## Game Show Problem

**Moderator:** Marilyn

### Game Show Problem

Game Show Problem

(This material in this article was originally published in PARADE magazine in 1990 and 1991.)

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?

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.

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!)

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.

Thanks, M.I.T. I needed that!

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.

Oh hush, now.

(This material in this article was originally published in PARADE magazine in 1990 and 1991.)

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

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?

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

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.

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!)

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

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.

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

Thanks, M.I.T. I needed that!

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

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
- Site Admin
**Posts:**11**Joined:**Mon May 01, 2006 12:16 pm

Gosh, this brings back so many memories! What I don't think came through at the time was the fact that I and so many mathematicians were mortified at the behavior of so many in our community. Not only that they got a probability problem wrong, but that they were behaving so childishly. I didn't realize how nasty it really got behind the scenes until I read The Power of Logical Thinking. I regret that I didn't write a letter of support at the time.

But let me take this opportunity to voice something that annoys me: Whenever this problem is discussed it is sometimes qualified with something like, "Marilyn was essentially right." NO. Marilyn was right. Period. She makes it clear in her first answer that the host always opens a losing door. How else would you interpret the problem for Heaven's Sake! You're not going to complain that game show hosts are being unfairly stereotyped! (Everyone knows game show hosts are a tricky lot, anyway.)

I think to really get the problem you have to imagine at a lot of trials. I would invite you to look at my example in the Monty Hall Dilemma thread concerning the multiple choice test with 300 questions. It is equivalent to the Monty Hall Dilemma.

But let me take this opportunity to voice something that annoys me: Whenever this problem is discussed it is sometimes qualified with something like, "Marilyn was essentially right." NO. Marilyn was right. Period. She makes it clear in her first answer that the host always opens a losing door. How else would you interpret the problem for Heaven's Sake! You're not going to complain that game show hosts are being unfairly stereotyped! (Everyone knows game show hosts are a tricky lot, anyway.)

I think to really get the problem you have to imagine at a lot of trials. I would invite you to look at my example in the Monty Hall Dilemma thread concerning the multiple choice test with 300 questions. It is equivalent to the Monty Hall Dilemma.

Kyle

- kyle
- Intellectual
**Posts:**119**Joined:**Wed May 10, 2006 4:09 pm**Location:**Louisiana

That red and white chart makes things blazingly clear, even for me, and I have a math learning disability.

"A new scientific idea does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it." -Max Planck

- Vosh
- Intellectual
**Posts:**384**Joined:**Sun May 28, 2006 1:49 pm**Location:**Earth, just visiting on my way somewhere...

Take that College PhDs and mathemeticans! Marilyn vos Savant is better at thinking than anyone. Everytime I read about that problem I'm convinced Marilyn is indeed the smartest person in the world

Bill Gates, Richest person in the world, George W. Bush, most powerful person in the world, Marilyn vos Savant smartest person in the world.

- Shade
- Intellectual
**Posts:**108**Joined:**Thu Apr 27, 2006 10:44 pm**Location:**New York

Aieeee! We're getting killed by language here, not mathematics!

Let's not forget that probability is an exact science. Controversies arise when problems are not precisely stated. I dare say that true mathematicians will go to great pains to make sure the problem is precisely stated.

There are two DIFFERENT problems in play here. The first, which Marilyn has in mind, is:

(a) Player chooses one of the three doors at random

(b) Host, who knows what is behind the doors, deliberately selects a losing door from the remaining two and opens it.

This problem has been correctly analyzed by Marilyn. If the player switches to the remaining unopened door, the player wins with probability 2/3.

The second and entirely different problem is:

(a) Player chooses one of the three doors at random

(b) Host opens one of the two remaining doors at random, which happens to be empty.

It is the second problem which is causing confusion. Marilyn alludes to this second problem by describing the "UFO" landing on stage and randomly selecting a door. In the case of the second problem, there is no advantage to switching. You will win with probability 1/2 whether or not you switch.

In order to see this experimentally, try the following game, based on the one Marilyn outlined above. 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 randomly selects one of the two remaining cups by throwing the die, and selecting leftmost remaining cup if the number on the die is even, or rightmost if it is odd. If the host happens to select the cup with the penny, the game is a "push" and is disregarded. Lastly, the contestant "stays" and lifts up his original cup to see if it covers the penny. If it covers the penny, you have a win, otherwise you have a loss. Play until you have a total of two hundred wins and losses. If, as hypothesized, the probability of winning is 1/2 whether you "stay" or "switch," the number of wins and losses will be about equal.

