BL5229
Projects: Option 1
Simulation: The Monty Hall problem
Marilyn vos Savant is famous for being the woman with the highest reported IQ (she is listed in
the Guiness book of record for this). She maintains a column in an American magazine,
“Parade”, in which she challenges people to submit questions for which she will provide answers.
In 1991, she was asked the following question:
“Suppose you're on a game show, and you're given the choice of three doors. Behind one door is
a car, behind the others, goats. You pick a door, say #1, and the host, who knows what's behind
the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick
door #2?" Is it to your advantage to switch your choice of doors?”
This question was named the “Monty Hall problem”, in reference to a TV game show, “Let’s
make a deal” hosted by Monty Hall. Marilyn vos Savant answered that it is to your advantage to
switch as it has a 2/3 chance of success, compared to 1/3 if you did not switch. Her answer
however led to a large controversy, with more than 10,000 people writing to “Parade” (including
many mathematicians) stating that her answer is wrong, as once a door has been revealed, each
remaining door has a 50% chance of being the door with the car.
Which of these two answers is right? We are going to simulate the Monty Hall problem with a
Matlab program to try to solve this controversy. Below, I give you the frame of the program:
complete it, implement it, and test it with different numbers of trials.
Before we start:
- The three doors will be numbered 1, 2, and 3. Note that with this scheme, the sum of the
labels of the three doors is always 6.
- We will use the Matlab function randi(N) to select a random integer number between
1 and N, as well as the Matlab function randperm(N) to generate a random
permutation of the numbers between 1 and N.
Matlab program:
%
% This small program simulates the Monty Hall problem
%
% It performs N dummy experiment and computes the chances of winning the car if we
switch % the choice for the door, or not.
%
% Define number of trials, and initialize the number of wins, for switching or not:
%
Ntrial = 10000;
Nwin_switch = 0;
Nwin_noswitch = 0;
%
% Start trials
%
for i = 1:Ntrial
%
%
Pick at random the door that corresponds to the car; store the corresponding
%
door number in the variable car, and the two remaining door numbers in the array
%
goat (size 2).
%
It is best to use the Matlab function randperm
%
car =
goat(1) =
goat(2) =
%
%
%
Pick a random door: initial choice of the user, and store in the variable choice.
choice =
%
%
%
%
%
%
Pick the door that Monty will open, which corresponds to a goat: if the user had
picked the door with the car, Monty picks at random one of the two doors corresponding
to a goat, while if the user had picked a door with a goat, Monty has only one choice.
Name this door monty.
monty =
%
%
%
Now checks if the user wins under two scenarios:
%
2) The user switches to the remaining door:
1) The user keeps the door she had initially chosen:
%
end trials
%
end
%
% Compute probabilities:
%
proba_switch = 100*Nwin_switch/Ntrial
proba_noswitch = 100*Nwin_noswitch/Ntrial
Problem:
-
Complete the program above and run it for multiple values of Ntrial. Based on those
results, should the use switch or not?
Can you find a mathematical explanation for this?
Extend this program such that it can deal with N doors, with 1 door corresponding to the
car, and N-1 doors corresponding to goats. Once the user had picked one door, Monty
opens N-2 doors with goats behind. Compute the probability of winning if the user
decides to switch to the remaining door. Plot this probability as a function of N. Could
you have predicted the result?
Please provide both the source code of the program you wrote, and a report describing the
results.