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 deemphasize) 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.
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 TI83 simulation of the game board is included. Additionally, Javascript simulations of Plinko for javascriptenabled browsers Netscape 4+ and Internet Explorer 5+ is included as an option. See the links at the end of the demo. 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.)
When students play the game, some obvious questions naturally arise:
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 TI83, TI89, or browser simulations are used. Finally, activities for further individual or group investigation are presented. The links below will take you to the PLINKO Simulations Plinko Board (5 begin and end columns, 13 rows) TI83: The TI83 simulation code and sample output may be viewed and downloaded from here. TI89: The TI89 simulation code and sample output may be viewed and downloaded from here. Javascript: 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, PrenticeHall, 1997. [2] Lemon, Patricia, "Pascal's TrianglePatterns, Paths, and Plinko," The Mathematics Teacher, Vol. 90, No. 4, April, 1997, pp. 270273. [3] Haws, LaDawn, "Plinko, Probability, and Pascal," The Mathematics Teacher, Vol. 88, No. 4, April, 1995, pp. 282285. Additional Plinko Links
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 quinncux) 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 5050 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. http://www.qualitytng.com/ Credits:
This demo was submitted by We appreciate helpful suggestions from Susie Lanier regarding how the Plinko game can be used at various levels. TI83 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. TI89 program code was adapted by Lila Roberts.


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