Let's recognize that we're not arguing about the ability to answer basic probability questions, a skill which most college freshmen can learn. Instead, we are arguing about which of two distinct problems we are solving based on personal interpretation of an ill-defined statement. Marilyn has the "edge" in this argument, because the statement of the problem mentions that the host "knows what's behind the doors." Unfortunately, the statement of the problem doesn't indicate whether the host acted on this knowledge, or how. For all we know, the host might have chosen randomly despite his knowledge. In order to adequately define the problem we must infer that the host made use of the knowledge and deliberately avoided the winning door if applicable. Only after this ambiguity was clarified did the conciliatory mail flow in. It is a pity that the statement of the problem was not more precise at the beginning.

Think about this. If, instead of the initial problem, we simply chose to discuss Marilyn's "cup" experiment or the variation on it that I just described, would we have any controversy? I would think not, simply because the problems are described unambiguously, and these problems, when described in unambiguous language, are easily analyzed by the techniques that we learn in the first few lessons of "Probability 101."

*******************************

I know I've rambled a bit, but one final point. I recall a similar acrimonious discussion involving Marilyn in the late 1990's regarding the following problem which I will paraphrase:

A woman has two children. She takes her son out in the stroller. What is the probability that her other child is a daughter, assuming male and female children occur with equal likelihood?

Don't even try to answer that without further clarification. It is AMBIGUOUS! The answer depends on how she chose which of her two children to take in the stroller! There are at least two valid interpretations of this question, each of which has a different answer. I'll do my best to quickly analyze each.

Now, I think we can all agree that in families with exactly two children, the "two sons" outcome occurs 1/4 of the time, the "two daughters" outcome occurs 1/4 of the time, and the "one of each" outcome occurs 1/2 of the time. So consider the two distinct problems:

(a) The mother randomly selects one of her children to take in the stroller, and

(b) The mother selects a son, if she has one, to take in the stroller, otherwise she takes her daughter.

For (a):

(1) 1/4 the mother with two daughters takes a daughter

(2) 1/4 the mother with two sons takes a son

(3) 1/4 the mother with "one of each" takes a daughter

(4) 1/4 the mother with "one of each" takes a son

If you observed a son in the cart, you observed either (2) or (4) with equal likelihood. In (2) the child at home is a son, in (4) the child at home is a daughter. Therefore the child at home is a daughter with probability 1/2.

For (b)

(1) 1/4 the mother with two daughters takes a daughter

(2) 1/4 the mother with two sons takes a son

(3) 1/2 the mother with "one of each" takes a son

If you observed a son in the cart, you observed either (2) or (3). Outcome (3) is twice as likely as outcome (2) therefore the child at home is a daughter with probability 2/3.

I do seem to remember Marilyn (who insisted the answer was 2/3) battling it out with the "professors" (who insisted the answer was 1/2). Personally, I've got to believe that all involved are intelligent enough to see the ambiguity. Perhaps nobody can resist the mudslinging!

Let's not forget that probability is an exact science. Controversies arise when problems are not precisely stated. I dare say that true mathematicians will go to great pains to make sure the problem is precisely stated.

There are two DIFFERENT problems in play here. The first, which Marilyn has in mind, is:

(a) Player chooses one of the three doors at random

(b) Host, who knows what is behind the doors, deliberately selects a losing door from the remaining two and opens it.

This problem has been correctly analyzed by Marilyn. If the player switches to the remaining unopened door, the player wins with probability 2/3.

The second and entirely different problem is:

(a) Player chooses one of the three doors at random

(b) Host opens one of the two remaining doors at random, which happens to be empty.

It is the second problem which is causing confusion. Marilyn alludes to this second problem by describing the "UFO" landing on stage and randomly selecting a door. In the case of the second problem, there is no advantage to switching. You will win with probability 1/2 whether or not you switch.

