Probability from a TV Game

Objectives:  Plinko is the most popular game of chance on the TV game show The Price is Right.  This demo uses Plinko to illustrate how mathematics is useful for predicting a strategy that gives the highest probability for a big win.

Level: The demo was developed as an activity for a Mathematics for Elementary and Middle School Teachers but is appropriate for any course in which basic probability is discussed.  In putting together this demo, we attempted to be as complete as possible, but the presentation here covers many topics in much detail. Depending on your course coverage, level of the students, etc., you may wish to skip (or de-emphasize) some of the details.  For more advanced classes you may wish to discuss the details completely or after introducing the computational details with a smaller board, you may ask your students to investigate details for the larger board as an individual/group project.  In any case, a discussion of the "smaller board" problem is appropriate.

Here are some suggestions for using Plinko at various levels.

• Middle School:  Use the Plinko board to motivate a study of probability and to find experimental probabilities.  Calculator simulations may also be used.

• High School:  Use the Plinko board to motivate a study of probability.  Use the board and calculator simulations to find experimental probabilities.  Have students explore the board in more detail.  For example, counting the number of paths from each slot and use path counts to make predictions on likelihood a chip will fall into a particular slot.  Smaller boards are useful to facilitate a discussion of tree diagrams and using tree diagrams to compute theoretical probabilities.  A complete discussion of the big Plinko board is probably not appropriate for average classes at this level.

• Discrete Math Class (undergraduate level):  Use the Plinko board to motivate a study of probability.  Use the board and calculator simulations to find experimental probabilities.  Have students explore the board in more detail.  For example, counting the number of paths from each slot and use path counts to make predictions on likelihood a chip will fall into a particular slot.  Smaller boards are useful to facilitate a discussion of tree diagrams and using tree diagrams to compute theoretical probabilities.  At this level, investigation of theoretical probabilities for the big Plinko board can be presented in a class discussion or individual/group projects.

• Mathematics for Elementary and Middle School Teachers (undergraduate level):  Present material appropriate for middle school or high school students with an emphasis on simulations and experimental/theoretical probabilities. Although students are usually interested in how to compute the theoretical probabilities for the big Plinko board, time often does not permit a detailed in-class discussion.  This is a nice activity that can inspire pre-service teachers to bring motivating examples into their own classroom.

• Probability (Undergraduate):  All of the material should be accessible for students at this level.  Playing the Plinko game as a motivational tool followed by an in-class discussion of the details of a smaller board can lead to a nice individual/group project for investigation of the larger board.

Prerequisites: The following basic ideas are involved in the analysis of the Plinko game:  counting strategies, Pascal's triangle, probability (experimental and theoretical), simulations, tree diagrams, and expected value.  This demo can be used as an activity to introduce/motivate these topics and then can be revisited as an interesting application of the topics that are appropriate for your course.

Platform/Equipment:  The sights and sounds of a real Plinko board game are exciting and engaging for students so it is recommended that you build a board to use with this demo.  For the student activities that accompany the demo, a TI-83 simulation of the game board is included.  Additionally, Javascript simulations of Plinko for javascript-enabled browsers Netscape 4+ and Internet Explorer 5+ is included as an option.   See the links at the end of the demo.

Instructor's Notes:

General Introduction

The Plinko board (Figure 1) is a maze consisting of thirteen rows of pegs.  The number of pegs in each row alternates between ten and nine.  The first row of ten pegs forms nine slots.  Chips enter the maze through a slot that is chosen by the contestant.  As a chip hits an interior peg, it has an equally likely chance of falling to the left or right except when it hits the side of the board or a peg on the outer boundary.   In the latter case, the chip is forced to fall in a direction so that it remains in play on the board.  Chips come to rest in slots labeled with the dollar amounts (from left to right), \$100, \$500, \$1000, \$100, \$0, \$10000, \$0, \$1000, \$500, \$100, at the bottom of the board.  Even students who are not familiar with the Plinko game will catch on quickly after a brief demonstration.

(On The Price is Right, a contestant plays a minimum of 1 to a maximum of 5 chips, depending on the outcome of a pricing game that is played before Plinko.)

 Figure 1.  Schematic of Plinko Board

When students play the game, some obvious questions naturally arise:

1. Where should I put my chips to maximize my chances of winning \$10,000?

2. Why are there zeros on either side of the \$10,000 slot?

3. How much money can I reasonably expect to win?

Although students may have some intuitive ideas about the answer to Question #1, several important mathematical concepts can help to make the answer precise and to help answer the other two.

