Let’s Make a Deal: Where’s the Math?
Description of The Monty Hall Problem (from http://montyhallproblem.com/):
The Monty Hall Problem gets its name from the TV game show, “Let’s Make A Deal,” hosted by Monty
Hall. The scenario is such: you are given the opportunity to select one closed door of three, behind one
of which there is a prize. The other two doors hide “goats” (or some other such “non–prize”), or nothing
at all. Once you have made your selection, Monty Hall will open one of the remaining doors, revealing
that it does not contain the prize. He then asks you if you would like to switch your selection to the
other unopened door, or stay with your original choice. Here is the problem: Does it matter if you
switch?
So, where’s the math?
The seemingly simple probability calculation represented by the problem is much more complex than it
seems at first glance. It is eye opening to students that it does in fact matter if you choose to switch
after the host reveals one of the remaining two doors. The solution to the problem requires students to
grapple with conditional probability: how to update an initial probability assessment given new
information. The lesson provides a nice way to discuss experimental vs. theoretical probability, and to
introduce mathematical proof in a more formal way. For more advanced students, Bayes’ Theorem
could also be introduced in the presentation of the solution.
How can you explore this game with a class?
Note: This lesson is focused more on the proof process. There are several more traditional lessons that
focus more on data collection and go less formally into the theoretical proof. An example of such a
lesson can be found here: http://www.ms.uky.edu/algebracubed/lessons/Monty_Hall_Lesson.pdf
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Introduce the students to the problem by having them play the game. It is easy to create doors,
prizes, and zonks. There are also a number of fun looking web applets that allow you to play
individual trials of the game (e.g., http://math.ucsd.edu/~crypto/Monty/monty.html)
Start by presenting the problem to students and have them present their initial responses to you.
They should be asked to justify their answer to whether or not one should switch. Most of them will
likely say that it doesn’t matter if you switch because, for each door, there is a 50% chance that the
prize is behind that door.
After gathering the initial responses, experimentally test their instincts using an applet that lets you
run multiple trials, such as http://mste.illinois.edu/reese/monty/MontyGame5.html. If all goes well,
switching will win about 2/3 of the time, whereas not switching will win about 1/3 of the time.
Ask students to explain why the experimental results came out so different from their initial
theoretical assessment. Was this just an incredible statistical anomaly or should they rethink how to
determine the correct theoretical probability?
Demonstrate that the correct solution is 2/3. Since the result is counter-intuitive, use a couple of
methods to describe the solution to make sure that students get it. This is an excellent opportunity
to highlight experimental vs. theoretical probability and talk about mathematical proof, especially
enumerative-style proof.
o Proof sketch: It is clear that the probability of the prize being behind the door you chose
is 1/3. The host showing a different door has nothing to do with this—you will pick the
correct door at the start 1/3 of the time. That means that 2/3 of the time the prize is
behind one of the other two doors. Here’s the catch… By revealing a zonk behind one of
the two doors not chosen, the host effectively showed you the door that the prize is
behind in those 2 out of 3 cases that the prize isn’t behind the door you chose.
Therefore 2/3 of the time the prize is behind the door to which you can switch.
o Enumerative proof: Enumerate all of the possibilities and count up the wins to get the
correct probability. Have students complete the handout. It may be helpful to lead the
students through the first case.
Suppose you choose door #1 and decide to switch. Here are the three equally likely
outcomes:
1. The prize is behind door #1. Doors #2 and #3 contain zonks. You switch. Zonk!
2. The prize is behind door #2. The host reveals door #3. You switch. Prize!
3. The prize is behind door #3. The host reveals door #2. You switch. Prize!
It should be clear that the same three possibilities emerge (with door numbers
switched), if you initially choose door #2 or door #3. In each case, you win 2 out of three
times when you switch.
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Extension: Bayes’ Theorem can be used to prove the 2/3 result as well. In fact, the Monty Hall
Problem is an excellent way to introduce students to Bayes’ Theorem. It is sometimes difficult for
students to set up and apply Bayes’ Theorem correctly. It is fairly straightforward to set up the
Monty Hall Problem in Bayes’ Theorem, so it is ideal to introduce in this setting.
Extension: Ask students to think about how the game would work if there were five doors (and one
prize) rather than three. Would switching make a difference in that case? Have them demonstrate
their results.
Why might this work well with gifted students?
This is an excellent lesson for gifted students as the result challenges their intuitions. That the solution is
counter-intuitive challenges them to reflect upon where their initial line of thinking went astray. This
process of meta-thinking both draws in and challenges gifted students. It also allows you to discuss
experimental versus theoretical probability and introduce students to a more rigorous notion of proof.
This lesson challenges gifted students to think about probability at a more complex level than is often
introduced. It engages students in thinking about how initial probabilities can change as a result of new
information.
Resources for The Monty Hall Problem:
A general overview with the interesting history of the problem:
http://www.letsmakeadeal.com/problem.htm
Monty Hall Problem applets:
http://math.ucsd.edu/~crypto/Monty/monty.html
http://mste.illinois.edu/reese/monty/MontyGame5.html
The Math Forum page with links to several resources:
http://mathforum.org/dr/math/faq/faq.monty.hall.html
The Monty Hall Problem
How often will you win if you choose to switch?
Contestant chooses door #___
Prize is behind door #___. Host reveals door #___.
Contestant wins/loses. (circle one)
Prize is behind door #___. Host reveals door #___.
Contestant wins/loses.
Prize is behind door #___. Host reveals door #___.
Contestant wins/loses.
Contestant chooses door #___
Prize is behind door #___. Host reveals door #___.
Contestant wins/loses.
Prize is behind door #___. Host reveals door #___.
Contestant wins/loses.
Prize is behind door #___. Host reveals door #___.
Contestant wins/loses.
Contestant chooses door #___
Prize is behind door #___. Host reveals door #___.
Contestant wins/loses.
Prize is behind door #___. Host reveals door #___.
Contestant wins/loses.
Prize is behind door #___. Host reveals door #___.
Contestant wins/loses.
Total number of contestant wins/number of possibilities: ___/___
Deal or No Deal: Where’s the Math?
Description of Deal or No Deal (from http://www.nbc.com/Deal_or_No_Deal/about/index.shtml ) :
“Deal or No Deal" is an exhilarating hit game show where contestants play and deal for a top prize of $1
million in a high-energy contest of nerves, instincts and raw intuition.
Each night, the game of odds and chance unfolds when a contestant is confronted with 26 sealed
briefcases full of varying amounts of cash—ranging from a measly penny to $1 million. Without knowing
the amount in each briefcase, the contestant picks one—his to keep, if he chooses—until its unsealing at
game's end.
The risk element kicks in when the player must then instinctively eliminate the remaining 25 cases—
which are opened and the amount of cash inside revealed. The pressure mounts as in each round, after
a pre-determined number of cases are opened, the participant is tempted by a mysterious entity known
only as "the Banker" to accept an offer of cash in exchange for what might be contained in the
contestant's chosen briefcase—prompting Mandel to ask the all-important question—Deal or No Deal?
As each case is opened, the likelihood of the player having a valuable cash amount in his or her own case
decreases or increases. Viewers will see if, truly, fortune favors the bold. The contestant knows that as
long as the larger cash prizes haven't been opened, the Banker's deals will only get higher. And if the
conflicted contestant accidentally opens a case with a bigger cash value—the Banker's offer could
suddenly evaporate. “
So, where’s the math?
The big idea at play in this game is EXPECTED VALUE. As the contestant moves through each round, to
weigh his or her options, they could look at the expected value or payoff of the remaining cases. This
really is the arithmetic mean of the remaining values. On the simplest level, if the banker offers a payout
less than this, they would respond with “no deal.” If it’s higher, they should take the deal. Generally
though, at the beginning of the game, the banker’s offers are below the expected payoff. As the game
goes on, the offers approach the expected, and often exceed it towards the end.
How can you explore this game with a class?
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Explore the concept of EXPECTED VALUE using the game as a springboard. Discuss what
expected value is, how it is calculated, and how it might be used in other places.