In order to see this experimentally, try the following game, based on the one Marilyn outlined above. 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 randomly selects one of the two remaining cups by throwing the die, and selecting leftmost remaining cup if the number on the die is even, or rightmost if it is odd. If the host happens to select the cup with the penny, the game is a "push" and is disregarded. Lastly, the contestant "stays" and lifts up his original cup to see if it covers the penny. If it covers the penny, you have a win, otherwise you have a loss. Play until you have a total of two hundred wins and losses. If, as hypothesized, the probability of winning is 1/2 whether you "stay" or "switch," the number of wins and losses will be about equal.

Let's recognize that we're not arguing about the ability to answer basic probability questions, a skill which most college freshmen can learn. Instead, we are arguing about which of two distinct problems we are solving based on personal interpretation of an ill-defined statement. Marilyn has the "edge" in this argument, because the statement of the problem mentions that the host "knows what's behind the doors." Unfortunately, the statement of the problem doesn't indicate whether the host acted on this knowledge, or how. For all we know, the host might have chosen randomly despite his knowledge. In order to adequately define the problem we must infer that the host made use of the knowledge and deliberately avoided the winning door if applicable. Only after this ambiguity was clarified did the conciliatory mail flow in. It is a pity that the statement of the problem was not more precise at the beginning.

Think about this. If, instead of the initial problem, we simply chose to discuss Marilyn's "cup" experiment or the variation on it that I just described, would we have any controversy? I would think not, simply because the problems are described unambiguously, and these problems, when described in unambiguous language, are easily analyzed by the techniques that we learn in the first few lessons of "Probability 101."

*******************************

I know I've rambled a bit, but one final point. I recall a similar acrimonious discussion involving Marilyn in the late 1990's regarding the following problem which I will paraphrase:

A woman has two children. She takes her son out in the stroller. What is the probability that her other child is a daughter, assuming male and female children occur with equal likelihood?

Don't even try to answer that without further clarification. It is AMBIGUOUS! The answer depends on how she chose which of her two children to take in the stroller! There are at least two valid interpretations of this question, each of which has a different answer. I'll do my best to quickly analyze each.

Now, I think we can all agree that in families with exactly two children, the "two sons" outcome occurs 1/4 of the time, the "two daughters" outcome occurs 1/4 of the time, and the "one of each" outcome occurs 1/2 of the time. So consider the two distinct problems:

(a) The mother randomly selects one of her children to take in the stroller, and

(b) The mother selects a son, if she has one, to take in the stroller, otherwise she takes her daughter.

For (a):

(1) 1/4 the mother with two daughters takes a daughter

(2) 1/4 the mother with two sons takes a son

(3) 1/4 the mother with "one of each" takes a daughter

(4) 1/4 the mother with "one of each" takes a son

If you observed a son in the cart, you observed either (2) or (4) with equal likelihood. In (2) the child at home is a son, in (4) the child at home is a daughter. Therefore the child at home is a daughter with probability 1/2.

For (b)

(1) 1/4 the mother with two daughters takes a daughter

(2) 1/4 the mother with two sons takes a son

(3) 1/2 the mother with "one of each" takes a son

If you observed a son in the cart, you observed either (2) or (3). Outcome (3) is twice as likely as outcome (2) therefore the child at home is a daughter with probability 2/3.

I do seem to remember Marilyn (who insisted the answer was 2/3) battling it out with the "professors" (who insisted the answer was 1/2). Personally, I've got to believe that all involved are intelligent enough to see the ambiguity. Perhaps nobody can resist the mudslinging!

- niles
- Scholar
**Posts:**45**Joined:**Thu Jul 06, 2006 2:32 am

Niles,

I suspect the reason none of the conciliatory and nonconciliatory (except yours) hint at the trouble being an ambiguously stated puzzle is because saying, "the host knows what is behind the doors" doesn't make clear whether or not he flipped a coin when he actually opened a door is what we, in the business, refer to as "rather a stretch". All you've managed to do for your college profs is accuse them of having the reading comprehension of a retard.

Now, you are not a retard, so don't you think your fantastic stretching and rather plain and transparent attempt to find something, anything, wrong with the puzzle itself, with such desperation that you don't care that everyone can see the scene you're creating and can tell you're just in a state, is a result of ego, and not necessarilly a short coming with raw intelligence?

Here is another puzzle. The thing about puzzles is that they typically do not hold your hand through the problem. The point of a puzzle is there are things in it that can trip you up.

