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.
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 as an option. See the links at the end of the demo.
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.
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
Schematic of Plinko
When students play the game,
some obvious questions naturally arise:
Where should I put my chips to
maximize my chances of winning $10,000?
Why are there zeros on either side
of the $10,000 slot?
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
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
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
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.
The TI-89 simulation code and sample output may be viewed and downloaded
5 Trials, user selects starting
100 trials from a single starting column selected by
100 trials from random starting column
 Musser, Gary L. and William F.
Burger, Mathematics for Elementary Teachers: A Contemporary Approach,
Fourth Edition, Prentice-Hall, 1997.
 Lemon, Patricia,
"Pascal's Triangle--Patterns, Paths, and Plinko," The
Mathematics Teacher, Vol. 90, No. 4, April, 1997, pp. 270-273.
 Haws, LaDawn, "Plinko,
Probability, and Pascal," The Mathematics Teacher, Vol.
88, No. 4, April, 1995, pp. 282-285.
Additional Plinko Links
ending columns with 15 rows.
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
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.
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.
This demo was submitted by
Susie Lanier and Sharon
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.
program code for Plinko simulations was submitted by Sharon Barrs.
appreciate the help of John Cason (contestant), Susie Lanier, Sharon and
code for Plinko simulations was developed by Lila Roberts. TI-89
program code was adapted by Lila Roberts.