The Monty Hall Problem
Thurber and Oliver
The Problem
In the way-back, Let’s Make a Deal was a TV show hosted by Monty Hall. A number of years after Hall stopped hosting the show, in an article published in Parade, Marilyn vos Savant discussed a problem that arose based upon one of the games that was played. Her description of the problem follows.
From Parade (1990 and 1991)
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?
–Craig F. Whitaker Columbia, Maryland
Answer
Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?
Response
Good heavens! With so much learned opposition, I’ll bet this one is going to keep math classes all over the country busy on Monday.
My original answer is correct. But first, let me explain why your answer is wrong. The winning odds of 1/3 on the first choice can’t go up to 1/2 just because the host opens a losing door. To illustrate this, let’s say we play a shell game. You look away, and I put a pea under one of three shells. Then I ask you to put your finger on a shell. The odds that your choice contains a pea are 1/3, agreed? Then I simply lift up an empty shell from the remaining other two. As I can (and will) do this regardless of what you’ve chosen, we’ve learned nothing to allow us to revise the odds on the shell under your finger.
The benefits of switching are readily proven by playing through the six games that exhaust all the possibilities. For the first three games, you choose #1 and “switch” each time, for the second three games, you choose #1 and “stay” each time, and the host always opens a loser. Here are the results.
p_load(knitr)
x<- matrix(c(
"GAME 1", "AUTO", "GOAT", "GOAT", "Switch and you lose.",
"GAME 2", "GOAT", "AUTO", "GOAT", "Switch and you win.",
"GAME 3", "GOAT", "GOAT", "AUTO", "Switch and you win.",
"GAME 4", "AUTO", "GOAT", "GOAT", "Stay and you win.",
"GAME 5", "GOAT", "AUTO", "GOAT", "Stay and you lose.",
"GAME 6", "GOAT", "GOAT", "AUTO", "Stay and you lose."
), byrow=TRUE, ncol=5)
x <- data.frame(x)
colnames(x) <- c("Game", "DOOR_1", "DOOR_2", "DOOR_3", "RESULT")
rownames(x) <- x$Game
x <- x[,-1]
kable(x)| DOOR_1 | DOOR_2 | DOOR_3 | RESULT | |
|---|---|---|---|---|
| GAME 1 | AUTO | GOAT | GOAT | Switch and you lose. |
| GAME 2 | GOAT | AUTO | GOAT | Switch and you win. |
| GAME 3 | GOAT | GOAT | AUTO | Switch and you win. |
| GAME 4 | AUTO | GOAT | GOAT | Stay and you win. |
| GAME 5 | GOAT | AUTO | GOAT | Stay and you lose. |
| GAME 6 | GOAT | GOAT | AUTO | Stay and you lose. |
When you switch, you win 2/3 of the time and lose 1/3, but when you don’t switch, you only win 1/3 of the time and lose 2/3. You can try it yourself and see.
Alternatively, you can actually play the game with another person acting as the host with three playing cards—two jokers for the goat and an ace for the prize. However, doing this a few hundred times to get statistically valid results can get a little tedious, so perhaps you can assign it as extra credit—or for punishment! (That’ll get their goats!)
