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GAMBLING, CASINOS, AND GAME SIMULATION Author(s): ROBERT L. HEINY Source: The Mathematics Teacher, Vol. 74, No. 2 (February 1981), pp. 139-143 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/27962356 . Accessed: 13/09/2014 06:30 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Mathematics Teacher. http://www.jstor.org This content downloaded from 99.54.109.148 on Sat, 13 Sep 2014 06:30:49 AM All use subject to JSTOR Terms and Conditions

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Page 1: GAMBLING, CASINOS, AND GAME SIMULATION

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 .

Accessed: 13/09/2014 06:30

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Mathematics Teacher.

http://www.jstor.org

This content downloaded from 99.54.109.148 on Sat, 13 Sep 2014 06:30:49 AMAll use subject to JSTOR Terms and Conditions

Page 2: GAMBLING, CASINOS, AND GAME SIMULATION

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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Page 3: GAMBLING, CASINOS, AND GAME SIMULATION

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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Page 4: GAMBLING, CASINOS, AND GAME SIMULATION

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).

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Page 5: GAMBLING, CASINOS, AND GAME SIMULATION

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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Page 6: GAMBLING, CASINOS, AND GAME SIMULATION

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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