wager in which a player places a chip at the intersection of four adjacent squares.

The player is betting on any of the four numbers to hit.

In the picture on the left, the player has money on 4, 5, 7, and 8.

What is the probability of winning a corner bet?

What is the probability of winning a corner bet?

Since the probability of the ball landing on any number is 1/38 the probability of the ball landing on either 4, 5, 7 or 8 is 4/38 = 2/19 or 10.53%.

The probability of not winning is 34/38 = 17/19 or 89.47%.

The odds are 17:2 and the payout is 8:1 which means that you will make $8 if you hit on a $1 bet.

We will revisit this example but first let’s introduce some new terminology that will help us translate these probability statements into dollars and cents.

Since the probability of the ball landing on any number is 1/38 the probability of the ball landing on either 4, 5, 7 or 8 is 4/38 = 2/19 or 10.53%.

The probability of not winning is 34/38 = 17/19 or 89.47%.

The odds are 17:2 and the payout is 8:1 which means that you will make $8 if you hit on a $1 bet.

We will revisit this example but first let’s introduce some new terminology that will help us translate these probability statements into dollars and cents.

House Advantage & Expectation Value

House Advantage & Expectation Value

House AdvantageHouse Advantage

House advantage (also called “house edge”) is the long-term advantage the casino holds over the player.

House advantage quantifies the fundamental advantage that the casino holds over the gambler. Crucially, it’s the one factor that the casino has total control over.

Quantitatively, it is the ratio of the average loss to the initial bet.

House advantage (also called “house edge”) is the long-term advantage the casino holds over the player.

House advantage quantifies the fundamental advantage that the casino holds over the gambler. Crucially, it’s the one factor that the casino has total control over.

Quantitatively, it is the ratio of the average loss to the initial bet.

The following brief video provides some more background on house advantage…

QuickTime™ and a decompressor

are needed to see this picture.

Expected Value & House AdvantageExpected Value & House Advantage

Quantitatively, Quantitatively,

€

House advantage = Expected value of a bet

Amount of the bet

The “expected value of a bet” is known simply as expected value (E.V.) It is sometimes known as “theoretical win” or just “win”.

The “expected value of a bet” is known simply as expected value (E.V.) It is sometimes known as “theoretical win” or just “win”.

Expected Value & House Advantage (cont.)

Expected Value & House Advantage (cont.)

The expected value is given by the formula The expected value is given by the formula

Back to our corner bet example, what would be the expected value of a $1 bet?

Back to our corner bet example, what would be the expected value of a $1 bet?

€

E.V. = Payout × P(Win) - Wager × P(Loss)

€

E.V. = $8 × (2/19) - $1× (17/19) = $ - 0.0526

On a $1 bet, you expect to lose about a nickel. On a $1 bet, you expect to lose about a nickel.

Expected Value & House Advantage (cont.)

Expected Value & House Advantage (cont.)

The house advantage can now be calculated using our previous formula

The house advantage can now be calculated using our previous formula

€

In the next section, we will explore how the probabilistic elements of casino gambling can used to calculate exactly how much money can be made by the casino on a player.

In the next section, we will explore how the probabilistic elements of casino gambling can used to calculate exactly how much money can be made by the casino on a player.

€

House advantage = -Expected value

Total Wager×100%

=0.0526

1×100% = 5.26%

The Algebra of Casino Gaming

The Algebra of Casino Gaming

Win, Hold, and Drop

Handle vs. DropHandle vs. Drop Handle is the total money amount of money

gambled. Or mathematically, Handle is the total money amount of money

gambled. Or mathematically,

€

Handle = Number of Hands Player × Average Bet

In practice, it is impossible to keep track of all wagers by every player in the casino.

In practice, it is impossible to keep track of all wagers by every player in the casino.

Instead, casinos pay attention to the “drop” - the total money spent on chips or lost in cash.

Instead, casinos pay attention to the “drop” - the total money spent on chips or lost in cash.

Hold PercentageHold Percentage The hold percentage is the ratio of the chips

the casino keeps (“win”) to the drop as a percentage:

The hold percentage is the ratio of the chips the casino keeps (“win”) to the drop as a percentage:

€

Hold Percentage = Win

Drop×100%

For example, you buy $100 in chips at a craps table. The $100 is deposited into a locked box. You leave the table with $85 which means the casino “won” $15. The hold percentage in this case is 15%.

For example, you buy $100 in chips at a craps table. The $100 is deposited into a locked box. You leave the table with $85 which means the casino “won” $15. The hold percentage in this case is 15%.

What Constitutes Winning?What Constitutes Winning? In the last part of our presentation, we will examine how

casinos assess the relative profitability of different

games.

In the last part of our presentation, we will examine how casinos assess the relative profitability of different

games. Since the amount of money the casino can make off of a

gambler on a certain table game - the win or theoretical win - depends on the amount of money bet (which itself a function of the rate of play, the duration of play, and the amount of money bet on a hand) as well as the house advantage, we can create a single equation that is expresses the win as a function of rate of play, duration of play, average bet, and the house advantage:

Since the amount of money the casino can make off of a gambler on a certain table game - the win or theoretical win - depends on the amount of money bet (which itself a function of the rate of play, the duration of play, and the amount of money bet on a hand) as well as the house advantage, we can create a single equation that is expresses the win as a function of rate of play, duration of play, average bet, and the house advantage:

€

Win = Amount Bet × House Advantage

= (Rate of Play× Playing Time × Average Bet) × House Advantage

What Constitutes Winning? (cont.)

What Constitutes Winning? (cont.)

Let’s suppose the casino wants to answer the following question:

Consider two players - one playing baccarat and one playing roulette. If a casino knows how many bets per hour each player is making and how long they will play, how much would each player have to bet on average for the casino to make the same amount of money on each player?