Comments
You’re in error, but Albert Einstein earned a dearer place in the hearts of people after he admitted his errors. – Frank Rose, Ph.D., University of Michigan
I have been a faithful reader of your column, and I have not, until now, had any reason to doubt you. However, in this matter (for which I do have expertise), your answer is clearly at odds with the truth. – James Rauff, Ph.D., Millikin University
May I suggest that you obtain and refer to a standard textbook on probability before you try to answer a question of this type again? – Charles Reid, Ph.D. University of Florida
I am sure you will receive many letters on this topic from high school and college students. Perhaps you should keep a few addresses for help with future columns. – W. Robert Smith, Ph.D., Georgia State University
You are utterly incorrect about the game show question, and I hope this controversy will call some public attention to the serious national crisis in mathematical education. If you can admit your error, you will have contributed constructively towards the solution of a deplorable situation. How many irate mathematicians are needed to get you to change your mind? – E. Ray Bobo, Ph.D., Georgetown University
I am in shock that after being corrected by at least three mathematicians, you still do not see your mistake. – Kent Ford, Dickinson State University
Maybe women look at math problems differently than men. – Don Edwards, Sunriver, Oregon
You are the goat! – Glenn Calkins, Western State College
You made a mistake, but look at the positive side. If all those Ph.D.’s were wrong, the country would be in some very serious trouble. – Everett Harman, Ph.D., U.S. Army Research Institute
Response
Gasp! If this controversy continues, even the postman won’t be able to fit into the mailroom. I’m receiving thousands of letters, nearly all insisting that I’m wrong, including the Deputy Director of the Center for Defense Information and a Research Mathematical Statistician from the National Institutes of Health! Of the letters from the general public, 92% are against my answer, and and of the letters from universities, 65% are against my answer. Overall, nine out of ten readers completely disagree with my reply.
Now we’re receiving far more mail, and even newspaper columnists are joining in the fray! The day after the second column appeared, lights started flashing here at the magazine. Telephone calls poured into the switchboard, fax machines churned out copy, and the mailroom began to sink under its own weight. Incredulous at the response, we read wild accusations of intellectual irresponsibility, and, as the days went by, we were even more incredulous to read embarrassed retractions from some of those same people!
So let’s look at it again, remembering that the original answer defines certain conditions, the most significant of which is that the host always opens a losing door on purpose. (There’s no way he can always open a losing door by chance!) Anything else is a different question.
The original answer is still correct, and the key to it lies in the question, “Should you switch?” Suppose we pause at that point, and a UFO settles down onto the stage. A little green woman emerges, and the host asks her to point to one of the two unopened doors. The chances that she’ll randomly choose the one with the prize are 1/2, all right. But that’s because she lacks the advantage the original contestant had—the help of the host. (Try to forget any particular television show.)
When you first choose door #1 from three, there’s a 1/3 chance that the prize is behind that one and a 2/3 chance that it’s behind one of the others. But then the host steps in and gives you a clue. If the prize is behind #2, the host shows you #3, and if the prize is behind #3, the host shows you #2. So when you switch, you win if the prize is behind #2 or #3. You win either way! But if you don’t switch, you win only if the prize is behind door #1.
And as this problem is of such intense interest, I’m willing to put my thinking to the test with a nationwide experiment. This is a call to math classes all across the country. Set up a probability trial exactly as outlined below and send me a chart of all the games along with a cover letter repeating just how you did it so we can make sure the methods are consistent.
One student plays the contestant, and another, the host. Label three paper cups #1, #2, and #3. While the contestant looks away, the host randomly hides a penny under a cup by throwing a die until a 1, 2, or 3 comes up. Next, the contestant randomly points to a cup by throwing a die the same way. Then the host purposely lifts up a losing cup from the two unchosen. Lastly, the contestant “stays” and lifts up his original cup to see if it covers the penny. Play “not switching” two hundred times and keep track of how often the contestant wins.
Then test the other strategy. Play the game the same way until the last instruction, at which point the contestant instead “switches” and lifts up the cup not chosen by anyone to see if it covers the penny. Play “switching” two hundred times, also.
And here’s one last letter.
Comment
- You are indeed correct. My colleagues at work had a ball with this problem, and I dare say that most of them, including me at first, thought you were wrong! – Seth Kalson, Ph.D., Massachusetts Institute of Technology
Response
Thanks, M.I.T. I needed that!
