Pose the following situation to the students:
During a certain game show, contestants are shown three
closed doors. One of the doors has a big prize behind it, and the other
two have junk behind them. The contestants are asked to pick a door,
which remains closed to them. Then the game show host, Monty, opens one
of the other two doors and reveals the contents to the contestant.
Monty always chooses a door with a gag gift behind it. The contestants
are then given the option to stick with their original choice or to
switch to the other unopened door.
In groups, students can discuss what they might do.
As a class, discuss the following strategies.
If you were the contestant, which of the following strategies would you choose, and why?
Strategy 1 (stick): Stick with the original door
Strategy 2 (flip): Flip a coin, stick if it shows heads, switch if it shows tails
Strategy 3 (switch): Switch to the other door
An Instructional Approach
A good way to attempt to solve probability problems is to proceed from guesses to experiments to computer simulations
to theoretical models.
Students should first make a guess about the best strategy--stick,
flip, or switch. Then, the students should carry out an experiment,
modeling the decision task with coins, dice, spinners, or random
numbers. Next, the use of a computer program that simulates the
hands-on experiment may be in order to perform large numbers of trials
for the experiment. Finally, depending on the background of the
students and the difficulty of the problem, it may be appropriate to
model the problem theoretically.
In Monty's dilemma, the simulation step is absolutely crucial
to enable our students to understand exactly what occurs within each of
the three strategies. Using an equal-area spinner and a coin, we can
model each of the three strategies. To determine which of the three
strategies yields the best chance of winning the prize, we should play
each strategy many, many times and keep a record of the outcomes. Let
us suppose that the prize is actually hidden behind door A, as shown in
the figure below. Remember that Monty must show us one of the doors
without a prize behind it.
Spinner for Monty Simulation
Alternatively, students may use the Adjustable Spinner interactive.
Let's walk through a simulation of each strategy.
The Stick Strategy
- Suppose that the spinner lands on door B. What does Monty do? (Shows you door C) What do you do? (Stick) Do you win or lose?
- Suppose that the spinner lands on door C. What does Monty do? (Shows you door B) What do you do? (Stick) Do you win or lose?
- Suppose that the spinner lands on door A. In this instance
Monty shows you either door B or door C. What do you do? (Stick) Do you
win or lose?
The Flip Strategy
- Suppose that the spinner lands on door B. Then Monty opens
door C. You then flip a coin to decide whether to stick with door B or
switch to door A. What is your chance of winning the prize?
- Suppose that the spinner lands on door C. Monty opens door
B. You flip to decide between A and C. Again, what is your chance of
winning the prize?
- Suppose that the spinner lands on door A. Monty opens B or
C. You flip to decide between A and the other door. Once more, what is
your chance of winning the prize?
The Switch Strategy
- Suppose that the spinner lands on door B. Monty opens door C. You switch to the unopened door. Do you win or lose?
- Suppose that the spinner lands on door C. Monty opens door B. You switch. Do you win or lose?
- Suppose that the spinner lands on door A. Monty shows you door B or C. You switch. Do you win or lose?
(Teacher Note: If you desire a somewhat simpler version of this activity, eliminate the flip strategy.)
We recommend that each student use the spinner to run at least
100 trials under each of the three strategies. This activity has a very
powerful effect on their understanding of the problem. After many
trials, the relative frequency of winning under each strategy gives an
experimental estimate for the theoretical probability of winning the
prize under each strategy.
Go to the Simple Monty Hall Simulation
to use an interactive applet to simulate individual trials of this
situation. You will need to manually flip a coin at the proper time to
simulate the flip strategy. To simulate a large number of trials, go to
the Generalized Monty Hall Simulation.
(Note, however, that you can only simulate the stick or switch
strategies with the applet.) The results of playing Monty's Dilemma
100,000 times under each of the three strategies using a computer
simulation are presented in the table below.
Monty's Dilemma: 100,000 Trials
What is the reason for the outcome? The big questions in Monty's Dilemma are (1)
What do you know and (2) When do you know it?
In problem solving, looking at an extreme case can often lend
valuable insight. Suppose Monty offered us a choice of 100 doors and
behind only one of these doors was a fabulous prize. Then, after
hearing your selection, Monty announces that the prize is either behind
your original choice or behind another door, which he identifies at
this time. Given the opportunity to switch choices, what would you do?
In this situation, more people are likely to favor the
"switch" strategy. Their reasoning may go like this. "My original
choice has a 99 percent chance of being wrong. That means that the
other door Monty offers has a 99 percent chance of being right!" A
computer simulation of this modified problem supports this analysis.
Similarly, in the instance of the three doors, since the probability is
2/3 that your original choice was wrong, the probability is 2/3 that
the great prize lies behind the door that remains after Monty opens a
door to a gag gift. Nevertheless, many people are quite convinced that
the probability is at best 1/2, even after they are offered the choice
to change doors. Let us undertake a careful analysis of the spinner
simulations using tree diagrams.
Once again, assume that the prize is behind door A. We spin
the spinner to make our initial choice. There is a 1/3 probability of
(randomly) choosing each door. Monty must reveal a gag gift behind a
remaining door. In the event we chose door A initially (the one with
the prize), Monty can show us either B or C. We assume that the
probability is 1/2 that Monty will show us B and 1/2 that he will show
us C. If we chose B, Monty must reveal door C and leave A. If we chose
C, Monty must reveal door B and leave A. Thus, initially, we obtain the
tree diagram in the figure below.
We can complete this diagram in three different ways,
corresponding to each of the three strategies, stick, flip, or switch.
The stick and switch strategies determine immediately which door is
finally chosen along each of the branches. The flip tree requires
adding two more branches at every node, each with a probability of 1/2
that we either keep our original choice or take Monty's offer to
switch. The three trees are shown in the figures below, with the final
probabilities computed for each end branch by the usual multiplication
principle. Again, without loss of generality, we assume that the prize
is behind door A.
Stick-strategy tree diagram for Monty's dilemma
Switch-strategy tree diagram for Monty's dilemma
Flip-strategy tree diagram for Monty's dilemma
When we tabulate the probabilities of ending up with a final
choice of door A, B, or C, we find that the three strategies all yield
a different result, as shown in the table below.
Analysis of the Stick, Flip, and Switch Strategies
Actually, the way most people respond initially to this problem
is not all that surprising. Misconceptions of chance are common among
students. Since this particular problem involves the process of
revising our probability estimates in the light of new information
about the doors, it involves conditional probabilities.