Let’s suppose the casino wants to answer the following question:

Consider two players - one playing baccarat and one playing roulette. If a casino knows how many bets per hour each player is making and how long they will play, how much would each player have to bet on average for the casino to make the same amount of money on each player?

What Constitutes Winning? (cont.)What Constitutes Winning? (cont.) Let’s suppose the baccarat player bets at a rate of 80

hands/hr and the roulette player bets at a rate 38 hands/hr. The baccarat player bets $100 per hand. They both play for 1 hour. How much would the roulette player have to bet on average for the casino to make the same amount of money on each? We have studied this kind of problem in our class so we are well-equipped to solve it.

Let’s suppose the baccarat player bets at a rate of 80 hands/hr and the roulette player bets at a rate 38 hands/hr. The baccarat player bets $100 per hand. They both play for 1 hour. How much would the roulette player have to bet on average for the casino to make the same amount of money on each? We have studied this kind of problem in our class so we are well-equipped to solve it.

€

(80 hands/hr)(1 hr)(1.24%)($100) = (38 hands/hr)(1 hr)(5.26%)(avg. bet roulette player)

avg bet roulette player =(80 hands/hr)(1 hr)(1.24%)($100)

(38 hands/hr)(1 hr)(5.26%)= $49.63 ≈ $50

These types of calculations are used in establishing comp levels for gamblers.

These types of calculations are used in establishing comp levels for gamblers.

What Factors Affect Win?What Factors Affect Win? Rate of play (hands per hour)

Skill and experience of dealer Operating standards Effective of management and employee motivation

Rate of play (hands per hour) Skill and experience of dealer Operating standards Effective of management and employee motivation

Average bet Betting limits Programs for repeat players

Average bet Betting limits Programs for repeat players

Duration of play Casino’s amenities and service at the table

Duration of play Casino’s amenities and service at the table

SummarySummary We have explored how probability and algebra underpin

the success of the casino gaming industry and are also used for assessing profitability.

We have defined and illustrated fundamental concepts of casino gaming - house advantage, expectation value, win, handle, drop, hold, and hold percentage.

We have applied these ideas in a number of actual casino scenarios to see how they work.

We have seen the usefulness of mathematics in the casino industry and also the incredible complexity of the financial dynamics of casino gaming even when dealing with just a single gambler.

While Hollywood films show the world of Las Vegas as a parade of gangsters and showgirls, deadbeats and conmen, they never show the actual star of the show because the star of the show isn’t a person or a place - it’s mathematics.

We have explored how probability and algebra underpin the success of the casino gaming industry and are also used for assessing profitability.

We have defined and illustrated fundamental concepts of casino gaming - house advantage, expectation value, win, handle, drop, hold, and hold percentage.

We have applied these ideas in a number of actual casino scenarios to see how they work.

We have seen the usefulness of mathematics in the casino industry and also the incredible complexity of the financial dynamics of casino gaming even when dealing with just a single gambler.

While Hollywood films show the world of Las Vegas as a parade of gangsters and showgirls, deadbeats and conmen, they never show the actual star of the show because the star of the show isn’t a person or a place - it’s mathematics.

ReferencesReferences

Blackwood, Kevin. Casino gambling for dummies. Hoboken, NJ: Wiley Pub. Inc., 2006. Print.

Grabianowski, Ed. "HowStuffWorks - How Casinos Work" HowStuffWorks. HowStuffWorks, 1 May 2007. Web. 24 Apr. 2014. <http://entertainment.howstuffworks.com/casino.htm>.

Hannum, Robert. "UNLV Center for Gaming Research: Casino Mathematics." Casino Mathematics. UNLV Center for Gaming Research: , 5 June 2012. Web. 24 Apr. 2014. <http://gaming.unlv.edu/casinomath.html#he>.

Moe, Al. “How Casinos Make Money”. About.com Casino Gambling. About.com, 4 Mar. 2014. Web. 24 Apr. 2014. <http://casinogambling.about.com/od/casinos101/a/How-Casinos-Make-Money.htm>.

Blackwood, Kevin. Casino gambling for dummies. Hoboken, NJ: Wiley Pub. Inc., 2006. Print.

Grabianowski, Ed. "HowStuffWorks - How Casinos Work" HowStuffWorks. HowStuffWorks, 1 May 2007. Web. 24 Apr. 2014. <http://entertainment.howstuffworks.com/casino.htm>.

Hannum, Robert. "UNLV Center for Gaming Research: Casino Mathematics." Casino Mathematics. UNLV Center for Gaming Research: , 5 June 2012. Web. 24 Apr. 2014. <http://gaming.unlv.edu/casinomath.html#he>.

Moe, Al. “How Casinos Make Money”. About.com Casino Gambling. About.com, 4 Mar. 2014. Web. 24 Apr. 2014. <http://casinogambling.about.com/od/casinos101/a/How-Casinos-Make-Money.htm>.

ReferencesReferences

Provost, Gary. High stakes: inside the new Las Vegas. New York: Truman Talley Books/Dutton, 1994. Print.

Siva, Rama. "Introduction to Casino Mathematics." Introduction to Casino Mathematics. SlideShare, 3 Oct. 2011. Web. 24 Apr. 2014. <http://www.slideshare.net/ramasiva1/introduction-to-casino-mathematics>.

Provost, Gary. High stakes: inside the new Las Vegas. New York: Truman Talley Books/Dutton, 1994. Print.

Siva, Rama. "Introduction to Casino Mathematics." Introduction to Casino Mathematics. SlideShare, 3 Oct. 2011. Web. 24 Apr. 2014. <http://www.slideshare.net/ramasiva1/introduction-to-casino-mathematics>.