### Product: Games and Statistics

```CHAPTER 9
PRODUCT: GAMES and
STATISTICS
Robert C. Hannum
University of Denver
Hashimoto: Casino Management: A
Strategic Approach
Introduction
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Overview of Casino Games
Some casino games are pure chance.
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No amount of skill or strategy can alter the odds
Roulette, craps, baccarat, keno, the big-six wheel
of fortune, slot machines
Some games have an element of skill.
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An element of skill can affect the house advantage
Blackjack, video poker, and the four popular pokerbased table games: Caribbean Stud poker, Let It
Ride, Three Card poker, and Pai Gow poker
Hashimoto: Casino Management: A
Strategic Approach
Summary of Major Casino Games
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Roulette: players try to guess which number
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Two types of wheels
will occur as the outcome of the spin of a ball
around a numbered wheel.
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The double-zero, common in the United States
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The single-zero, favored in Europe
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contains 38 pockets, numbered 1 thru 36, 0, & 00
18 red, 18 black; 0 and 00 are green
lacks 00; has only 37 numbers
Game is essentially the same regardless of which
wheel is used.
that of the double-zero.
Hashimoto: Casino Management: A
Strategic Approach
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Craps: betting on the outcome of roll of dice.
“Roll” determined as the sum of values of the
two dice thrown (except for certain bets).
Wagers have myriad possibilities; each type has
different odds, payoff, and house advantage.
 The Pass Line – first roll of game or round is
called the “come-out” roll. Pass line wager wins
even money if come-out roll is a 7 or 11; loses if it
is a 2, 3, or 12 (called craps).
 Odds – can be taken after a point has been
established on the pass line wager; can also be
taken on “come,” or given on “don’t pass” and
“don’t come” after a point is established.
Hashimoto: Casino Management: A
Strategic Approach

Craps: Wagers (cont’d)
 Come
Bet – essentially a pass line bet on any roll
 Don’t
Pass Bet – works opposite of pass line bet,
after a point is established. It turns the next roll
into a come-out roll that wins on a 7 or 11, loses
on a 2, 3, or 12, and otherwise establishes a point.
except winning is barred on come-out roll of 12.
Bettor wins on come-out roll of 2 or 3, loses on a 7
or 11, ties on a 12. On point rolls, wins if 7 is
rolled before the point, loses if point is rolled
before a 7. Like the pass line bet, odds can be
given once a point is established.
Hashimoto: Casino Management: A
Strategic Approach

Craps: Wagers (cont’d)

Don’t Come Bet – essentially a don’t pass bet made
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Buy & Lay Bets – similar to place, don’t place bets
after a point has been established.
 Place & Don’t Place Bets – a place bet on one of
the point numbers is a bet that the number will roll
before a 7.
--A don’t place bet is the opposite: a bet a 7 will roll
before the number.
--Payoffs for both are less than true odds.
except that the payoffs are at true odds. A 5%
commission is charged on the amount of a buy bet
and on the possible winning amount of a lay bet.
Hashimoto: Casino Management: A
Strategic Approach

Craps: Wagers (cont’d)

Field Bet – next roll will be 2, 3, 4, 9,10, 11,12.
Payoff: 2 to 1 for 2 or 12, even on 3, 4, 9, 10, or 11.

Big 6 & Big 8 – bet that 6 will roll before a 7, or that an 8
will roll before a 7.
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Hardways – wager that selected number will roll with
doubles before any other way. Payoff: 7 to 1 for hard 4 & 10;
9 to 1 for hard 6 & 8.
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Any Craps – one-roll bet that the next roll will be 2, 3, or,
12. If not, the wager is lost. Payoff: 7 to 1.
Any Seven – one-roll bet; 7 on next roll wins 4 to 1.
 Craps & Eleven (C&E) – one-roll bet that roll is 2, 3, 11,

12. Pays 3 to 1 on the 2, 3, 12; 7 to 1 on 11.
Hashimoto: Casino Management: A
Strategic Approach

Craps: Wagers (cont’d)
 Two
or Twelve – two separate one-roll bets,
wager that next roll will be a 2 or that next roll will
be a 12. Each typically pays 30 to 1.
 Three or Eleven – Like two or twelve, two
separate one-roll bets. Each typically pays 15 to 1.
 Horn Bet – one-roll bet that next roll will be a 2,
3, 11, or 12.
 Horn High Bet – horn bet made in multiples of
five with the extra unit wagered on the “high”
number designated by the bettor.
 Hop Bet – next roll will be a certain combination.
Two types: “easy” or two-way hop, such as 6-1;
“hard” or one-way hop, such as 2-2.
Hashimoto: Casino Management: A
Strategic Approach

Blackjack or “21”: the most popular and
largest revenue-generating casino table game.
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Popularity stems from simple play.
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Player can guide his own destiny to some extent.
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Correct perception that the game can be beaten.
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Players are in friendly competition against casino.
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Objective: to beat the dealer by having a total
higher than the dealer’s without exceeding 21.
The object is not, as some people think, to get
as close to 21 as possible.
Hashimoto: Casino Management: A
Strategic Approach

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Baccarat
Results of consecutive hands are mathematically
dependent; it is not possible to take advantage
of this dependency.
A game of pure chance, unlike blackjack.
Relatively easy game to play, since hitting and
drawing rules are fixed and no decisions are
required (or even possible) once play begins.
Players need only decide whether to bet on the
banker or the player (there is also a tie bet).
Hashimoto: Casino Management: A
Strategic Approach

Caribbean Stud Poker
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Standard deck of 52; uses usual 5-card rankings.
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Let It Ride
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Poker-type game; standard deck of 52, player
places 3 equal-sized bets before cards are dealt.
