Middle School (Grades 6-8) Mathematics Lesson
Experimental and Theoretical Probability Using a Monte Carlo Simulation
Monty’s Dilemma: Should You Stick or Switch?
Lesson from NCTM Illuminations Website
Submitted by Lisa Scott
scottl@billings.k12.mt.us
Monty’s Dilemma is a lesson designed to develop the theoretical probability, using a
simulation, for the chance of winning what’s behind door number 1, 2, or 3 on the game
show “Let’s Make a Deal”. The students simulate the game show using an applet from
Shodor Education Foundation’s website, Shodor Interactive
(http://www.shodor.org/interactivate/index.html).
The students use experimental probability and the Law of Large Numbers to make and test
conjectures about the theoretical probabilities of winning if they stay with their original
door choice or switch to a new door after seeing what is behind one door.
You can view the original NCTM Illuminations lesson plan at
http://illuminations.nctm.org/index_d.aspx?id=377.
This is a fun way to learn about Monte Carlo simulations, experimental and theoretical
probabilities, and the Law of Large Numbers.
Monty’s Dilemma: Should You Stick or Switch?
Experimental Probability
I.
Objective: The learner will use a Monte Carlo simulation to determine
theoretical probabilities by examining experimental probabilities and using the
Law of Large Numbers. The learner will use proportionality and a basic
understanding of probability to make and test conjectures about the results of
experiments and simulations. The learner will compute probabilities for simple
compound events, using such methods as tree diagrams and area models. The
learner will apply and adapt a variety of appropriate strategies to solve problems.
II. MT Content Standards Addressed in This Lesson: MT Content Standard 1,
Benchmarks 1, 2, 3, & 4, MT Content Standard 2, Benchmarks 2 & 3, MT Content
Standard 6, Benchmarks 1, 2, 4, & 5
III. Materials: Computer and lightbook, access to internet, spinners and copies of a
spinner divided into three equal areas, coins.
IV. Lesson: Monty’s Dilemma: Should You Stick or Switch?
a. State the objectives of the lesson.
b. Describe the game show “Let’s Make a Deal” hosted by Monty Hall. Read the
situation from “Monty Hall, Three Doors Exploration Questions”.
c. Ask students to predict what is better: to switch doors or to stay with the
original choice. Write down your prediction and the reason for your
prediction. By a show of hands, how many would stay? How many would
switch? Share some of the reasons with the class.
d. Go to the interactive applet to simulate individual trials of this game.
http://www.shodor.org/interactivate/activities/monty3/index.html. Play the
game with the students. Ask a student to choose a door and click on it.
Monty will show another door with a pig behind it. Ask the student if they
want to “stick” with their original choice of doors or “switch” to the other
door. Click on their second door choice, if they see the boat and prizes they
win a piece of candy. If they see the other pig, they win nothing.
e. After playing the applet simulation, ask students if they want to change their
predictions. Write down their new predictions and reason for the change. Or
write down no change and why they decided not to change.
f. Hand out the spinners and the copies of the spinner divided into three equal
areas.
g. First the class will simulate the choice of “sticking” to the original door. The
prize will always be behind door A. Students spin the spinner. The area it
lands on is the students “door” choice. Monty shows you the other door
V.
without a prize behind it. However, you don’t switch from your original door
choice. Ask students to tally mark the number of games where they won the
prize and the number of total games played. Find the experimental
probability after playing 25 times.
h. Collect the class data and obtain a class experimental probability. Compare
your probability to the class probability. Which do you think is more
accurate? Why? This introduces the “Law of Large Numbers”. The more
trials performed, the closer the experimental probability will come to the
theoretical probability.
i. What is the experimental probability of winning if you stick to your original
door? What is the experimental probability of losing if you stick to your
original door? How did you obtain your answer?
j. Use the applet for a large number of trials to demonstrate the Law of Large
Numbers and how the experimental probability approaches a constant amount.
http://www.shodor.org/interactivate/activities/monty/index.html
k. Now the class will simulate the choice of “switching” to the other door after
Monty has shown the first door with a pig. The prize will always be behind
door A. Students spin the spinner. The area it lands on is the student’s first
“door” choice. Monty shows you the other door without the prize behind it.
This time, you switch to the remaining door. Tally mark the number of games
won and the total number of games played. Find the experimental probability
after playing 25 times.
l. Collect the class data and obtain a class experimental probability. Compare
your probability to the class probability.
m. What is the experimental probability of winning if you switch to the other
door? What is the experimental probability of losing? How did you obtain
your answer?
n. Use the applet for a large number of trials to demonstrate the Law of Large
Numbers and how the experimental probability approaches a constant amount.
http://www.shodor.org/interactivate/activities/monty/index.html
o. Does sticking or switching give a better chance of winning the prize? Why?
How does this compare to your prediction? Write down your answers.
p. Determine the theoretical probability of winning by using the areas on the
spinner and walking through the simulations of each strategy. Compare the
theoretical probabilities to the experimental probabilities. Write down your
observations.
Summary: Misconceptions of chance are common. Since this particular problem
involves the process of revising our probability estimates in the light of new
VI.
information about the doors, it involves conditional probabilities. Strong
evidence points to the fact that people are especially prone to misconceptions of
chance that involve conditional probabilities. A good way to solve probability
problems is to proceed from predictions to experiments to simulations to
theoretical models. These steps are very helpful in understanding exactly what
occurs within each of the strategies.
Assessment: Students turn in their write-ups about their predictions, changing
their predictions, the results of their experimental probabilities and the class
probabilities, and the theoretical probabilities.
a. Extension: Suppose you enter a sweepstakes and subsequently receive an
announcement that after a random drawing out of a million entries, the
winning ticket number is one of six listed numbers and your entry is among the
six. What is the probability that you hold the winning ticket? Suppose you
are given the opportunity to change your entry to any of those six listed.
What should you do and what are your chances of winning?
i. Make a prediction, design and perform and experiment, find the
experimental probability for a large number of trials, find the
theoretical probability. Write up your results.
Spinner for Monty's Simulation
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1
2
Monty’s Dilemma Assignment
1. Write a few sentences explaining whether you would “stay” with your original door choice
or “switch” to a new door after seeing one of the choices. Explain why you would “stay” or
“switch”.
2. Keep a tallied list of your experiment for “staying”. Copy the class list of experimental
outcomes for “staying”. Write the experimental probability of “staying”. Explain how we
determined this probability.
3. Write the experimental probability of “switching”. Explain how we determined the
experimental probability.
4. Write a few sentences comparing your original prediction in #1 to how you feel now. Would
you “stay” or “switch”? Explain why. Use the experimental and theoretical probabilities we
obtained during this lesson in your response.