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GAMBLING, CASINOS, AND GAME SIMULATIONAuthor(s): ROBERT L. HEINYSource: The Mathematics Teacher, Vol. 74, No. 2 (February 1981), pp. 139-143Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27962356 .
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GAMBLING, CASINOS, AND GAME SIMULATION
By ROBERT L. HEINY University of Northern Colorado
Greeley, CO 80639
In 1973, the University of Northern Col
orado (UNC) initiated a new under
graduate general education program for
the purpose of encouraging innovative courses and interdisciplinary cooperation. The university has never required mathe
matics as a part of its general education
program. However, the Mathematics De
partment at UNC is becoming more in
volved with service courses for other de
partments. In the hope of encouraging more students to take some mathematics, I
developed a course entitled "Gambling, Casinos, and Game Simulation." This ar
ticle explains the objectives, the content, and the intent of the course.
Objectives
Students are not required to have any
college mathematics as a prerequisite for
the course, although a background of at
least one year of high school algebra is en
couraged. The course is individualized so
that students with a good mathematics
background can pursue topics in more
depth. The probability foundations of most
games can be developed by the advanced students. Enrollment is limited to twenty six students to facilitate individualization.
The major objectives are (1) to give the
student experience in making rational deci
sions when confronted with alternatives in a situation that can be described by a prob
ability model, (2) to introduce simple prob
ability concepts through games of chance, and (3) to demonstrate some practical uses
of mathematics in the real world. The stu
dents are asked to participate in dis
cussions, reading, and laboratory experi ences in order to reinforce the concepts that
are presented. In addition to games of
chance, insurance premiums and invest
ment situations are explored and discussed.
There are various other outcomes real
ized by the course. For example, computer simulation is a laboratory technique that
introduces students to computers and com
puter-aided instruction. Students are made aware of the odds (and poor expected re
turns) of games of chance. They also learn
some of the mathematics of finance when
insurance is discussed.
Content
Two books are required for the course:
Ed Thorp's Beat the Dealer (1973) and Bill
Friedman's Golden Guide to Gambling
(1972). Both are paperbacks and cost $2.95. In addition, twelve handouts are distrib uted as we progress from topic to topic.
(Copies of the handouts are available to in
terested readers.) I will list the topics as I
cover them in the course with a brief ex
planation of material covered for each.
Attitude and information test
On the first day of class I give a general examination covering some common mis
conceptions about gambling and probabil ity. Also included are several open-ended
questions relating to insurance. Students are asked what they expect to gain from the course. On the last day of class a posttest is
given asking the same questions. (Copies of
the test are available to interested readers.)
Numbers racket
The first topic examined is the numbers
racket. Probabilities and expected gains are
deferred until a later date. We play the
numbers in the class throughout the quar ter. After the game is explained to the stu
dents, each one selects three numbers: one
between 0 and 999, one between 0 and 99, and one between 0 and 9, inclusive. The
February 1981 139
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payoffs are explained for each number, and records are kept of the winning and losing numbers for the entire quarter.
The winning three-digit number is the last three digits of the Dow-Jones Indus trial closing average on Monday (payoff odds 600:1). The winning two-digit num
ber is the last two digits of the Dow-Jones Industrial closing average on Tuesday (payoff odds 60:1). Finally, the winning one-digit number is the last digit of the Dow-Jones Industrial closing average on
Wednesday (payoff odds 6:1). The sample size is so small that no infer
ences can be made concerning the three
digit or two-digit pool. However, students are interested in seeing if one can "win"
$600 or $60 on these numbers. In four
quarters no one has picked a winning three-digit number and only five students have successfully picked a winning two
digit number. The one-digit pool is in
structive, since the sample size is sufficient to check on the theoretical derivations. The
imaginary bookie should clear 30 percent of the money wagered on the numbers racket. As seen in table 1, the bookie has done slightly better in the four classes by clearing 34 percent. (The total profit and total payoff columns do not add to 830 be cause the wager is not considered as part of the payoff.)