Create (or have students create) a spreadsheet to help them calculate the expected value for
each round of the game as they play. Discuss strategies for game play based on these
calculations. For example, if the banker always offered the expected value of the amounts
remaining in the suitcases, would any strategy give you a higher expected winning than just
accepting the banker's first offer? If the banker didn’t offer the expected amount, what
strategies might you employ? Why might you not want to use expected value to play this game?
Experiment with the game and the calculated expected values by completing trials of the Deal or
No Deal game online and work to determine how often (in each round, overall, etc.) the banker
offers the contestant a deal less than or greater than the expected value. Can a percentage of
the expected value offered for each round be calculated? How might this be integrated into an
expected value spreadsheet to estimate what is a “good” banker’s offer and what is not? Have
students create a graph/presentations of the results of their experimentation. If you were the
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banker, how would determine what to offer in order to minimize the amount of money given
away?
Have students calculate the standard deviation of each round and respond to various discussion
questions such as: Suppose the banker offered you $200,000 in the when you have the following
values on the board:
o Scenario 1: $100, $1,000, $50,000, $100,000, $300,000, $750,000
o Scenario 2: $0.01, $1, $5, $10, $1,000,000
o Scenario 3: $100,000, $200,000, $300,000
In which scenario would you accept it? Why? Does the standard deviation provide you with
an indication of why you might or might not accept the offer?
Have students create a new game that they could use the same concept of expected value to
explore roles similar to “the contestant’s” and “the banker’s.”
Why will this work well with gifted students?
The idea of a game is often very motivating for any student. For mathematically talented students, the
outlets for open-ended exploration and taking the basic concept involved to higher levels are pretty
varied. Thus, lessons with this game at the center can be differentiated nicely. The handouts following
can also be adapted to provide more or less guidance depending on the ability-level of the students you
have. Things can be approached in a more structured way with younger students, whereas the
organization of work and structure can be left more open-ended for older students.
The topic of expected value in this lesson is pretty basic on the surface; however, it can easily be
extended to encourage gifted students to think about the concept and game-play strategies from a
variety of perspectives, really honing in on their analytical skills. Further connections can be made to
their previous study (or a mini-lesson could be built in appropriately) of probability, percentage, and
standard deviation. The basic game can also be explored by students in an open-ended manner
encouraging them to make connections between other games and the concept, or even creating their
own game. This will appeal to their sense of creativity and need for structuring their own learning.
Resource for Playing Deal or No Deal Online:
http://www.nbc.com/Deal_or_No_Deal/game/flash.shtml (flash required)
http://www.brothersoft.com/games/deal-or-no-deal.html Free download (different prize amounts than
NBC game)
HANDOUTS FOLLOWING:
What’s the Deal with Deal or No Deal?- A Discussion Guide/Student Work Record
Directions for Creating a Sample “Payoff” Spreadsheet
Name:_______________________________________________ Date: ________________________
What’s the Deal with Deal or No Deal?
Part 1: Expected Value
What is expected value?
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How is expected value calculated?
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Calculate
the expected value of rolling a regular six-sided die.
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How does expected value play a role in the Deal or No Deal game?
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What other applications of expected value can you think of?
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Part 2: Playing Deal or No Deal- The Contestant Perspective
1. Create a spreadsheet (see attached directions) in Microsoft Excel to determine the expected
value of a payoff in a given round.
2. If the banker always offered the expected value of the amounts remaining in the suitcases,
would any strategy give you a higher expected winning than just accepting the banker's first
offer? Explain.
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3. If the banker didn’t offer the expected amount, what strategies might you employ?
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4. Why might you not want to use expected value to play this game?
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Part 3: Playing Deal or No Deal- The Banker’s Perspective
Play at least 10 games of Deal or No Deal here (Be sure not to accept the banker’s offer until the end so
you can record all needed data): http://www.nbc.com/Deal_or_No_Deal/game/flash.shtml
Calculate the Expected Payoff for each round as well as the Banker’s Offer. Record your results below:
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Trial 6
Trial 7
Trial 8
Trial 9
Trial 10
Round 0
Round 0
Round 1
Round 1
Round 2
Round 2
Round 3
Round 3
Round 4
Round 4
Expected
Payoff
Banker’s
Offer
Expected
Payoff
Banker’s
Offer
Expected
Payoff
Banker’s
Offer
Expected
Payoff
Banker’s
Offer
Expected
Payoff
Banker’s
Offer
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Trial 6
Trial 7
Trial 8
Trial 9
Trial 10
Round 5
Round 5
Round 6
Round 6
Round 7
Round 7
Round 8
Round 8
Round 9
Round 9
Expected
Payoff
Banker’s
Offer
Expected
Payoff
Banker’s
Offer
Expected
Payoff
Banker’s
Offer
Expected
Payoff
Banker’s
Offer
Expected
Payoff
Banker’s
Offer
Part 3: Playing Deal or No Deal- The Banker’s Perspective- Discussion Questions
1. Can a percentage of the expected value offered for each round be calculated? How might this be
integrated into an expected value spreadsheet to estimate what is a “good” banker’s offer and
what is not?
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2. If you were the banker, how would determine what to offer in order to minimize the amount of
money given away?
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Part Four: Extensions
1. Calculate the expected value and standard deviation of the remaining amounts in the
following scenarios:
Amounts in remaining cases
Expected Value
Standard deviation
a. $100, $1,000, $50,000, $100,000,
$300,000, $750,000
b. $0.01, $1, $5, $10, $1,000,000
c.
$100,000, $200,000, $300,000
2. Suppose the banker offered you $200,000 in the three scenarios from the problem above.
In which scenario would you accept it? Why? Does the standard deviation provide you with
an indication of why you might or might not accept the offer?
3. Describe a new game that you could use the same concept of expected value. What would
the “contestant perspective” look like compared to “the banker” perspective.
Directions for Creating a Sample “Payoff” Spreadsheet
First, here is a basic run-down of the game:
Game Directions:
1.) Choose 1 suitcase (your case- remains unopened).
2.) Open 6 suitcases.
3.) Open 5 suitcases
4.) Open 4 suitcases
5.) Open 3 suitcases
6.) Open 2 suitcases
7.) Open 1 suitcase
8.) Open 1 suitcase
9.) Open 1 suitcase
10.) Open 1 suitcase
Open your own suitcase- That's what you win!
Round
0
1
2
3
4
5
6
7
8
9
Total Number of
Cases Unopened
26
20
15
11
8
6
5
4
3
2
To set-up your spreadsheet, follow the steps below:
1.) In Row 1, enter your title (such as “Deal or No Deal Expected Payoffs”).
2.) In Row 2, enter the following headings in each column as indicated:
A. Suitcase Values
B. Open/Shut
C. Probability
D. Avg. Expected Payout
3.) In Rows 3-28 in Column A, enter the appropriate suitcase amounts:
4.) For Column B, enter 1 as the default for each row. As you play a round of the game, you
will change this to 0 if the suitcase is opened.
5.) For Column C, you will enter the probability of choosing that case in the current round.
For example, all start with probability of 1/26. To enter the probability in a cell type
=1/26 replacing the 1/26 with the appropriate probability as you move through the
round. To prepare for this, list the appropriate probability next to each Round Number
below (Use the “Total Number of Unopened Cases” from the rundown of the game
given above to help you):
a. Round 0 _____
Round 5 _____
b. Round 1 _____
Round 6 _____
c. Round 2 _____
Round 7 _____
d. Round 3 _____
Round 8 _____
e. Round 4 _____
Round 9 _____
6.) For Column D, you will need to enter the formula to calculate the expected payout for
the suitcase value in that row.
a.) What will that forumula be in words? ____________________________________
b.) What will that formula be using values in Row 3? ___________________________
c.) To enter that into Excel correctly, type (for each row as appropriate- so change the
Row # to correspond with the row you are in):
=A3*B3*C3
d.) How does the above “Excel Formula” match what you wrote for letters “a” and “b”?
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7.) For the final row, Row 29, enter “Total Expected Payoff” in Column A. In Column D, you
will need to add all of the individual expected payoffs up. To do this, enter the formula:
=SUM(D3:D28)
Sample Spreadsheet- Round 0
Deal or No Deal: Expected Payoff Spreadsheet
Suitcase Values
Open/Shut
Probability
$0.01
1
0.038461538
$0.00
$1.00
1
0.038461538
$0.04
$5.00
1
0.038461538
$0.19
$10.00
1
0.038461538
$0.38
$25.00
1
0.038461538
$0.96
$50.00
1
0.038461538
$1.92
$75.00
1
0.038461538
$2.88
$100.00
1
0.038461538
$3.85
$200.00
1
0.038461538
$7.69
$300.00
1
0.038461538
$11.54
$400.00
1
0.038461538
$15.38
$500.00
1
0.038461538
$19.23
$750.00
1
0.038461538
$28.85
$1,000.00
1
0.038461538
$38.46
$5,000.00
1
0.038461538
$192.31
$10,000.00
1
0.038461538
$384.62
$25,000.00
1
0.038461538
$961.54
$50,000.00
1
0.038461538
$1,923.08
$75,000.00
1
0.038461538
$2,884.62
$100,000.00
1
0.038461538
$3,846.15
$200,000.00
1
0.038461538
$7,692.31
$300,000.00
1
0.038461538
$11,538.46
$400,000.00
1
0.038461538
$15,384.62
$500,000.00
1
0.038461538
$19,230.77
$750,000.00
1
0.038461538
$28,846.15
$1,000,000.00
1
0.038461538
$38,461.54
Total Expected Payoff
Avg. Expected Payout
$131,477.54
Price is Right and Plinko: Where’s the Math?
Description of Plinko (from http://gscentral.net/plinko.htm ):
“Plinko is by far the most celebrated and enjoyed pricing game ever played on The Price is Right. The
game is simple. A contestant is shown four small prizes. For each prize, s/he sees two digits, but only
one is right. Either the first digit shown is the correct first digit or the last digit shown is the correct last
digit. For each small item guessed correctly, the contestant wins a chip. Bob gives the contestant a free
chip just for playing the game, so five chips can be earned. The contestant climbs to the top of the Plinko
board and drops the chips on a pegged board. The pegs send the chip skyrocketing all over the board
until they land in slots representing money amounts at the bottom. The slots are, from left to right;
$100, $500, $1000, $0, $5000, $0, $1000, $500, $100. A maximum of $25,000 can be won. After all the
chips have dropped, the contestant wins whatever money s/he has earned to that point.
In August, 1996, CBS aired a 25th Anniversary Special in primetime to celebrate The Price is Right's long
run. On that special episode, Plinko was played with a special $10,000 slot replacing the $5000 slot. This
slot became $10,000 permanently during the 27th season, meaning that a total of $50,000 can be won.”
So, where’s the math?
While playing Plinko begins with a pricing game, the more mathematically interesting aspect of the
game is in examining the Plinko board. There are a lot of probability concepts at play in the looking at
the Plinko board as contestants weigh the best “entry points,” the most likely prize amounts, etc.
Both experimental and theoretical probability can be explored with students as well as the process of
developing sample spaces for the various outcomes using tree diagrams, path counts and formulas, and
patterns found in Pascal’s triangle. Depending on the levels of your students, the game can be explored
by just looking experimental probability; taking it up a notch and exploring theoretical probability,
conditional probability, and multiplicative and additive probability properties; or taking it even further
to explore questions such as what happens if the board were even larger. Expected value can also be
calculated and examined.
How can you explore this game with a class?
(lesson adapted from: http://mathdemos.gcsu.edu/mathdemos/plinko/)
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Start by motivating the students to get interested in playing Plinko by playing a video clip through
YouTube or otherwise. Then, discuss how the game works.
Ask students what questions might be able to be answered with a mathematical analysis of the
board. For example, where would a contestant drop a chip to have the greatest chance of a large
prize? What prize amount do you think a contestant can expect to win? Why are the prize values
placed where they are?
Then, students will need to get familiar with the Plinko board itself. Provide a handout of the Plinko
board and have students analyze (as a class or with a partner) different characteristics of the board.
How many pegs are there? What is the pattern? How many rows/columns does the board have?
What will happen when a chip is dropped? Why are the triangles on the sides shaded in?
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Once a discussion of board characteristics is completed and students have a general sense of the
game, they can move on to complete different investigations playing the game. Students can play
online using a simulation or you can work to create a class set of Plinko boards. Directions for this
and links to simulations can be found in the “resources” section of this handout (pg. 5).
Depending on the level of your students, you can then approach the lessons and extensions below
as appropriate. It may be that you can differentiate by having one group work on one lesson while
another does a different lesson. You may choose just to work through experimental probability or to
skip experimental probability and jump right to theoretical probability. Please note that the
theoretical probability lesson would likely require some pre-requisite teaching in that realm.
o
Lesson 1: Experimental Probability
 Introduce or review the basic idea of experimental probability
 Experimental probability is the number of positive outcomes over the
total number of trials or
P(EVENT) = TOTAL TIMES EVENT HAPPENS
TOTAL # OF TRIALS
 Discuss some basic examples such as figuring out the experimental
probability of rolling a two on six-sided die.
 Have students explore using experiments/simulations the basic question of,
“What is the probability of winning each amount when dropping a single chip?”
 In small groups, have each student do five trials and gather their group data. In
their groups, they then can answer several follow up questions such as what is
the experimental probability for each prize amount.
 Students can then combine their data with the class data to explore how
including more data affects the probabilities. Questions such as, “Which
calculation, group or class, is more accurate?,” can be explored as well how
“accuracy” is measured. This then can motivate moving on to looking at
theoretical probability. A discussion can also be held about what we might be
assuming with the simulations/experiments (e.g., if using an actual board that
all chips are perfectly identical and the board is constructed with perfect
measurements)
 EXTENSION OF CONCEPTS: Other experiments can be done to further explore
the game and the concept of experimental probability if you choose. Students
can develop their own questions to answer using an experiment and develop an
experiment. For example, they might choose to have each group member drop
all chips in the same starting slot and determine the experimental probability of
each prize amount for the individual slot. This could then be extended to look at
expected value for that individual slot based on the experimental data.
o
Lesson 2: Theoretical Probability
 Begin by introducing or reviewing the concept of theoretical probability. Keep in
mind that for Plinko, the theoretical probability of a particular prize for a chip
that is dropped from a particular column is conditional, that is, not all outcomes
are equally likely and depend upon where the chip is dropped from originally.
 The basic idea of theoretical probability when all outcomes are equally likely is
the total number of ways an event can happen divided by the total possible
outcomes. For example, the theoretical probability of rolling a two or four on
six-sided die is 2/6 or 1/3 since there are two ways to get a two or four out of six
possible outcomes.
 If this is the first time students are working with theoretical probability, you will
also want to go over the concept the conditional probability. You can spark this
discussion by looking at the Plinko board and asking students how they might
determine theoretical probability of a given outcome when dropping a chip in
Slot B (but don’t expect a correct answer!). Then, discuss if these probabilities
would change if we dropped the chip from a different slot. Why would this be?
You might need to spend some time covering conditional probability, the
multiplicative property of probability, and the additive property as well as
independent and dependent events of probability before getting into the
investigation following. Again, this will depend on your students’ prior
experiences with these topics.
 From there, explore a smaller version of the Plinko board to develop theoretical
probabilities. This can include creating tree diagrams to explore possibilities on
the board as well as discussing and using multiplicative and additive properties
of probabilities. Again, if students have not used additive or multiplicative
properties of probability these need to be reviewed first.
 From here, students can look at counting paths on the small board to determine
probability and move to developing a formula that can be used with path counts
to determine theoretical probability for an ending slot given a starting slot.
 Students can then transition to looking at how the “path counts” match up to
Pascal’s triangle. It should be noted by purely counting paths, it will
(misleadingly) seem that there are 12 possibilities for the small board, which
does not match the probability fraction denominators of 8 and 16 found in the
calculated probabilities of the ending slots in the tree diagram. Students must
think about all the possibilities. So, if a chip bounces off the wall when it might
have moved forward if the wall wasn’t there, how do we account for that? Do
we need to? In the end, to account for this, students need to adjust the path
counts at the end to reflect the chips that hit the edge and those that do not.
 Discussion can then move towards how theoretical probabilities can be
determined for the full scale board both when a particular starting slot is
chosen. Students can use Pascal’s triangle superimposed over the board (and
adjusting for “edges”) to determine the total number of possible paths for slots.


