Geometry.Net - Theorems_And_Conjectures: Monty Hall Problem

The Infamous Monty Hall Problem
Short introduction for beginners.
http://www.comedia.com/hot/monty.html
The Monty Hall Problem
The monty hall problem. Check the references. About Obstinacy, Comprehension, andthe monty hall problem An Exchange with a Skeptical Reader. Footnotes
http://www.io.com/~kmellis/monty.html

Extractions: The Monty Hall Problem gets its name from the TV game show, "Let's Make A Deal," hosted by Monty Hall (See Footnote 1 . The scenario is such: you are given the opportunity to select one closed door of three , behind one of which there is a prize. The other two doors hide "goats" (or some other such "non-prize"), or nothing at all. Once you have made your selection, Monty Hall will open one of the remaining doors, revealing that it does not contain the prize (See Footnote 2 . He then asks you if you would like to switch your selection to the other unopened door, or stay with your original choice. Here is the problem: Does it matter if you switch? This problem is quite interesting, because the answer is felt by most people - including mathematicians - to be counter-intuitive. For most, the "solution" is immediately obvious (they believe); and that is the end of it. But it's not. Because most of the time, that "obvious" solution is incorrect. The correct solution is not obvious. Further, I've found that many persons have difficulty grasping the validity of the correct solution even after several explanations. Thus, this short paper. Before I continue, you may wish to attempt to solve this problem by yourself. You've a good chance to do so, because you now know not to trust your instincts in this; and that you should consider the problem very carefully. Try it.
Monty Hall Problem Web Sites
The WWW Tackles The monty hall problem. Discourse on the monty hall problem
http://math.rice.edu/~ddonovan/montyurl.html
The Monty Hall Problem
The monty hall problem Game show setting. There are 3 doors, behind one of which is a prize. Monty Hall, the host, asks you to pick a door, any door. You pick door A (say).
http://astro.uchicago.edu/rranch/vkashyap/Misc/mh.html

Extractions: Game show setting. There are 3 doors, behind one of which is a prize. Monty Hall, the host, asks you to pick a door, any door. You pick door A (say). Monty opens door B (say) and shows voila there is nothing behind door B. Gives you the choice of either sticking with your original choice of door A, or switching to door C. Should you switch? Yes. In other words, the probability that the prize is behind door C is higher when Monty opens door B, and you SHOULD switch! kashyap@ockham.uchicago.edu
The Monty Hall Problem
Contains a look at the strategy for working out the solution, a page on the monty hall problem debate and a different story to show the maths.
http://www.icdc.com/~samba/marlright/monty.htm

Extractions: Marilyn gives the correct solution to a simple but tricky problem. She is viciously attacked by an army of highly-placed academics who insist that she is wrong. She holds her ground. They finally admit their error and are repentant. Monty Hall is the MC of a television quiz show called Let's Make A Deal. One of the games he offers uses three doors. Behind one door is an expensive automobile and behind each of the others is a goat. Nobody but Monty and the stage crew know which door leads to the automobile. The essence of the game is that the contestant is permitted to choose one of the three doors. If he chooses the right door he wins the automobile. If he chooses a wrong one he gets goat cheese. There is a twist. After the contestant has chosen a door Monty asks him if he wants to change his mind. To help him, or perhaps to confuse him, he opens one of the doors not chosen and shows that it conceals one of the goats. The question is, should you stick with the door you chose originally or should you accept the switch offered by Monty, and what is the probability of winning the automobile in each case? There has been some discussion about whether or not Monty always offered the switch and if his doing so depended on whether the contestant chose the right door to begin with. Monty said that he did not always offer the switch. Sometimes he would offer it to lure the contestant away from the right door, and to keep the audience off balance sometimes he would bluff and offer the switch when the contestant had chosen wrongly to see if he could make him be stubborn. If we knew the probability that Monty would offer the switch as a function of the accuracy of the contestant's choice we could use it in deciding the best strategy. That is the approach we would take in attempting to crack the game commercially. In presenting it as a puzzle, however, it is customary to assume that Monty always offers the switch regardless. Decide on your strategy and turn the page when you are ready.
A New Approach To The Monty Hall Problem
Introduces the problem and tries to look at the problem in a new light.
http://www.reenigne.org/maths/montyhall.html

Extractions: Reams and reams have been written about the Monty Hall problem, but no-one seems to have mentioned a simple fact which, once realised, makes the whole thing seem intuitive. The Monty Hall show is a (possibly fictional, I'm not sure) TV gameshow. One couple have beaten all the others to the final round with their incredible skill at answering questions on general knowledge and popular culture, and now have a chance to win a Brand New Car. There are three doors. The host explains that earlier, before the couple arrived, a producer on the show rolled a dice. If a 1 or a 4 was rolled, the car was placed behind the red door. If a 2 or a 5 was rolled, it was placed behind the blue door and if a 3 or a 6 was rolled, it was placed behind the yellow door. The host invites the couple to pick which door they think the car is behind. He then opens one of the other two doors and there's no car behind the door! (He knows where the car is, so he can always arrange for this to happen). Then the host asks the couple if they want to change their mind about which door they think the car is behind. Should they change? