An investigation of the mathematics of casino gaming particularly how quantities like house advantage, expected value, win, hold, drop, and hold percentage are used by casinos.
2. The 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?
3. “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.
4. 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…
5. Probability & 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.
6. Probability & Roulette (cont.)
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
7. 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
Let’s take a look at how the laws of
probability apply to a certain roulette
wager - the corner bet
8. 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.
9. 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.
11. House 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.
14. Expected Value & House Advantage
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”.
15. Expected Value & House Advantage
(cont.)
The expected value is given by the formula
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.
16. Expected Value & House Advantage
(cont.)
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.
House advantage =
-Expected value
Total Wager
×100%
=
0.0526
1
×100% = 5.26%
18. Handle vs. Drop
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.
Instead, casinos pay attention to the “drop” -
the total money spent on chips or lost in cash.
19. Hold 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%.
20. What Constitutes Winning?
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:
Win = Amount Bet × House Advantage
= (Rate of Play × Playing Time × Average Bet)× House Advantage
21. 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? ♣
22. 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.
(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.
23. What Factors Affect Win?
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
Duration of play
Casino’s amenities and service at the table
24. Summary
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.
25. References
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>.
26. References
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>.