It would be best to have different groups of students work on paths for a given a
slot as well. Keep in mind the last four slots can be calculated based on
symmetry with the first four slots. Solutions to the theoretical probabilities of
each path can be found at:
http://mathdemos.gcsu.edu/mathdemos/plinko/bigboardpathcount.html
Finally, students can answer the questions of what starting slot is best, that is,
which one might offer the biggest payoff? What strategies could they employ as
a contestant to opt to win the biggest the prize? Why might the game designers
have placed prize amounts the way they did? If they were designing the came to
try to minimize prize amounts, how might they place the prizes? If they were
trying to maximize prize amounts, how might they place the prizes? What is the
expected value of each starting slot?
EXTENSION OF CONCEPTS: Students can look at calculating probabilities on the
large and/or small board if they were to randomly select a starting slot and
discuss the pros and cons of this for game play.
In looking at the large board, students can also look at how they can use path
counts/tree diagrams to get accurate theoretical probabilities and develop
formulas based on these. This will involve determining (for each starting slot)
how many paths there are and how many of those paths involve hitting and
edge. The website,
http://mathdemos.gcsu.edu/mathdemos/plinko/bigboard_details.html, has a
detailed explanation how this would work.
Expected value can also be saved as an extension as can the theoretical
calculations for the larger board. Pascal’s triangle can also be further explored if
not done so previously from looking at more patterns in it to using it to expand
binomials. The Law of Large Numbers (see link in “resources”) can also be
explored in comparing experimental and theoretical data.
Another area students can explore is a similar game created by Galton—entitled
Quincunx. Unlike Plinko, there are not walls to contend with and students can
look at the distribution of where chips fall. Interestingly, the chips tend to fall in
a normal distribution. Links to explore Quincunx are included in the “resources”
section.
Finally, these lessons can lead to a study of more advanced topics such as
stochastic processes and the central limit theorem.
Why might this work well with gifted students?
This lesson works well with gifted students because it provides an opportunity to create open-ended
scenarios as well as differentiated instruction. Once students start experimenting with the game, you
can decide how much you want to structure their further explorations. Students can explore their own
questions by developing their own experiments or organizing their own theoretical analyses. You can
provide more or less structure depending on the student. This flexibility is what really allows an
instructor to cater this topic to gifted students. This also appeals to the capability and interest of gifted
students to apply high level thinking skills in a non-routine manner. In exploring theoretical probability
in particular, there are aspects that are counterintuitive for many students. Finally, because it is game,
gifted students will enjoy taking their mathematical knowledge and applying to a (fun) realistic situation.
Resources for PLINKO and related topics:
 Detailed lessons related to Plinko (from which this lesson was derived):
http://mathdemos.gcsu.edu/mathdemos/plinko/








This site also includes directions for programming TI-83/84 calculators to run a PLINKO
simulation as well as to internet links to a simulation.
The site and materials were developed by Susie Lanier and Sharon Barrs.
How to make a PLINKO board: http://www.ehow.com/how_5085611_build-plinko-board.html
YouTube clip of PLINKO: http://www.youtube.com/watch?v=oYmXRbkM8tw (Search “Plinko” on
YouTube to find others.)
Free download of PLINKO game (does not have columns labeled):
http://www.freewarefiles.com/Plinko-V_program_22033.html
Online simulation of PLINKO(9 x 9 PLINKO board- not the full scale board):
http://www.mredkj.com/javascript/plinko.html
Galton’s Quincunx Exploration:
http://www.math.psu.edu/dlittle/java/probability/plinko/index.html
Article about an 11th grade finite math’s class foray into Pascal’s Triangle and Plinko: “Pascal’s
Triangle—patterns, paths, and Plinko.” By Patricia Lemon. Featured in Mathematics Teacher
v90n4 (April 1997).
Another take on exploring PLINKO:
http://statweb.calpoly.edu/mcarlton/gameshows/index.html
The site also offers 5 other game show math activities and solutions!
A little on the Law of Large Numbers: http://mathforum.org/library/drmath/view/52799.html
HANDOUTS FOLLOWING:
Student Handout for exploring PLINKO- includes notes pages, discussion guides, and activity guides
Student Handout with selected suggested responses
Name: ________________________________
Image of PLINKO board 1
Date:_____________
Record your observations about the PLINKO board below:
EXPLORING PLINKO
(lessons adapted from: http://mathdemos.gcsu.edu/mathdemos/plinko/)
Experimental Probability Investigation:
Experimental Probability is defined as:
Here are two examples of experimental probability:
Example 1:
Example 2:
Let’s investigate: What is the probability of getting each prize amount when a chip is
dropped on the PLINKO board?
Using your PLINKO board or a computer simulation, play PLINKO by dropping five
chips. Use the table following to record your results and those of your group members.
Then, answer the discussion questions below.
DISCUSSION QUESTIONS
Based on your group’s data, answer the questions below:
1.) What is the experimental probability of dropping a single chip and winning
$10,000? $1,000? $500? $100? $0?
2.) What might have affected your results and your answers to number one?
3.) Is a particular “starting slot” better than others to win a bigger prize? Will this slot
always be best? What slot is likely most often “the best”?
Add your data to the class data, then answer the questions below:
1.) How did the experimental probabilities for each prize amount change?
2.) Why did the values change (if they did)? Why didn’t they change (if they didn’t)?
3.) Which experimental probability calculation, that based on small group or class
data, do you think is more accurate? Why?
4.) How might we determine which is more accurate?
PLINKO Experimental Data for Small Group
Name
Starting Slot
Ending Slot
Prize Amount
Theoretical Probability Investigation:
Theoretical probability is defined as:
Here are two examples of theoretical probability:
Example 1:
Example 2:
Additive Property of Probability:
Multiplicative Property of Probability:
Conditional probability is defined as:
Here are two examples of conditional probability:
Example 1:
Example 2:
Theoretical Probability and PLINKO
Before looking at the full scale PLINKO board, let’s look at a smaller version to determine
theoretical probability.
Questions for discussion:
1.) Are all prize outcomes equally likely? Why or why not?
2.) How might we figure out how many possible paths there are?
Theoretical Probability: Small PLINKO Board
Let’s make a tree diagram to determine the theoretical probability of getting a given prize on
the small PLINKO board. Let’s start with Slot B.
Draw a tree diagram below to show all of the possible paths a chip can take when dropped
from Slot B.
SLOT B
What is the P(A’Ι B)?
What is the P(B’Ι B)?
What is the P(C’Ι B)?
With your group, determine the theoretical probability of getting each prize amount for either
starting SLOT A or starting SLOT C. Start by drawing a tree diagram.
SLOT ____
What is the P(A’Ι ___)?
What is the P(B’Ι ___)?
What is the P(C’Ι ___)?
Exploring the small PLINKO board further: Class/Small Group Discussion Guide
1.) Why are the denominators in the tree diagram 8 and 16 when it seems there are only 12
possible paths?
2.) Using the multiplicative property of probability, what will the denominator of our
probability fraction be for paths that do not hit an edge given there is a ½ probability for
each of the four parts of the path?
3.) Using the multiplicative property of probability, what will the denominator of our
probability fraction be for paths that do hit an edge given there is a ½ probability for
three parts of the path and a probability of 1 for the other part?
So, in determining the probability of a given ending slot when starting with Slot B, we need to
think about how to account for paths that do not hit an edge (and hence are out 16) vs. paths
that do hit and edge (and hence are out of 8).
How can you summarize calculating the probability for a given ending slot given a starting slot
in a formula?
-
-
We know to find the probability we need to have the numerator and denominator of
the probability fraction.
For the numerators, let’s call the number of paths that DO hit an edge a, and those
that do NOT b.
For the denominators, we know that for paths that do not hit an edge the
denominator will be 24 using the multiplicative property (which can be generalized
to 2n where n is the total parts of each path.)
For denominators for paths that do hit an edge, the denominator will be 24-1 or 23
(which can be generalized to 2n-x where n is the total parts of each path and x is the
number of times a chip hits the edge.)
Thus, a formula for calculating the probability of an ending slot given a starting slot would be:
Small PLINKO Board Discussion Questions and Summary:
Using the formula above, fill in the table below to summarize the probabilities of getting each
prize amount based on the starting slot.