A man is looking at a painting on a wall. He says, "brothers and sisters have I none, but this man's father is my father's son." Who is he looking at?

It might be silly, and a stretch, to kvetch that the problem doesn't make clear that the man is not a transvestite...

I can hear someone saying that here at last must be the very last way anyone can come up with to nay say the puzzle and M's adventure with it, that surely this last bit of, well, there's no other word I can think of or it, stretching, surely means every possibility has been exhausted.

But I know ppl. and their credulity should not be underestimated. So I'll just wait for the next one. If I say to myself, "there can't be others, that would be insane" one will come along and I'll be depressed, and I don't wanna be depressed.

I suspect the reason none of the conciliatory and nonconciliatory (except yours) hint at the trouble being an ambiguously stated puzzle is because saying, "the host knows what is behind the doors" doesn't make clear whether or not he flipped a coin when he actually opened a door is what we, in the business, refer to as "rather a stretch". All you've managed to do for your college profs is accuse them of having the reading comprehension of a retard.

Now, you are not a retard, so don't you think your fantastic stretching and rather plain and transparent attempt to find something, anything, wrong with the puzzle itself, with such desperation that you don't care that everyone can see the scene you're creating and can tell you're just in a state, is a result of ego, and not necessarilly a short coming with raw intelligence?

Here is another puzzle. The thing about puzzles is that they typically do not hold your hand through the problem. The point of a puzzle is there are things in it that can trip you up.

A man is looking at a painting on a wall. He says, "brothers and sisters have I none, but this man's father is my father's son." Who is he looking at?

It might be silly, and a stretch, to kvetch that the problem doesn't make clear that the man is not a transvestite...

I can hear someone saying that here at last must be the very last way anyone can come up with to nay say the puzzle and M's adventure with it, that surely this last bit of, well, there's no other word I can think of or it, stretching, surely means every possibility has been exhausted.

But I know ppl. and their credulity should not be underestimated. So I'll just wait for the next one. If I say to myself, "there can't be others, that would be insane" one will come along and I'll be depressed, and I don't wanna be depressed.

"A new scientific idea does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it." -Max Planck

- Vosh
- Intellectual
**Posts:**384**Joined:**Sun May 28, 2006 1:49 pm**Location:**Earth, just visiting on my way somewhere...

Vosh,

I'm not quite sure how to respond to this. My purpose was to present my analysis of why so many people in the field of probability were at odds with Marilyn. My analysis, which is merely my opinion, is that they misinterpreted the problem and solved a similar but different problem. Clearly you believe that my explanation is a "stretch" and if so I am sure you have the ability to present a well-reasoned argument which is supported by more than an "everyone can see" assertion.

I don't agree with your assessment of my response as "non-conciliatory" since I stated quite clearly that Marilyn had correctly analyzed the first problem. I think that it is clear without my saying so that although the problem was ill-defined, Marilyn immediately clarified her assumptions and proceeded correctly based upon them. On the other hand, some people who work in the field of probability have a predisposition to view problems as conditional probability problems, and would tend to formulate and analyze the alternate problem (where the door is opened at random, and the player has conditional knowledge of that outcome), stopping only to make sure that the original statement of the problem does not exclude that interpretation.

As for ego, certainly there is no shortage of ego in the mean-spirited responses which Marilyn chose to highlight in her posting. Marilyn exhibits extraordinary grace in her eloquent and straightforward response to those criticisms, with never a word of condescension or anger. Although I cannot hope to attain Marilyn's high genius, I can certainly try to learn from her ego-free demeanor. I would never expect to read the "r" word in any of Marilyn's responses, and I question why you would use it, negated or not, to describe me or my comments on those who misinterpreted the problem.

The only reason I decided to comment was that I read the thread, and it seemed to me that people were happily jumping to the conclusion that all of those who wrote in to disagree with Marilyn in 1990-91 were incompetent in spite of whatever impressive titles or degrees they might have. The comments were overwhelmingly of the "Marilyn was right, PhD's were wrong, ha, ha" genre. I thought, perhaps incorrectly, that a little bit of reasoned analysis might be welcome.

I'm not quite sure how to respond to this. My purpose was to present my analysis of why so many people in the field of probability were at odds with Marilyn. My analysis, which is merely my opinion, is that they misinterpreted the problem and solved a similar but different problem. Clearly you believe that my explanation is a "stretch" and if so I am sure you have the ability to present a well-reasoned argument which is supported by more than an "everyone can see" assertion.