The game has been successful not only for increasing the interest level of the class, but also as a long-term experiment that al lows the student to investigate empirical re sults versus theoretical results.
Insurance
Students are given copies of a booklet entitled Sets, Probability, and Statistics, dis tributed by the Institute of Life Insurance.
Fifty copies of this booklet can be obtained free of charge for class use by writing the Institute of Life Insurance, 277 Park Ave.,
New York, NY 10017. This booklet in troduces the basic probability concepts to be used in the course. These concepts in clude the definition of probability, union, intersection and complement properties, conditional probability, and independent and dependent trials. Once these concepts are covered, we introduce the methods needed to calculate premiums for insurance
policies using mortality tables. The stu dents are required to work all problems and hand them in. On completion of the
booklet, they study expected values, given the various alternatives and probabilities. For example, the one-digit number game has alternatives of winning $6 or losing $1 with respective probabilities of 1/10 and
9/10. The expected gain (EG) is [EG =
6(1/10) + (-1) (9/10) = -0.30]. Thus, if one dollar is wagered on each trial over a
period of time, the bettor would expect to lose an average of 30 cents for each dollar
wagered. The bookie would clear 30 per cent of the money bet.
E
The game of E is explained to the
students, and each receives a copy of the
payoff sheets used in Las Vegas casinos.
(See fig. 1 for an example of the informa tion given.) The expected values are given to the students, but not derived, since this
game is an example of the hypergeometric probability law, and many of the students are not ready for the counting techniques involved. Those students capable of han
dling the mathematics are asked to pursue the derivations.
In order to demonstrate the futility of the
TABLE 1 Numbers Racket (One Digit)
Experimental Results vs. Theoretical Results
Experiment Theoretical
(Expected)
No. of Bets Placed
830 830
No. of
Winning Bets
78
83
Total Intake
830 830
Total
Payoff (6:1) 468 498
Total Profit
284 249
Total Profit Total Intake
34% 30%
140 Mathematics Teacher
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HOW TO PLAY E TO WIN $25 000.00
(Limit each game to aggregate players) Mark any number of spots from 1 to 15. Twenty balls are drawn each game out of the 80 numbered balls that correspond to the numbered spots on a ticket.
The amount you win depends on the type of ticket
played and the number of spots caught. MARK ALL TICKETS WITH "X."
Consult the charts to determine the costs and pay offs of the different tickets.
I 2 X456789 10 II 12 13 M 15 16 W 18 19 20 21 22 23 24 25 26 27 28 3 30 31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50 51 52 5G 54 55 56 57 58 59 60 61 62 63 64 65 66 07 68 69 70 TA 72 73 74 75 76 77 78 79 80
SAMPLE TICKET FOR 1.20 (Mark 8 Spots)
CATCH 5 SPOTS?WIN. 10.00
CATCH 6 SPOTS?WIN. 100.00 CATCH 7 SPOTS?WIN. 2 200.00
CATCH 8 SPOTS?WIN.25 000.00
TABLE OF TICKET COSTS AND PAYOFFS
MARK 1 SPOT_60 Minimum
Catch Play60$ Play 1.20 Play 3.00
2 Win 7.50 15.00 37.50
MARK 2 SPOTS 60 Minimum
Catch Play60$ Play 1.20 Play 3.00
2 Win 7.50 15.00 37.5CM
MARK 3 SPOTS 60$ Minimum
Catch Play60<P Play 1.20 Play 3.00
2 Win .60 1.20 3.00
3 Win 26.00 52.00 130.00
MARK 4 SPOTS 60 Minimum Catch Play60<P Play 1.20 Play3.0d
2 Win .60 1.20 3.0d
3 Win 2.50 5.00 12.5
4 Win 70.00 140.00 350.0d
MARK 5 SPOTS 60 Minimum
Catch Play60$ Play 1.20 Play3.0d
3 Win 1.00 2.00 5.0Q
4 Win 14.00 28.00 70.oq
5 Win 300.00 600.00 1 500.00?
The more expensive the ticket, the better the chance to win $25 000.