Organization of the Demo

This demo covers many bases and makes full use of graphics and tables.  To help shorten download times, it is presented by links to several different web pages.

The demo starts with activities using a Plinko board and/or calculator or browser simulations to help students become familiar with experimental probabilities.  A discussion about potential disadvantages associated with experimental probability help to motivate a study of theoretical probability.  Because of the complexities in analysis of the Plinko game, if path counting strategies and probabilities are to be discussed, we strongly recommend that you present an analysis for a smaller Plinko board (included with this demo) because techniques for the small board extend to the larger board and are much easier to develop.

The discussion on theoretical probability begins with an investigation of a smaller Plinko board (three starting and ending columns with five rows).  Probability tree diagrams and a strategy for path counting are introduced.  Analysis of conditional probabilities for a chip landing in an ending column given that it started in a particular column is discussed as well as probabilities associated with a random starting column.  Possible connections between experimental and theoretical probabilities are investigated via activities in which a browser simulation is used.

After the small board analysis, attention turns to the "original" Plinko board (5 beginning and ending columns with thirteen rows).  Theoretical probabilities are computed using a procedure involving Pascal's Triangle.  Counting paths and conditional probability computations based on path counts are discussed for a chip starting in the middle slot.  Expected value (in the context of computing expected winnings) is discussed and computed.  Connections between experimental and theoretical probabilities are investigated via activities in which TI-83, TI-89, or browser simulations are used.

Finally, activities for further individual or group investigation are presented.

The links below will take you to the

Plinko Board (5 begin and end columns, 13 rows)

TI-83: The TI-83 simulation code and sample output may be viewed and downloaded from here.

TI-89:  The TI-89 simulation code and sample output may be viewed and downloaded from here.

Javascript:
NOTE: The Javascript routines have been tested in Internet Explorer 5 and Netscape Navigator 4.7. The browser must be Javascript enabled. We recommended that your display resolution is 800 x 600.

5 Trials, user selects starting columns

100 trials from a single starting column selected by user

100 trials from random starting column

References

[1] Musser, Gary L. and William F. Burger, Mathematics for Elementary Teachers: A Contemporary Approach, Fourth Edition, Prentice-Hall, 1997.

[2] Lemon, Patricia,  "Pascal's Triangle--Patterns, Paths, and Plinko," The Mathematics Teacher, Vol. 90, No. 4, April, 1997, pp. 270-273.

[3] Haws, LaDawn, "Plinko, Probability, and Pascal," The Mathematics Teacher, Vol. 88, No. 4, April, 1995, pp. 282-285.

• http://www.geocities.com/sailoranch/barker.html
This site has a Javascript Plinko game having 9 beginning and ending columns with 15 rows.

• http://statistics.unl.edu/faculty/bilder/stat2023/excel.htm
This site has several activities associated with a statistics class.  In the listing of topics/resources for Chapter 5, there are Excel worksheets with an analysis of the probability distributions for a Plinko game as well as a Javascript routine for a game that has 6 starting and 5 ending columns with 10 rows.  The simulation runs 100 trials from a user-selected starting column.

Plinko boards are available for purchase from various commercial game vendors.  A web search can identify some of these with little effort.

Galton's Quincunx

Plinko is very similar to the game called Quincunx.  The Quincunx (pronounced quinn-cux) was developed by a mathematician named Galton in the late 1800's. The device works by dropping a series of acrylic balls, or beads, through rows of located pins. Each bead, as it hits a pin, has a 50-50 chance of falling to the left or right. When the beads pass through all the of pins they fall into a slot or cell. The shape of the beads' distribution forms what looks like a bell shaped or 'normal curve'.

There are several web addresses that discuss different variations of the Quincunx game.  Some have Java applets to demonstrate the board.  Some of these URLs are listed below.

Credits:  This demo was submitted by

Susie Lanier and Sharon Barrs
Mathematics and Computer Science Department
Georgia Southern University
Statesboro, GA 30460

and is included in Demos with Positive Impact with their permission.

We appreciate helpful suggestions from Susie Lanier regarding how the Plinko game can be used at various levels. TI-83 program code for Plinko simulations was submitted by Sharon Barrs.  We appreciate the help of John Cason (contestant), Susie Lanier, Sharon and Keith Barrs in developing the Plinko movies. Javascript code for Plinko simulations was developed by Lila Roberts.  TI-89 program code was adapted by Lila Roberts.

LFR 5/28/02  Last updated 9/15/2010 (DRH)