Comments from Elementary, Middle, and High Schools
In a recent column, you called on math classes around the country to perform an experiment that would confirm your response to a game show problem. My eighth grade classes tried it, and I don’t really understand how to set up an equation for your theory, but it definitely does work! You’ll have to help rewrite the chapters on probability. – Pat Gross, Ascension School, Chesterfield, Missouri
Our class, with unbridled enthusiasm, is proud to announce that our data support your position. Thank you so much for your faith in America’s educators to solve this. – Jackie Charles, Henry Grady Elementary, Tampa, Florida
My class had a great time watching your theory come to life. I wish you could have been here to witness it. Their joy is what makes teaching worthwhile. – Pat Pascoli, Park View School, Wheeling, West Virginia
Seven groups worked on the probability problem. The numbers were impressive, and the students were astounded. – R. Burrichter, Webster Elementary School, St. Paul, Minnesota
The best part was seeing the looks on the students’ faces as their numbers were tallied. The results were thrilling! – Patricia Robinson, Ridge High School, Basking Ridge, New Jersey
You could hear the kids gasp one at a time, “Oh my gosh. She was right!” – Jane Griffith, Magnolia School, Oakdale, California
I must admit I doubted you until my fifth grade math class proved you right. All I can say is WOW! – John Witt, Westside Elementary, River Falls, Wisconsin
It’s a lesson we’ll never forget. – Andreas Kohler, Cherokee High School, Canton, Georgia
This experiment caused so much discussion among students and parents that I’m going to have the results on display at our school open house. – Nancy Transier, Bear Branch Elementary, Kingwood, Texas
My classes enjoyed this exercise and look forward to the next project you give America’s students. This is the stuff of real science. – Jerome Yeutter, Hebron Public Schools, Hebron, Nebraska
Thank you for supplying us with this wonderful project which lightened our lives during a particularly cheerless winter without snow. – Marcia Jones, Berkshire Country Day School, Lenox, Massachusetts
Thanks for that fun math problem. I really enjoyed it. It got me out of fractions for two days! Have any more? – Andrew Malinoski, Mabelle Avery School, Somers, Connecticut
I’m a fourth grade student, and I used your column for a science fair project. My test results showed that you were right. My science fair project won a red ribbon. – Elizabeth Olson, Edgar Road Elementary, Webster Groves, Missouri
I did your experiment for the Regional Science and Engineering Fair at the University of Evansville, and I won both third place and a special award from the Army called the “Certificate of Excellence”! – Analda House, Evansville Day School, Evansville, Indiana
I did your experiment on probability as part of a Science Fair project, and after extensive interview with the judges, I was awarded first place. – Adrienne Shelton, Holy Spirit School, Annandale, Virginia
Congratulations! You’ve discovered a new concept. At first I thought you were crazy, but then my computer teacher encouraged us to write a program, which was quite a challenge. I thought it was impossible, but you were right! – Anabella Sousa, Dominican Commercial High School, Jamaica, New York
The teachers in my graduate-level mathematics classes, most of whom thought you were wrong, conducted your experiment as a class project. Each of the twenty-five teachers had students in their middle or high school classes play at least 400 games. In all, we had 14,800 samples of the experiment, and we’re convinced that you were correct —the contestant should switch! – Eloise Rudy, Furman University, Greenville, South Carolina
You have taken over our Mathematics and Science Departments! We received a grant to establish a Multimedia Demonstration Project using state-of-the-art technology, and we set up a hypermedia laboratory network of computers, scanners, a CD-ROM player, laser disk players, monitors, and VCR’s. Your problem was presented to 240 students, who were introduced to it by their science teachers. They then established the experimental design while the mathematics teachers covered the area of probability. Most students and teachers initially disagreed with you, but during practice of the procedure, all began to see that the group that switched won more often. We intend to make this activity a permanent fixture in our curriculum. – Anthony Tamalonis, Arthur S. Somers Intermediate School 252, Brooklyn, New York
I also thought you were wrong, so I did your experiment, and you were exactly correct. (I used three cups to represent the three doors, but instead of a penny, I chose an aspirin tablet because I thought I might need to take it after my experiment.) – William Hunt, M.D., West Palm Beach, Florida