I don't agree with your assessment of my response as "non-conciliatory" since I stated quite clearly that Marilyn had correctly analyzed the first problem. I think that it is clear without my saying so that although the problem was ill-defined, Marilyn immediately clarified her assumptions and proceeded correctly based upon them. On the other hand, some people who work in the field of probability have a predisposition to view problems as conditional probability problems, and would tend to formulate and analyze the alternate problem (where the door is opened at random, and the player has conditional knowledge of that outcome), stopping only to make sure that the original statement of the problem does not exclude that interpretation.

As for ego, certainly there is no shortage of ego in the mean-spirited responses which Marilyn chose to highlight in her posting. Marilyn exhibits extraordinary grace in her eloquent and straightforward response to those criticisms, with never a word of condescension or anger. Although I cannot hope to attain Marilyn's high genius, I can certainly try to learn from her ego-free demeanor. I would never expect to read the "r" word in any of Marilyn's responses, and I question why you would use it, negated or not, to describe me or my comments on those who misinterpreted the problem.

The only reason I decided to comment was that I read the thread, and it seemed to me that people were happily jumping to the conclusion that all of those who wrote in to disagree with Marilyn in 1990-91 were incompetent in spite of whatever impressive titles or degrees they might have. The comments were overwhelmingly of the "Marilyn was right, PhD's were wrong, ha, ha" genre. I thought, perhaps incorrectly, that a little bit of reasoned analysis might be welcome.

- niles
- Scholar
**Posts:**45**Joined:**Thu Jul 06, 2006 2:32 am

You can't say "retarded"? You have to say, "the r word"?

Well, back to work on my space ship.

Well, back to work on my space ship.

"A new scientific idea does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it." -Max Planck

- Vosh
- Intellectual
**Posts:**384**Joined:**Sun May 28, 2006 1:49 pm**Location:**Earth, just visiting on my way somewhere...

I am inclined to side with Marilyn, but something still keeps me awake at night. Let's say you initially choose door 1, then the host shows a goat behind door 3.

Consider this argument for Marilyn's result:

With all doors closed, the probability that door 1 contains the car is 1/3. Thus, the probability that one of the other two doors contains the car is 2/3. When door 3 is opened to reveal a goat, we can safely set its probability of containing the car to 0. In effect, the entire 2/3 probability of both doors you did not select "shifts" entirely to the unopened door 2. Thus, switching to door 2 gives you a 2/3 probability of winning the car.

Consider this argument against Marilyn's result by similar reasoning:

With all doors closed, the probability that any two doors combined contain the car is 2/3. So it is with doors 1 and 3. However, when door 3 is opened to reveal a goat, we can safely set its probability of containing the car to 0. In effect, the entire 2/3 probability of (doors 1 and 3 combined) containing the car "shifts" entirely to door 1, the one you selected. Thus, staying at door 1 gives you a 2/3 probability of winning the car.

I know something is wrong with the second argument (or possibly both), but I can't pinpoint it precisely. I believe the reason people want to think the chance of each unopened door containing the car to be 1/2 is that they go through both of these arguments and see that the results are equal, then normalize the results by multiplying each by 3/4.

In any case, please show me the flaw in the second argument!

Consider this argument for Marilyn's result:

With all doors closed, the probability that door 1 contains the car is 1/3. Thus, the probability that one of the other two doors contains the car is 2/3. When door 3 is opened to reveal a goat, we can safely set its probability of containing the car to 0. In effect, the entire 2/3 probability of both doors you did not select "shifts" entirely to the unopened door 2. Thus, switching to door 2 gives you a 2/3 probability of winning the car.

Consider this argument against Marilyn's result by similar reasoning:

With all doors closed, the probability that any two doors combined contain the car is 2/3. So it is with doors 1 and 3. However, when door 3 is opened to reveal a goat, we can safely set its probability of containing the car to 0. In effect, the entire 2/3 probability of (doors 1 and 3 combined) containing the car "shifts" entirely to door 1, the one you selected. Thus, staying at door 1 gives you a 2/3 probability of winning the car.