Pai Gow Poker
Uses 53 cards – standard deck plus Joker. Joker
can fill a straight or flush, or it is an Ace. After
bets, players and dealer are dealt 7 cards, used to
make a 5-card hand and a 2-card hand. Five-card
hand must be equal to or higher than the two-card
hand.
Hashimoto: Casino Management: A
Strategic Approach
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Three Card Poker
Relatively new poker-based game; popular due
to its simplicity, fast action, and reasonably
Uses a single deck of 52 cards.
Each player’s hand is only three cards.
Rankings are slightly different from 5-card poker
rankings.
Hashimoto: Casino Management: A
Strategic Approach
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Slot Machines
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Term “slots” refers to:
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Slots are more popular in U.S. than table games:
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Video games now on the market
They produce 70%-75% of the gaming revenues
Typically fill 80% or more of the floor space
Four types of slots that are most popular:
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Video poker
Multi-line/multi-play video devices
Wide area progressives – linked together from many
casinos; progressive jackpot gets progressively larger
as more coins are dropped into the machine.
Hashimoto: Casino Management: A
Strategic Approach
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Keno
Players choose up to ten of the twenty numbers
which are randomly drawn by computer from the
numbers 1 through 80.
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Several types of keno tickets are available.
Casino advantage on keno is large – 25% to 35%
depending on the type of ticket and payout
schedule.
Hashimoto: Casino Management: A
Strategic Approach
Basic Casino Math
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Importance of Mathematics in Casino
Gaming
Relying mathematical principles assures that
casinos generate positive gross gaming revenues
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Many casino professionals fail to understand the
basic mathematics of the games and their
relationships to casino profitability.
Mathematics should also overcome the dangers of
superstitions.
Important for the casino operator to ensure the
reasonable expectations of the players are met.
Casino managers should also understand the
mathematical aspects relating to gaming regulation.
Hashimoto: Casino Management: A
Strategic Approach

Probabilities and Odds
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Probability = long run ratio of:
(# of times an outcome occurs) to
(# of times experiment is conducted).
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Odds = long run ratio of:
(# of times an outcome does not occur) to
(# of times an outcome occurs).
Hashimoto: Casino Management: A
Strategic Approach
•
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Represents the long run percentage of the
wagered money that will be retained by the
casino
The player’s chances of winning and the rate at
which she wins or loses money depends on:
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the game,
the rules in effect for that game, and
for some games, the level of skill.
The amount of money the player can expect to
win or lose in the long run – if the bet is made
over and over again – is called the player’s wager
expectation, or expected value (EV).
Hashimoto: Casino Management: A
Strategic Approach
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Some types of percentages in gaming:
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Win percentage
Theoretical win percentage
Hold percentage
Casino bosses sometimes use these terms
interchangeably, but there are fundamental
differences among them.
Some terms are interchangeable:
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House advantage (how casinos make money) is same as
theoretical win percentage; also called house edge and
expected win percentage.
For slot machines, hold percentage is (in principle) equivalent
to win percentage.
Hashimoto: Casino Management: A
Strategic Approach
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Win Rate Percentages: Summary
House advantage is also theoretical win percentage.
Hold percentage is win over drop:
Hold % = Win/Drop
Win percentage is win over handle:
Win % (actual) = Win/Handle
Win percentage approaches the house advantage as the
number of plays increases.
Hold percentage is equivalent to win percentage for slots
but not table games.
House advantage = Theoretical Win % =
Limit (Actual Win %) = Limit (Win/Handle).
Hold Percentage  House Edge.
Hashimoto: Casino Management: A
Strategic Approach
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Volatility and Risk
Risk depends on the house advantage, standard
deviation, bet size, and length of play.
Statistical theory is used to predict magnitude of
difference between win % and theoretical win
% for a given number of wagers.
The basis for analysis of volatility questions is a
statistical measure called the standard deviation
(essentially the average deviation of all possible
outcomes from the expected mean).
Hashimoto: Casino Management: A
Strategic Approach
Casino Games & Odds
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Table Games
Player skill affects the house advantage in
some games:
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Blackjack; also Caribbean Stud, Let It Ride, Pai
Gow, and Three Card poker.
Statistical advantage for these is usually
reported assuming optimal strategy.
In blackjack, a player who follows basic
strategy perfectly will maximize average gain,
or minimize average loss.
Hashimoto: Casino Management: A
Strategic Approach

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Gaming Devices
theoretical win % or theoretical hold %
Flip side is proportion of wagered money returned
to player: payback, or return percentage.
Theoretical hold percentage can be set to virtually
any value with the computer program that controls
the machine.
Video poker, a game with an element of skill, can
be favorable for the player if correct strategy is
used.
Hashimoto: Casino Management: A
Strategic Approach
Related Operations Issues
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Player Value and Comps
Player earning potential (player value,
player worth, theoretical win).
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Using house advantage, bet size, duration of play,
pace of the game, a casino can determine how
much it expects to win from a certain player.
Calculated using this formula:
Earning Potential = Average Bet  Hours Played 
Decisions per Hour  House Advantage
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Many casinos set comp policies by giving back a
set percentage of the player’s earning potential.
Hashimoto: Casino Management: A
Strategic Approach


Casino Pricing Mistakes
Casinos occasionally offer novel wagers, side
bets, increased payoffs, or rule variations.
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Effect – lowers the house advantage and the
effective price of the game for the player.
Sound from a marketing standpoint, but can be
disastrous if care is not taken to ensure the math
behind the promotion is sound.
Gaming Regulations and
Mathematics
Regulatory systems share common objectives:
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Keep the games fair and honest and assure that
players are paid accurately if they win.
Hashimoto: Casino Management: A
Strategic Approach