Starting SLOT
A
A’ Prize 1
Ending SLOT
B’ Prize 2
C’ Prize 3
B
C
Discussion Questions:
1.) How might we find the expected value or expected payoff for each starting slot? What
further information would we need to know?
2.) Calculate the expected payoff for each starting slot assuming that Prize 1 is $0, Prize 2 is
$1000, and Prize 3 is $50.
3.) Based on the calculated and expected payoffs, what is the best starting slot?
4.) If you had five chips to drop, what strategy would you use to drop them?
5.) How might you reallocate prize amounts so that SLOT A would provide the player with
highest expected value?
Linking the Small PLINKO Board and Pascal’s Triangle: Student Discussion Guide
Examine the number of paths from starting SLOT B. Draw in all the possible paths a chip can
take from B to each ending slot. The paths to SLOT A’ are drawn for you.
1.) Based on the diagram, how many possible paths are there to A’? ____ B’? _____ C’? ______
2.) How does this differ from our earlier probability calculations using the tree
diagram/formula?
3.) How well might this process or using a tree diagram/path count formula work for the large
PLINKO board? Why would it or wouldn’t work well?
4.) How might we use a more efficient way to count the number of possible paths to get to
each prize level?
Let’s look at the board a little closer.
We can write in the number of paths to get to a certain region in each cell. Then, look for
patterns.
1
1
1
1
2
3
3
3
1
6
3
1.) How were each of these numbers determined?
2.) What patterns do you notice?
3.) Are there really only 12 possibilities we need to consider when determining theoretical
probability? Why or why not? Consider examining your tree diagram/formula for SLOT B as
well.
Pascal’s Triangle & the PLINKO board: Class Discussion Guide
(from http://www.mathsisgoodforyou.com/artefacts/pascalstriangle.htm)
Above is an image of Pascal’s Triangle. How is this triangle formed?
What similarities do you notice between the triangle and the PLINKO board paths?
What differences do you see between the triangle and the PLINKO board paths? What makes
these differences?
How can we use Pascal’s Triangle to help us determine theoretical probabilities for the small
PLINKO board?
Using Pascal’s Triangle to Computer Theoretical Probability for the Small PLINKO board:
1.) Let’s review: if a chip does not encounter the edge of the board at all, what is the
probability of moving along each branch of a path? _____
2.) Given this and that there are four branches along each path, the probability of any one
path using the multiplicative property of probability would be what? _____
3.) How does this relate to the denominators in our original tree diagram and the formula
for slot B?
4.) In looking at Pascal’s triangle below, how does the bottom row differ from the bottom
row of the path counts on our small board?
5.) To reconcile this, place the values from Pascal’s triangle as shown above into each row
of the PLINKO board, but use the bottom row of the Pascal’s triangle above to replace
the bottom row of the values on the PLINKO board. You should have ones on the
outside of the board.
6.) To account for the fact that the edge plays a role in theoretical probability of some
paths, reflect the “extra” ones onto the middle row, making the six, an eight.
7.) Using these new values for A’, B’, and C’, and the denominator of 16, find the
probabilities following:
What is the P(A’Ι B)? ______
What is the P(B’ΙB)? ______
What is the P(C’Ι B)? ______
8.) How do these relate to the probabilities found with the tree diagram/formula?
EXTENSION: Use this same idea to compute the theoretical probability for each ending slot
starting with slot A or C.
Theoretical Probability on the large PLINKO board:
Your group was assigned one starting slot on the large board. Using methodology similar to
what we used for the small board, determine the theoretical probabilities of getting each prize
amount for your given starting slot.
Begin by using Pascal’s Triangle to help you count paths. Don’t forget to account for “the
edges.” Start by superimposing Pascal’s triangle, then go back and adjust the last row.
Regardless of your starting slot, the total paths will be the sum of the bottom row (or 2 to the
power of the total number of branches of a path).
Starting SLOT ______
P(A’Ι ___) = _____
P(B’Ι ___) = _____
P(C’Ι ___) = _____
P(D’Ι ___) = _____
P(E’Ι ___) = _____
P(F’Ι ___) = _____
P(G’Ι ___) = _____
P(H’Ι ___) = _____
P(I’ Ι ___) = _____
Record each groups’ calculations of theoretical probabilities for each starting slot below. Then,
calculate the expected payoff for each starting slot.
Starting SLOT
A
A’
100
B’
500
C’
1000
D’
0
Ending SLOT
E’
10,000
F’
0
G’
1000
H’
500
I’
100
Expected
Payoff
Expected Payoff Calculations:
B
C
D
E
F
G
H
I
Final Discussion Questions:
Based on the probabilities you and your classmates calculated, answer the questions following.
1.) What starting slot is best, that is, which one might offer the biggest payoff? Why might this be?
2.) What strategies could they employ as a contestant to opt to win the biggest the prize?
3.) Why might the game designers have placed prize amounts the way they did?
a. If they were designing the came to try to minimize prize amounts, how might they place
the prizes?
b.
If they were trying to maximize prize amounts, how might they place the prizes?
EXPLORING PLINKO- Some Suggested Responses
(lessons adapted from: http://mathdemos.gcsu.edu/mathdemos/plinko/)
Experimental Probability Investigation:
Experimental Probability is defined as:
Experimental probability of an event is the ratio of the total number of positive outcomes in an experiment to the
total number of trials completed in the experiment. This can be written as:
P(event) = positive outcomes
total # of trials
Here are two examples of experimental probability:
Example 1:
Example 2:
Bobby Lee rolled two dice 25 times. He got a sum of 6
eight times. The probability of a sum of 6 is 6/25.
P(sum of 6) = 6/25
Sally tossed a coin 30 times. She got heads 12 times.
The probability of getting heads (H) is 12/30 or 2/5.
P(H)= 2/5
Let’s investigate: What is the probability of getting each prize amount when a chip is
dropped on the PLINKO board?
Using your PLINKO board or a computer simulation, play PLINKO by dropping five
chips. Use the table following to record your results and those of your group members.
Then, answer the discussion questions below.
DISCUSSION QUESTIONS
Based on your group’s data, answer the questions below:
4.) What is the experimental probability of dropping a single chip and winning
$10,000? $1,000? $500? $100? $0? (responses based on individual data)
5.) What might have affected your results and your answers to number one?
-starting slots used
-weight of chips or flaws in board if physical board
-many more options for student responses…
6.) Is a particular “starting slot” better than others to win a bigger prize? Will this slot
always be best? What slot is likely most often “the best”?
Based on students’ individual data, they should have a conclusion about the best starting slot.
The rest of their response should indicate that they’d need to experiment more and/or figure out
all the possibilities to determine the overall best slot. Some students may have “gut feeling” about
the best slot(s) as well. No real right or wrong answer here, just look for signs of logical
reasoning.
Add your data to the class data, then answer the questions below:
5.) How did the experimental probabilities for each prize amount change?
Based on data. Check for reasoning.
6.) Why did the values change (if they did)? Why didn’t they change (if they didn’t)?
Based on data. Check for reasoning.
7.) Which experimental probability calculation, that based on small group or class
data, do you think is more accurate? Why?
Students, in most cases, will reason that the calculation based on class data is more accurate as
it includes more trials. Logically, the more trials an experiment has the closer it will reflect the
theoretical possibilities. If students have explored the law of large numbers prior to this lesson,
that can be tied in.
8.) How might we determine which is more accurate?
Student responses will vary. You should look for an indication of reasoning that if we find out what
is theoretically possible, then whichever experiment comes closer to that is more closely linked to
what would happen if we experimented time and time again.
Theoretical Probability Investigation:
Theoretical probability is defined as:
Theoretical probability of an event is simply put, the likelihood of an event occurring. It is the ratio of the number
of favorable outcomes to the total possible outcomes. It can be expressed as:
P(event)= favorable outcomes
total possible outcomes
Here are two examples of theoretical probability:
Example 1:
Example 2:
If a single six-sided number cube is rolled, the
probability of rolling a two or a four is 2/6 or 1/3.
A spinner has 5 sections in five different colors: red,
blue, green, yellow, and orange. The probability of
spinning the spinner and getting green (G) is 1/5.