I put my solution of the problem on the bulletin board in the physics department office at the Naval Academy, following it with a declaration that you were right. All morning I took a lot of criticism and abuse from my colleagues, but by late in the afternoon most of them came around. I even won a free dinner from one overconfident professor. – Eugene Mosca, Ph.D., U.S. Naval Academy, Annapolis, Maryland
After considerable discussion and vacillation here at the Los Alamos National Laboratory, two of my colleagues independently programmed the problem, and in 1,000,000 trials, switching paid off 66.7% of the time. The total running time on the computer was less than one second. – G.P. DeVault, Ph.D., Los Alamos National Laboratory, Los Alamos, New Mexico
One of my students wanted to know whether they were milk goats or stinky old bucks. Presumably that would redefine what a favorable outcome was! – Daphne Walton, Bayview Christian School, Norfolk, Virginia
Now ‘fess up. Did you really figure all this out, or did you get help from a mathematician? – Lawrence Bryan, San Jose, California
Response
Wow! What a response we received! It’s still coming in, but so many of you are so anxious to hear the results that we’ll stop tallying for a moment and take stock of the situation so far. We’ve received thousands of letters, and of the people who performed the experiment by hand as described, the results are close to unanimous: you win twice as often when you change doors. Nearly 100% of those readers now believe it pays to switch. (One is an eighth-grade math teacher who, despite data clearly supporting the position, simply refuses to believe it!)
But many people tried performing similar experiments on computers, fearlessly programming them in hundreds of different ways. Not surprisingly, they fared a little less well. Even so, about 97% of them now believe it pays to switch.
And plenty of people who didn’t perform the experiment wrote, too. Of the general public, about 56% now believe you should switch compared with only 8% before. And from academic institutions, about 71% now believe you should switch compared with only 35% before. (Many of them wrote to express utter amazement at the whole state of affairs, commenting that it altered their thinking dramatically, especially about the state of mathematical education in this country.) And a very small percentage of readers feel convinced that the furor is resulting from people not realizing that the host is opening a losing door on purpose. (But they haven’t read my mail! The great majority of people understand the conditions perfectly.)
And so we’ve made progress! Half of the readers whose letters were published in the previous columns have written to say they’ve changed their minds, and only this next one of them wrote to state that his position hadn’t changed at all.
“I still think you’re wrong. There is such a thing as female logic.”
– Don Edwards, Sunriver, Oregon
Oh hush, now.
Marilyn vos Savant
We’ve received thousands of letters, and of the people who performed the experiment by hand as described, the results are close to unanimous: you win twice as often when you change doors. Nearly 100% of those readers now believe it pays to switch.
Simulation
We can simulate playing the game using R’s random number generator.
montyhall <- function(games = 100, initial.pick = 1)
{
### Generate a matrix representing the number of games by the number of doors ###
### The car is represented by 1 and the goats are 0's ###
temp <- matrix(rep(c(0, 0, 1), games), ncol = 3,
byrow = T)
### Within each game (row) randomize the placement of the prizes ###
temp <- t(apply(temp, 1, sample))
### Use the switch method... ###
### If your initial pick was the car, you will be shown one of the two goats ###
### and you will switch to the other goat. If your initial pick was a goat, ###
### you will be shown the one other goat, and you will switch to the car. ###
wins.switch <- games - sum(temp[, initial.pick])
### Use the stick method... ###
### If your initial pick was the car, you will be shown one of the two goats ###
### and you will stick with the car. If your initial pick was a goat, you ###
### will be shown the other goat, and you will stick with the goat.
wins.stick <- sum(temp[, initial.pick])
### Compute the probabilities from the frequencies ###
prob.switch <- wins.switch/games
prob.stick <- wins.stick/games
### Return the results ###
temp <- c(wins.switch, prob.switch, wins.stick,
prob.stick)
names(temp) <- c("Switch Wins", "Switch Prob",
"Stick Wins", "Stick Prob")
temp
}
### Now run it ###
montyhall(10000)## Switch Wins Switch Prob Stick Wins Stick Prob
## 6637.0000 0.6637 3363.0000 0.3363Simulation with a Plot
Plotting the success probability shows convergence.