I know something is wrong with the second argument (or possibly both), but I can't pinpoint it precisely. I believe the reason people want to think the chance of each unopened door containing the car to be 1/2 is that they go through both of these arguments and see that the results are equal, then normalize the results by multiplying each by 3/4.

In any case, please show me the flaw in the second argument!

- Kenyai
- Thinker
**Posts:**2**Joined:**Wed Jul 12, 2006 8:10 pm**Location:**Earth

Forgive me for double-posting, but I think I've figured out the answer to my question. In the upper paragraph, doors 2 and 3 are not restricted from being opened by the rules. However, in the lower paragraph, door 1 is restricted from being opened by the rules, while door 3 is not. That is why the probabilities are different. Am I correct now?

- Kenyai
- Thinker
**Posts:**2**Joined:**Wed Jul 12, 2006 8:10 pm**Location:**Earth

### a formula and alternate explanation for your approval

I accidently posted this in the wrong forum, but reposted it here (duh):

this is how many people imagined the probabilities:

1. contestant picks door #1 (1/3 prob.)

2. host opens #3 UNKNOWINGLY (got lucky and opened the goat door)

3. each remaining door contains 1/2 chance)

here's the reality

1. contestant picks a door #1 (1/3 prob.)

2. host opens goat door KNOWINGLY

3. since door #2 and #3 each had a 1/3 chance, and the host removed one, you add those 2 doors to get the new probability of 2/3

Its the KNOWINGLY that changes the probabilties because the host is choosing, this IS NOT left to chance. That is why the probability of the remaining doors are added, because the removal of the doors is not random.

Here is the formula to calculate the probability for each remaining door, even if your gameshow had 100 doors:

(Original door chosen) / (Total doors)

X (Total doors - Original door chosen)

/ (Total doors still closed - Original door chosen)

I explain it in detail:

(Original door chosen) / (Total doors)

[this is the probability of a single door]

X (Total doors - Original door chosen)

[this is the probability of remaining doors]

/ (Total doors still closed - Original door chosen)

[this divides probabilty amongst remaining doors]

I dont have a proof yet, but I pretty much failed in math, so feel free to add to this.

this is how many people imagined the probabilities:

1. contestant picks door #1 (1/3 prob.)

2. host opens #3 UNKNOWINGLY (got lucky and opened the goat door)

3. each remaining door contains 1/2 chance)

here's the reality

1. contestant picks a door #1 (1/3 prob.)

2. host opens goat door KNOWINGLY

3. since door #2 and #3 each had a 1/3 chance, and the host removed one, you add those 2 doors to get the new probability of 2/3

Its the KNOWINGLY that changes the probabilties because the host is choosing, this IS NOT left to chance. That is why the probability of the remaining doors are added, because the removal of the doors is not random.

Here is the formula to calculate the probability for each remaining door, even if your gameshow had 100 doors:

(Original door chosen) / (Total doors)

X (Total doors - Original door chosen)

/ (Total doors still closed - Original door chosen)

I explain it in detail:

(Original door chosen) / (Total doors)

[this is the probability of a single door]

X (Total doors - Original door chosen)

[this is the probability of remaining doors]

/ (Total doors still closed - Original door chosen)

[this divides probabilty amongst remaining doors]

I dont have a proof yet, but I pretty much failed in math, so feel free to add to this.

I like Shopping carts

- hippypink
- Thinker
**Posts:**4**Joined:**Wed Jul 19, 2006 7:56 pm

Kenyai wrote:I am inclined to side with Marilyn, but something still keeps me awake at night. Let's say you initially choose door 1, then the host shows a goat behind door 3.

Consider this argument for Marilyn's result:

With all doors closed, the probability that door 1 contains the car is 1/3. Thus, the probability that one of the other two doors contains the car is 2/3. When door 3 is opened to reveal a goat, we can safely set its probability of containing the car to 0. In effect, the entire 2/3 probability of both doors you did not select "shifts" entirely to the unopened door 2. Thus, switching to door 2 gives you a 2/3 probability of winning the car.

Consider this argument against Marilyn's result by similar reasoning:

With all doors closed, the probability that any two doors combined contain the car is 2/3. So it is with doors 1 and 3. However, when door 3 is opened to reveal a goat, we can safely set its probability of containing the car to 0. In effect, the entire 2/3 probability of (doors 1 and 3 combined) containing the car "shifts" entirely to door 1, the one you selected. Thus, staying at door 1 gives you a 2/3 probability of winning the car.