P(2 or 4)= 1/3
P(G) = 1/5
(NOTE: The following should be a quick review of this if students have studied the concepts before. If they have
not, it is suggested this is studied before exploring PLINKO. The study should include discussion of independent
and dependent events. You can also judge by your students’ experience and comfort with probability in general
to decide how to best approach these concepts).
Additive Property of Probability:
If two events are mutually exclusive, then the probability of either event happening (A or B) is found by adding the
probability of event A to the probability of event B
EX: In looking at example one, we can reason that a two or a four on a six-sided number cube occurs two of the six
times. We can also write this using the additive property of probability.
P(2 or 4) = P(2) + P(4)
= 1/6 + 1/6
=2/6 or 1/3
Multiplicative Property of Probability:
To find the probability of two independent events occurring (one does not affect the other), you
multiply the probability of each event together.
P (A and B)= P(A) * P(B)
EX: Two sixed-sided number cubes are rolled. What is the probability of getting a 1 on the first and a 6 on
the second.
P( 1 and 6)= P(1) * P(6)
= 1/6 * 1/6
= 1/36
To find the probability of two dependent events occurring (one does affect the other), you multiply the
probability of each event together, but must adjust the second probability given the first event occured.
P (A and B)= P(A) * P(B Ι A )
EX: Four names are put into hat: Sally, Bobby Lee, Susie Q, and Ferdinand. What is the probability of
selecting Ferdinand (F) and then Sally (S) assuming Ferdinand’s name is not put back in the hat?
P( F and S)= P(F) * P(S Ι F)
= ¼ * 1/3 (Why 3? There are now only three names since Ferdinand’s was removed).
= 1/12
Conditional probability is defined as:
A theoretical probability is conditional when a probability of one event (B) is related to the outcome of another
event (A). Thus, it is finding the probability of B given A. This can be written as:
P(BΙ A)
This is read “the probability of B given A.”
Probability of conditional events is found by using the multiplicative property of probability (see above)
Here are two examples of conditional probability:
Example 1:
Example 2:
Two coins are tossed. The probability of getting heads
on the first coin and then tails on the second is
conditional.
P(H Ι T) = P(H) * P(T)
=½*½
=¼
Ten marbles are in a bag. Four are red and six are blue.
What is the probability of pulling a red marble, then a
blue marble (without putting the red one back).
P(B Ι R) = P(R) * P(B Ι R)
= 4/10 * 6/9 (remember there is one less marble now)
= 2/5 * 2/3
= 4/15
Theoretical Probability and PLINKO
Before looking at the full scale PLINKO board, let’s look at a smaller version to determine
theoretical probability.
Questions for discussion:
3.) Are all prize outcomes equally likely? Why or why not?
No. Because of the walls and placement of the slots, the chips will not end up in each of the
ending slots an equal number of times.
4.) How might we figure out how many possible paths there are?
-
Possible paths could be drawn and counted.
We could use a tree diagram or table.
Other answers possible as well….judge the reasonableness of student responses and
discuss accordingly.
Theoretical Probability: Small PLINKO Board
Let’s make a tree diagram to determine the theoretical probability of getting a given prize on
the small PLINKO board. Let’s start with Slot B.
Draw a tree diagram below to show all of the possible paths a chip can take when dropped
from Slot B.
Slot B
½
½
½
½
½
1
½
½
½
½
½
½
1
½
½
½
½
½
½
½
½
½
A’
B’
A’
B’
B’
C’
A’
B’
B’
C’
B’
C’
⅛
⅛
1/16
1/16
1/16
1/16
1/16
1/16 1/16
1/16
⅛
⅛
½
½
Using the multiplicative property of probability we can get the probability of each path shown by multiplying the
probabilities of each of the four branches of that path together. These are shown underneath each ending location
of the tree.
What is the P(A’Ι B)?
To get the total probability of A’ given B, we can simply add up the probabilities from the tree diagram related to
A’.
So, P(A’Ι B) = 1/8 + 1/16 + 1/16 = 1/4
What is the P(B’Ι B)?
P(B’Ι B) = 1/8 + 1/16 + 1/16 + 1/16 + 1/16 + 1/8 = 1/2
What is the P(C’Ι B)?
P(C’Ι B) = 1/16 + 1/16 + 1/8 = 1/4
With your group, determine the theoretical probability of getting each prize amount for either
starting SLOT A or starting SLOT C. Start by drawing a tree diagram
Link to: http://mathdemos.gcsu.edu/mathdemos/plinko/details_small.html to see results.
Probabilities for each starting slot/ending slot are shown on pg. 31
Exploring the small PLINKO board further: Class/Small Group Discussion Guide
1.) Why are the denominators in the tree diagram 8 and 16 when it seems there are only 12
possible paths?
They are based on the possibilities for that particular path. If a path doesn’t hit an edge, then theoretically,
there are 16 routes. If it hits an edge once, there are 8.
2.) Using the multiplicative property of probability, what will the denominator of our
probability fraction be for paths that do not hit an edge given there is a ½ probability for
each of the four parts of the path?
(1/2) (1/2) (1/2) (1/2) = 1/16, so for the denominator only this is 2 * 2* 2 * 2 = 24 = 16
3.) Using the multiplicative property of probability, what will the denominator of our
probability fraction be for paths that do hit an edge given there is a ½ probability for three
parts of the path and a probability of 1 for the other part?
(1/2) (1/2) (1) (1/2) = 1/8, so for the denominator only this is 2 * 2* 1 * 2 = 23 = 8
So, in determining the probability of a given ending slot when starting with Slot B, we need to
think about how to account for paths that do not hit an edge (and hence are out 16) vs. paths
that do hit and edge (and hence are out of 8).
How can you summarize calculating the probability for a given ending slot given a starting slot
in a formula?
-
-
We know to find the probability we need to have the numerator and denominator of
the probability fraction.
For the numerators, let’s call the number of paths that DO hit an edge a, and those
that do NOT b.
For the denominators, we know that for paths that do not hit an edge the
denominator will be 24 using the multiplicative property (which can be generalized
to 2n where n is the total parts of each path.)
For denominators for paths that do hit an edge, the denominator will be 24-1 or 23
(which can be generalized to 2n-x where n is the total parts of each path and x is the
number of times a chip hits the edge.)
Thus, a formula for calculating the probability of an ending slot (E’) given a starting slot (S)
would be:
P (E’│S) = a + b
2n-x
2n
In looking at A’, this formula would like:
P (A’│B)= a + b
24-1
24
or
P (A’│B)= 2a + b = 2a + b
24
24
16
Small PLINKO Board Discussion Questions and Summary:
Using the formula above, fill in the table below to summarize the probabilities of getting each
prize amount based on the starting slot.
A’ Prize 1
A
3/8
B’ Prize 2
C’ Prize 3
Starting SLOT
B
1/4
C
1/8
1/2
1/2
1/2
1/8
1/4
3/8
Ending SLOT
Discussion Questions:
6.) How might we find the expected value or expected payoff for each starting slot? What
further information would we need to know?
To find expected value, you would take each prize amount and multiply by the total probability of the
prize. We would need to know the actual prize amounts for the small PLINKO board in order to calculate
these.
7.) Calculate the expected payoff for each starting slot assuming that Prize 1 is $0, Prize 2 is
$1000, and Prize 3 is $50.
SLOT A = (3/8)(0) + ½ (1000) + (1/8)(50)= 0 + 500 + 6.25= $506.25
SLOT B= (1/4)(0) + ½ (1000) + ¼(50) = 0 + 500 + 12.5 = $512.50
SLOT C= (1/8)(0) + ½ (1000) + (3/8)(50)= 0 + 500 +18.75 = $518.75
8.) Based on the calculated and expected payoffs, what is the best starting slot?
SLOT C
9.) If you had five chips to drop, what strategy would you use to drop them?
Drop them all in SLOT C to maximize winnings.
10.) How might you reallocate prize amounts so that SLOT A would provide the player with
highest expected value?
Answers can vary. If you swap prize 1 for prize 3 and keep prize 2, then SLOT A’s expected value would
move would be the highest.
