Title
Winning Roulette (Jo Ann Ellerbrock)
Goals
(1) Students will investigate basic probabilities.
(2) Students will investigate basic counting techniques.
(3) Students will explore methods of determining the expected value of a single bet.
(4) Students will explore the question of how gambling can be a business for a casino.
(5) Students may explore the normal distribution and the Central Limit Theorem (optional)
Abstract
How can gambling be a profitable business for a casino? Individual gamblers cannot predict whether they will win or lose, but the casino is a business which consistently makes money. Why is this since some people win large amounts of money? This activity can be used to introduce students to probability and counting theory. More advanced students can use it to look at how expected values and the normal distribution assure that the more money bet, the more money the casino is guaranteed to take in.
Problem Statement
The Internet is loaded with home sites of people who make such claims as, "By using our money management method at the roulette wheel, you can lose more spins that you win and still come out ahead in cash winnings." (Search "Roulette" on Lycos for more.) Can their claims be true, or are the only people getting rich with such claims the ones who sell their methods? How do casinos make money from gambling? And what are your--or my--chances for winning?
Instructor Suggestions
(1) The roulette wheel can easily be simulated with appropriately numbered and colored chips in a bag.
(2) A web site with a "Guide to Roulette for Beginners" is http://www.avc-net.com/saros/rules.htm
(3) The published disadvantage on an American wheel is 5.26%. For a French style wheel, it is 2.7%. This can be verified by examining the expected value of a single bet on red: E=(1)(18/38)+(-1)(20/38). Students may generate a distribution of winnings in repeated bets on red or black via simulation, or you may tell them that the spread of outcomes observed in a previous simulation was -0.47 to 0.37. These outcomes should span about three standard deviations on each side of the mean and result in standard deviation of 0.14. with a mean of the expected value of a single bet. For a further explanation of this, see For All Practical Purposes, pp. 198-200.
(4) One web site ( http://www.omen.com.au/~advance) claims they have a method of breaking even if a player wins only a third of their bets. Evaluating this claim provides a nice extension for more advanced students as well as an opportunity for students to communicate a position about the feasibility of such a claim.
Materials
Winning Roulette activity sheet, pencil, paper, calculator, tiles and bag for simulation (optional)
Time
This activity will take one to three days depending on students' background and whether or not it is set up with a simulation.
Mathematics Concepts
Discrete Mathematics Concepts
Counting techniques, probability, expected value
Related Mathematics Concepts
Normal distributions, Central Limit Theory, mathematical modeling
NCTM Standards Addressed
Problem Solving, Reasoning, Algebra, Probability, Statistics, Discrete Math
Colorado Model Content Standards Addressed
Data Collection and Analysis (3), Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
One purpose of this activity is to provide some probabilities that are easier to compute than poker hand probabilities yet with the same level of interest so that they could be used in an Algebra I level class. The extension to sampling distributions and the Central Limit Theorem add a degree of difficulty so that the activity can be used by higher level classes as well.
Further Investigation
Other bets which are available are (1) one of four numbers sharing a common corner, with a payoff of 9 for 1, (2) six numbers sharing two contiguous streets, with a payoff of 6 for 1, (3) one of a dozen numbers (1-12, 13-24, or 25-36) with a payoff of 3 for 1, (4) small number (1-18) or large number (19-36), with a payoff of 2 for 1, (5) even or odd numbers, with a payoff of 2 for 1. The only bet in the game for which the house has a bigger advantage than the standard 5.26% is a bet that covers 0, 00, 1, 2, 3 and pays 18 for 1. The house edge for this bet is 7.89%.
Variations/Comments
References/Resources
Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete mathematics through applications. New York: W.H. Freeman and Company
Kenney, M.J., & Hirsch, C.R. (Eds.). (1991). Discrete mathematics across the curriculum, K-12. Reston, VA: National Council of Teachers of Mathematics.
Steen, Lynn A. (Ed.). (1991). For All Practical Purposes. New York: W.H. Freeman and Company