Fig. 1
game, I use a table of random numbers to
pick the twenty winning numbers between 1 and 80 inclusive. The students choose their numbers prior to my selections. This
laboratory approach is very successful in
demonstrating how intuition and a prize of
$25 000 can entice the unsuspecting person into playing a game with very poor payoffs.
Craps
I spend a week on the game of craps. First we discuss the standard bets of PASS DON'T PASS, COME-DONT COME, Big 6 or 8, field bets, proposition bets, and
place bets. A player can win in craps by rolling a
sum of 7 or 11 on the first toss of a pair of dice. The player loses on the first roll if a 2, 3, or 12 is rolled. If a 4, 5, 6, 8, 9, or 10 is rolled on the first toss, this sum becomes
the player's point. To win, the player must roll this point again before rolling a 7. Oth erwise the player loses.
A bet of PASS means that the gambler is
betting that the player will win. If the gam bler wants to bet the player will lose, a bet of DON'T PASS is placed. The bets of COME and DON'T COME are variations of the two bets above. If a gambler wants to make a wager on the player after the player has begun the game, a COME bet is
placed. The next roll of the player is treated as the first roll for the gambler's wager.
In the Big 6 or 8 bet, the gambler selects
either 6 or 8 and wagers that this number will occur on a roll before the sum of 7.
The other so-called side bets of field, prop osition, and place bets are similar to the
Big 6 or 8 bet. The reader should consult Friedman (1972).
February 1981 141
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Next we derive the probabilities of win
ning or losing on these bets. These deriva tions provide a demonstration of one of the
applications of the geometric series, a topic covered in many high school algebra classes. The expected values demonstrate that the side bets such as Big 6 or 8, field
bets, proposition bets, or place bets should never be taken if the player wishes to mini mize his losses. The casino will take only 1.4 percent of the money bet on PASS or
COME bets, compared with 1.5 to 16.67
percent on the various side bets. The students play the game in a labora
tory session on a computer terminal. I have
programmed the department's HP-2114B
computer to play craps. The students are
able to obtain firsthand knowledge of the
empirical results as compared to the theo retical derivations.
Dog track
Each quarter, I take the class to the Clo verleaf Dog Track in Loveland, Colorado. The person in charge of odds-making takes the students through the facility and ex
plains the entire operation, including how
payoffs are determined. In 1975 the track converted to a computer for calculating payoffs. Prior to that time, the track calcu lated odds manually.
The students are amazed to find that the odds posted on the tote board before a race are only approximate. The final payoffs are
calculated after the race has been run.
They also find that the track does not take a "bath" when a long shot comes in. On the
contrary, the track finds that the resulting publicity from a big payoff to a bettor is
good for business. In Colorado, the track receives between
10 and 11 percent of the money wagered, and the state receives 5 percent. So, dis
regarding any knowledge of the dogs, the
gambler is playing a game where 16 per cent of the money wagered is not returned to the bettors.
Blackjack
The final game we discuss is blackjack, or "21." The students are asked to read
Beat the Dealer, in which Ed Thorp dis cusses several systems to help the player win at blackjack.
The strategies and probabilities pre sented in the book are discussed and tried in actual play with cards and on the com
puter. The appendix to Thorp's book gives probabilities derived by computer simula tion. These probabilities are based on hun dreds of thousands of hands dealt by the
computer. This game is an excellent ex
ample of how the computer has allowed
previously unsolved problems to be at
tacked with reasonable success. The theo retical derivation of probabilities con
cerning all possible outcomes in blackjack is too mind-boggling to imagine. In this
day and age, workable solutions are pos sible through simulation on a computer. This point is made vividly to the students as the game of blackjack is explored.