Linking the Small PLINKO Board and Pascal’s Triangle: Student Discussion Guide
Examine the number of paths from starting SLOT B. Draw in all the possible paths a chip can
take from B to each ending slot. The paths to SLOT A’ are drawn for you.
1.) How many possible paths are there to A’? 3 B’? 6
C’? 3
2.) How does this differ from our earlier probability calculations using the tree
diagram/formula?
The numbers do not match the numerators of the probability fractions.
3.) How well might this process or using a tree diagram work for the large PLINKO board? Why
would it or wouldn’t work well?
Not well. It would be difficult, if not impossible, to keep track of all of the different paths.
4.) How might we use a more efficient way to count the number of possible paths to get to
each prize level?
Look for patterns to find a mathematical way of counting/computing the total paths.
Let’s look at the board a little closer.
We can write in the number of paths to get to a certain region in each cell. Then, look for
patterns.
1
1
1
1
2
3
3
3
1
6
3
1.) How were each of these numbers determined?
By counting the total number of ways to get to each “diamond cell.”You can also add the two numbers above a cell
to get the value for that cell.
2.) What patterns do you notice?
Answers vary.
For most, you add the two number above the cell to get the one below. The exception is the edges. The left and
right side are the same (i.e., 1, 2, 1; 3, 6, 3)
3.) Are there really only 12 possibilities we need to consider when determining theoretical
probability? Why or why not? Consider examining your tree diagram/formula for SLOT B as
well.
It seems there are only 12, but if a chip doesn’t hit a wall, then the theoretical probability would be out of 16 since
every path has four parts, and the probability of traveling down each part of the path is ½. The counts alone do not
account for this difference.
Pascal’s Triangle & the PLINKO board: Class Discussion Guide
(from http://www.mathsisgoodforyou.com/artefacts/pascalstriangle.htm)
Above is an image of Pascal’s Triangle. How is this triangle formed?
Answers will vary.
By adding two adjacent values to get the value below it.
What similarities do you notice between the triangle and the PLINKO board paths?
Answers will vary.
The path counts are calculated in a similar way.
What differences do you see between the triangle and the PLINKO board paths? What makes
these differences?
Answers will vary.
The final row and bottom part of the board don’t match up because the PLINKO is rectangular so the triangle does
not fit nicely in it.
How can we use Pascal’s Triangle to help us determine theoretical probabilities for the small
PLINKO board?
Answers will vary.
It can help us count paths in a systematic way.
Using Pascal’s Triangle to Computer Theoretical Probability for the Small PLINKO board:
1.) Let’s review: if a chip does not encounter the edge of the board at all, what is the
probability of moving along each branch of a path? 1/2
2.) Given this and that there are four branches along each path, the probability of any one
path using the multiplicative property of probability would be what? (1/2)4 = 1/16
3.) How does this relate to the denominators in our original tree diagram and the formula
for slot B?
The denominators for the paths that did not hit edges were 16.
4.) In looking at Pascal’s triangle below, how does the bottom row differ from the bottom
row of the path counts on our small board?
The bottom row on the small board doesn’t match Pascal’s triangle since the path counts took away paths
where edges interfered. Thus, it reads 3, 6, 3 where the triangle has a 4, 6, 4 instead.
5.) To reconcile this, place the values from Pascal’s triangle into each row of the PLINKO
board as shown above, but use the bottom row of the Pascal’s triangle above to replace
the bottom row of the values on the PLINKO board. You should have ones on the
outside of the board.
1
1
1
4
4
1
6.) To account for the fact that the edge plays a role in theoretical probability of some
paths, reflect the “extra” ones onto the middle row, making the six, an eight.
7.) Using these new values for A’, B’, and C’, and the denominator of 16, find the
probabilities following:
What is the P(A’Ι B)? 4/(4 + 8 + 4) = 4/16 = 1/4
What is the P(B’ΙB)? 8/(4 + 8 + 4) = 8/16 = 1/2
What is the P(C’Ι B)? 4/(4 + 8 + 4) = 4/16 = 1/4
8.) How do these relate to the probabilities found with the tree diagram/formula?
They match exactly!
EXTENSION: Use this same idea to compute the theoretical probability for each ending slot
starting with slot A or C.
Theoretical Probability on the large PLINKO board:
Your group was assigned one starting slot on the large board. Using methodology similar to
what we used for the small board, determine the theoretical probabilities of getting each prize
amount for your given starting slot.
Begin by using Pascal’s Triangle to help you count paths. Don’t forget to account for “the
edges.” Start by superimposing Pascal’s triangle, then go back and adjust. Regardless of your
starting slot, the total paths will be the sum of the bottom row (or 2 to the power of the total
number of branches of a path).
See results here: http://mathdemos.gcsu.edu/mathdemos/plinko/bigboard_probs.html
Record each groups’ calculations of theoretical probabilities for each starting slot below. Then,
calculate the expected payoff for each starting slot.
Starting SLOT
A’
100
B’
500
C’
1000
D’
0
Ending SLOT
E’
10,000
F’
0
G’
1000
H’
500
I’
100
Expected
Payoff
A
B
C
D
E
F
G
0.226
0.193
0.121
0.054
0.016
0.003
0.000
0
H
0
I
0.387
0.346
0.247
0.137
0.057
0.016
0.003
0.000
0
0.242
0.247
0.241
0.196
0.121
0.054
0.016
0.003
0.000
0.107
0.137
0.196
0.226
0.193
0.121
0.054
0.016
0.006
0.032
0.057
0.121
0.193
0.226
0.193
0.121
0.057
0.032
0.006
0.016
0.054
0.121
0.193
0.225
0.196
0.137
0.107
0.000
0.003
0.016
0.054
0.121
0.196
0.241
0.247
0.242
0
0.000
0.003
0.016
0.057
0.137
0.247
0.346
0.387
0
0
0.000
0.003
0.016
0.054
0.121
0.193
0.226
$778.10
$1012.80
$1605.10
$2262.20
$2561.70
$2262.20
$1605.10
$1012.80
$778.10
Expected Payoff Calculations:
EXAMPLE: SLOT A
(0.226)(100)+ (0.387)(500)+ (0.242)(1000)+ (0.107)(0) + (0.032)(10,000)+ (0.006)(0) + (0.000)(1000) + (0)(500) +
0(100)= 22.6 + 193.50 + 242 + 320 = $778.10
Note: This can be simplified by adding probabilities of like prize amounts first.
Final Discussion Questions:
Based on the probabilities you and your classmates calculated, answer the questions following.
1.) What starting slot is best, that is, which one might offer the biggest payoff? Why might this be?
E. Few paths hit edges, so more ways to get prizes!
2.) What strategies could they employ as a contestant to opt to win the biggest the prize?
Always drop in the middle slot to win the most prize money.
3.) Why might the game designers have placed prize amounts the way they did?
Answers vary.
There is only one slot with 10,000 automatically making its probability less. The next ending slots with
higher probabilities of chip landing in them have a prize of zero. So it balances out a bit.
a. If they were designing the came to try to minimize prize amounts, how might they place
the prizes?
Put larger prizes on the outermost slots and lower prizes towards the middle.
b. If they were trying to maximize prize amounts, how might they place the prizes?
Put largest prizes in the middle slots.
Golden Balls: Where’s the Math?
Description of Golden Balls:
Golden Balls is a British television game show. Contestants must do whatever they can—be it lie, cheat
or play it straight—to avoid being voted off in each round and make it through the final showdown.
During the earlier rounds, the team of contestants accumulates a jackpot total while members are voted
off. Eventually, they get down to two contestants and move to the Split or Steal round.
Split or Steal
Both contestants have 2 balls in front of them. One of them has the word "Split" inside it and the other
has the word "Steal" inside it. If one of the contestants decides to split and the other steals, the player
who steals goes home with all of the money and the player who splits goes home with nothing. If both
players split, they both split the jackpot total. If both contestants steal, they both go home with nothing.
The host asks the players to look inside both of the golden balls, to make sure they know which one is
which. Once they know what they are, the host lets the players discuss what they are going to do for a
short while. He then asks the players to pick 1 of the 2 balls in front of them. The host then counts down
from 3, and the players reveal either their Split or their Steal ball.
So, where’s the math?