Probability isn't a description of how the game turned out, so it doesn't change when you lose or win. When you role a 6 sided die trying to get a 4 and you get a 2, your prob. of getting a 4 is not in that moment, zero. Probability doesn't describe what happened, it's a description of how many chances you have to win compared to the total number of things that can happen. 6 things can happen; you've got one chance to win. This doesn't change no matter what the dice does. So if you open one door and get a goat, it is still the case that the probability measurement for that door is 1/3; because probability isn't a description of what happened, it is merely the ratio that represents the number of chances to win compared to the total number of things that could happen. If you open two doors, then your number of chances to win is two out of three, and that is a measurement that remains true even if you get two goats.

The prob. only changes if you begin with a different game, with a different number of total possible outcomes and chances to win.

It really helps to visualize the total number of possibilities like this;

1) car,goat,goat

2) goat,car,goat

3) goat,goat,car

Playing once means that there can only be on favourable out come out of three total possible outcomes. You are either looking at 1, 2, or 3, you don't know which, but there are those three total possibilities. If you play once, the door you picked could be the ONE combo that has a car at that door, so you have one chance to win, one out of three total possible scenarios, so P = 1/3. Playing twice means picking doors that could be winning doors for two of the three combos, which means of the three combos, two of them are favourable, so P = 2/3. Probability is the no. of favourable events divided by the total number of existing events. If this was all there was to the puzzle, then, the soln. would be easy as pie!

It still is a simple prob. calculation, but the puzzle dazzles and distracts us with the extra added feature that the host ALWAYS KNOWINGLY reveals a goat door. So now if you switch what is the probability?

You now go back and look at that list, or table if you prefer, and see that if you switch, and keep in mind that someone ALWAYS reveals a goat door, then there are two chances, two combinations out of the three possible in the list, where you will win. 2/3!

- Vosh
- Intellectual
**Posts:**384**Joined:**Sun May 28, 2006 1:49 pm**Location:**Earth, just visiting on my way somewhere...

dbridges wrote:Anyone that does not believe you are correct (especially the learned scholars) should become familiar with the Bayes Theorem. Rev. Bayes was a true genius and every self-proclaimed mathematician would do well to learn from him.

Just the phrase, "Bayes theorem" scares me to death.

- Vosh
- Intellectual
**Posts:**384**Joined:**Sun May 28, 2006 1:49 pm**Location:**Earth, just visiting on my way somewhere...

Hello and greetings from Bulgaria!

Reading this was pretty funny. And one doesn't even need to know "advanced" mathematics and theorems to solve the problem.

This is a classical and beautiful example of how people fool themselves, even PhDs, etc... It takes no more than a minute for one to solve it, literally seeing the right answer.

In the case with the PhDs, they have fooled themselves due to a lack of attention and have given "hard" answers. It's that simple. What happens - with little observation, their classically trained minds have run for the first "idea" (classical factual knowledge) for solving probability problems, building up a false assumption; a key that just doesn't fit with the doors, because... the key is in the host.

Marilyn used a word on which I'd like to accentuate - "visualize".

"I'm very concerned with the general public's lack of mathematical skills." - Robert Sachs, Ph. D.

I'm very concerned with the non-general public's lack of seeing.

Best regards,

K.

Reading this was pretty funny. And one doesn't even need to know "advanced" mathematics and theorems to solve the problem.

This is a classical and beautiful example of how people fool themselves, even PhDs, etc... It takes no more than a minute for one to solve it, literally seeing the right answer.

In the case with the PhDs, they have fooled themselves due to a lack of attention and have given "hard" answers. It's that simple. What happens - with little observation, their classically trained minds have run for the first "idea" (classical factual knowledge) for solving probability problems, building up a false assumption; a key that just doesn't fit with the doors, because... the key is in the host.

Marilyn used a word on which I'd like to accentuate - "visualize".

"I'm very concerned with the general public's lack of mathematical skills." - Robert Sachs, Ph. D.

I'm very concerned with the non-general public's lack of seeing.

Best regards,

K.

Aut viam inveniam aut faciam.

- Kamen
- Thinker
**Posts:**12**Joined:**Sat Aug 05, 2006 3:36 pm**Location:**Bulgaria

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