Others
In addition to the specific topics men
tioned above, we also have time to discuss
roulette, baccarat, chuck-a-luck, slot ma
chines, sweepstakes, lotteries, and football
parlays. Handouts are provided that relate to the New York Stock Exchange and to investment possibilities. Copies for class use can be obtained free of charge by writ
ing New York Stock Exchange, 11 Wall
St., New York, NY 10005.
Conclusions and Implications
The ideas presented to the students in this course can be listed as follows: (1)
probability, (2) expected value, (3) comput ers, and (4) mathematics of finance.
Probability
The concept of probability is, I believe, a
part of nearly everyone's thinking. In many instances, probability and intuition are
closely related in a person's everyday func
tioning. However, intuition can easily mis lead people into believing that gambling in its various forms is a good investment.
The standard approach to probability in a course for general education or liberal arts is too complicated for the time allowed
142 Mathematics Teacher
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to cover the topic. Under this standard ap proach, the students are given formulas and rules with little opportunity to apply them. As a result, some of the students be come confused and discouraged.
In this course, probability is introduced from a frequency approach with few, if
any, formulas. Admittedly, this course does
not, and cannot, give students all the tools
they may need, but it does encourage them to be alert, to ask questions, and to seek the answers before acting.
Expected value
Expected value is a simple concept that can be valuable to the student in purchas ing commodities, buying insurance, gam bling, or performing everyday business ac tivities. Of course, many of the everyday decisions cannot be solved by the expected value process, but the type of thinking used to weigh alternatives is similar.
Students who enter the business world will find this concept valuable. For ex
ample, a construction firm may need to de cide whether it should transport its workers to a site if there is a 90 percent chance of rain. The firm must weigh the losses and
gains in light of the probability of rain.
Companies use this process when sub
mitting bids for projects. Expected values will be used by many
students in their occupation as well as their recreational pursuits. I feel it is important to expose students in all disciplines to this tool for decision making.
Computers
The use of computers is a spinoff from the way the course is taught. No program
ming techniques are taught, but an appre ciation for the uses of the computer is
gained. Approximately one-fifth of the stu
dents taking the course have expressed an
interest in the computer and have taken ad ditional course work in computer science.
By using the computer to simulate various
games, it is possible to emphasize the dif ference between intuition and the actual
probabilities associated with many games
of chance. The students have enjoyed this
portion of the course more than any other.
Mathematics of finance
The work with insurance and investing on a modest scale gives students an aware ness of the pitfalls and opportunities avail able to them in the area of personal fi nance. Topics include interest and profits in conjunction with insurance premiums.
Mortality tables are used to calculate the
premiums under several assumptions con
cerning the return of capital invested by the insurance company.
Again, it is not the intent to teach the mathematics of finance but merely to offer an overview in order to show students what
questions should be asked when investing. The concept of expected value is used here when calculating premiums based on actu arial tables.
Although it is impossible to cover every situation where rational decisions based on
probabilistic concerns are to be made, I have found that the majority of students are excited about the various areas covered in the course and have indicated their in tention to pursue these areas in further course work. I believe that students taking this course will have a better chance of
functioning efficiently in society.
REFERENCES
Friedman, Bill. Golden Guide to Gambling. Racine, Wis.: Golden Press, 1972.
Institute of Life Insurance. Sets, Probability, and Sta tistics. New York: The Institute, 1973.
New York Stock Exchange. You and the Investment World. New York: The Exchange, 1973.
Thorp, Ed. Beat the Dealer. New York: Random
House, 1973.
SOME PROBLEMS WITH FRACTIONS FOR THE MIDDLE SCHOOL
(Continued from page 104)
REFERENCE Carpenter, Thomas P., Mary Kay Corbitt, Henry
Kepner, Mary Montgomery Lindquist, and Robert E. Reys. "Results and Implications of the Second NAEP Mathematics Assessment: Elementary School." Arithmetic Teacher 27 (April 1980):10-12, 44-47.
February 1981 143
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