Similar to the Prisoner’s Dilemma, it is a finite payout (constant sum) game that can be used to
introduce the basics of Game Theory, the mathematical exploration of games and strategic behavior.
Students learn how to solve these games to find the optimal strategy for each. They explore key
concepts including payoff matrices, Nash equilibria, Pareto optimums, pure and mixed strategies,
dominant and weakly dominant strategies, and the tit-for-tat strategy.
How can you explore this game with a class?



Begin by showing the Golden Bills video clip to get the students’ interest. Ask whether it’s better to
split or steal and why.
Introduce Game Theory as a means to studying games and strategic behavior.
o Define a game.
o Define a strategy.
To introduce students to the basic components of solving a game and finding optimal strategies,
solve the Prisoner’s Dilemma along with the students. You can structure this exercise with more or
less teacher guidance depending on your students’ abilities and experiences with this kind of
thinking. In examining the Prisoner’s Dilemma, be sure to:
o Introduce the concept of a payoff matrix.
o Introduce the concept of a strictly dominant strategy.
o Introduce the concept of a Nash equilibrium.
o Introduce the notion of a Pareto optimum.

Once you’ve introduced these basic concepts, you can then move on to exploring other games and
expanding upon basic ideas introduced. Each of the games following can be worked through with
varying levels of direct teacher involvement.
 Analyze Golden balls using the concepts just developed.
 Introduce the concept of a weakly dominant strategy.
 Solve Golden Balls; show it’s better to steal and no contestants should ever win
money.
 Introduce Chicken to demonstrate that some games have multiple Nash equilibria.
 Introduce Matching Pennies.
 Introduce the concepts of pure and mixed strategies.
 With the students, use algebraic techniques to calculate an optimal strategy.
 Consider iterated games and introduce the tit-for-tat strategy.
 Finally, in a discussion format, draw some general conclusions about game theory and finite
payout games (also known as constant sum games).
 You can find or even make up your own finite payout games and analyze them together.
Practice using the algebraic technique developed in the Matching Pennies segment to solve
for optimal mixed strategies quantitatively. The books listed in the resources below may be
helpful, especially Game Theory and Strategy and The Mathematics of Poker. The former is
more introductory and general, while the latter will have more advanced applications.
Why might this work well with gifted students?
Gifted students need to engage in meta-thinking: not just following directions to do something, but
stepping back and analyzing how best to do it. In exploring game theory, students don’t just employ
strategies to play games, they think about how to find the best strategy. They’re often pleasantly
surprised to find that strategic behavior can be studied in a mathematical manner.
Resources for Golden Balls and other constant sum games:
You can find the video clip of the big steal in Golden balls here:
http://www.bing.com/videos/watch/video/golden-balls-100-000-split-or-steal-14-0308/d567841d4a26e1f6d5f8d567841d4a26e1f6d5f8280963121423?q=golden%20balls&FROM=LKVR5&GT1=LKVR5&FORM=LKVR24
There are resources on game theory for educators and students at the GameTheory.net website:
http://gametheory.net/
Wikipedia has articles on game theory (http://en.wikipedia.org/wiki/Game_theory) and the Prisoner’s
Dilemma (http://en.wikipedia.org/wiki/Prisoner%27s_dilemma).
There are some books that can be excellent resources:
Game Theory and Strategy (Straffin): http://www.amazon.com/Strategy-Mathematical-AssociationAmerica-Textbooks/dp/0883856379/ref=sr_1_1?ie=UTF8&s=books&qid=1289425654&sr=1-1#_
Thinking Strategically (Dixit and Nalebuff): http://www.amazon.com/Thinking-Strategically-CompetitiveBusiness-Professional/dp/039396101X/ref=sr_1_1?ie=UTF8&s=books&qid=1289425770&sr=1-1#_
The Mathematics of Poker (Chen and Ankenman): http://www.amazon.com/JerrodAnkenman/e/B001JPBTBE/ref=sr_ntt_srch_lnk_1?qid=1289425896&sr=1-1
HANDOUTS FOLLOWING:
Student Handout for exploring Golden Balls and other constant sum games (For answers to the
questions posed on this handout, refer to the PowerPoint presentation.)
Golden Balls and Game TheoryIntroduction and The Prisoner’s Dilemma
1.) A game is defined as an__________________________ containing the following elements:
1.
2.
3.
4.
2.) What is a strategy?
3.) Consider the following “Prisoner’s Dilemma” situation:
Two suspects are arrested by the police. The police have separated both prisoners, and visit each of them to
offer the following deal. If one testifies for the prosecution against the other and the other remains silent,
the betrayer goes free and the silent accomplice receives a ten-year sentence. If both remain silent, both
prisoners are sentenced to only one year in jail. If each accuses the other, each receives a five-year
sentence. Each prisoner must choose to betray the other or to remain silent. Each one is assured that the
other would not know about the betrayal before the end of the investigation.
What does the payoff matrix for this “game” look like?
Prisoner B
Stay Silent
Prisoner A
Betray
Stay Silent
Betray
4.) Looking at the payoff matrix, which strategy would you choose if you were Prisoner A? Why? Which
strategy would you choose if you were Prisoner B? Why?
5.) The answer to question 4 is a strictly dominant strategy. What does that mean?
6.) What is a Nash Equilibrium?
7.) Which square in the payoff matrix is the Nash Equilibrium for our Prisoner’s Dilemma situation?
8.) Is there a square in the payoff matrix that would be better for the two prisoners? If so, which one?
9.) What is a Pareto optimum?
10.) Why can’t the prisoners move from the Nash Equilibrium square to the Pareto optimum square?
Golden Balls
1.) Two contestants have reached the “Split or Steal” round of Golden Balls with £100,000 in the jackpot.
What does the payoff matrix look like for this game?
Contestant B
Split
Contestant A
Steal
Split
Steal
2.) If you were one of the contestants, which strategy would you choose? Why?
3.) The answer to question 2 is a weakly dominant strategy. What does that mean?
4.) Which square in the payoff matrix is the Nash Equilibrium for this game?
5.) Which square in the payoff matrix is the Pareto optimum for this game?
Chicken!
1.) Two friends decide to play “chicken” in the bumper cars ride at the amusement park:
Two bumper cars are speeding towards each other. Each driver can swerve away to avoid the collision or
stay the course. If one driver swerves and the other does not, the one who swerved loses pride. If they both
swerve, they both lose some pride, but not as much. If neither swerves, they collide and have sore necks for
the rest of the day.
What does the payoff matrix look like for this game, assuming we just rank the outcomes from 1
(worst) to 4 (best)?
Driver B
Swerve
Driver A
Stay
Swerve
Stay
2.) If you are driver A, what information do you need in order to know whether it’s better to swerve or
stay?
3.) Where in the payoff matrix are the Nash Equilibria?
Matching Pennies
1.) Two people are playing the game of matching pennies:
Each of two players simultaneously shows either a head (H) or a tail (T). If the pennies match, one player
wins both coins. If they do not match, then the other player wins both.
What does the payoff matrix for this game look like (where “H” is heads and “T” is tails)?
Player B
H
Player A
T
H
T
2.) If you are player A, what information do you need in order to know whether it’s best to show a head
or a tail?
3.) Since you can’t have that information in advance, what should you do? Why?
4.) The answer to question 3 is called a mixed strategy, as opposed to a pure strategy. What do each of
those terms mean?
5.) How often should you choose each action? Why?
6.) What does it mean to say a strategy is optimal?
Iterated Games and Cooperation
1.) If two players play a game like Golden Balls or the Prisoner’s Dilemma repeatedly, what two things
can they do to try to get the other player to cooperate and end up in the Pareto optimum square
instead of the Nash Equilibrium square of the payoff matrix?
1.
2.
2.) A strategy like the one described in question 1 is called a tit-for-tat strategy. For a tit-for-tat strategy
to work, it must have what properties? For each property, explain what it means.
Tit-for-Tat Strategy
Property
1.)
2.)
3.)
4.)
Explanation
3.) For a tit-for-tat strategy to work, the game also must go on indefinitely—it can’t last only a definite,
finite number of turns. Why?
4.) So what would you do if you were playing Golden Balls once?
5.) What would you do if you and your opponent were playing Golden Balls indefinitely many times?