montyhall2 <- function(games = 100, initial.pick = 1)
{
### Generate a matrix representing the number of games by the number of doors ###
### The car is represented by 1 and the goats are 0's ###
temp <- matrix(rep(c(0, 0, 1), games), ncol = 3,
byrow = T)
### Within each game (row) randomize the placement of the prizes ###
temp <- t(apply(temp, 1, sample))
### Use the switch method... ###
### If your initial pick was the car, you will be shown one of the two goats ###
### and you will switch to the other goat. If your initial pick was a goat, ###
### you will be shown the one other goat, and you will switch to the car. ###
wins.switch <- games - sum(temp[, initial.pick])
### Use the stick method... ###
### If your initial pick was the car, you will be shown one of the two goats ###
### and you will stick with the car. If your initial pick was a goat, you ###
### will be shown the other goat, and you will stick with the goat.
wins.stick <- sum(temp[, initial.pick])
### Compute the probabilities from the frequencies ###
prob.switch <- wins.switch/games
prob.stick <- wins.stick/games
game <- 1:games
cumwins.switch <- game - cumsum(temp[, initial.pick]
)
cumwins.stick <- cumsum(temp[, initial.pick])
cumprob.switch <- cumwins.switch/game
cumprob.stick <- cumwins.stick/game
plot(c(1, games), c(0, 1), type = "n", xlab = "Game",
ylab = "Probability of a Win")
title("Monty Hall Game")
lines(game, cumprob.switch, type = "s", lty = 1,
col = 1)
lines(game, cumprob.stick, type = "s", lty = 6,
col = 6)
legend(0.75 * games, 0.94999999999999996, c("Switch",
"Stick"), lty = c(1, 6), col = c(1, 6))
abline(h = 2/3, lty = 7, col = 7)
### Return the results ###
temp <- c(wins.switch, prob.switch, wins.stick,
prob.stick)
names(temp) <- c("Switch Wins", "Switch Prob",
"Stick Wins", "Stick Prob")
temp
}
### Run the new function to plot the simulation ###
montyhall2(20)
## Switch Wins Switch Prob Stick Wins Stick Prob
## 15.00 0.75 5.00 0.25montyhall2(10000)
## Switch Wins Switch Prob Stick Wins Stick Prob
## 6591.0000 0.6591 3409.0000 0.3409Apparently, Madilyn and all of the elementary school students had it right.
Proof
And now for a little probability. For either strategy, the probability of a “win” can be computed as \[P(C_2) = P(C_2|G_1) P(G_1) + P(C_2|C_1) P(C_1)\] For both strategies the probabilities for the first selection are
\(P(G_1) = 2/3\) and \(P(C_1) = 1/3\)
The conditioinal probabilities are different depending upon the strategy:
Stay: \(P(C_2|G_1) = 0\) and \(P(C_2|C_1) = 1\)
Switch: \(P(C_2|G_1) = 1\) and \(P(C_2|C_1) = 0\)
Substitution provides the final probabilities for the two strategies:
Stay: \(P(C_2) = 0 (2/3) + 1 (1/3) = 1/3\)
Switch: \(P(C_2) = 1 (2/3) + 0 (1/3) = 2/3\)
Not surprisingly, these values correspond to the results seen in the simulations above.
Comments
Since you seem to enjoy coming straight to the point, I’ll do the same. You blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to 1/2. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and in the future being more careful. – Robert Sachs, Ph.D., George Mason University
You blew it, and you blew it big! Since you seem to have difficulty grasping the basic principle at work here, I’ll explain. After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your selection or not, the odds are the same. There is enough mathematical illiteracy in this country, and we don’t need the world’s highest IQ propagating more. Shame! – Scott Smith, Ph.D., University of Florida
Your answer to the question is in error. But if it is any consolation, many of my academic colleagues have also been stumped by this problem. – Barry Pasternack, Ph.D., California Faculty Association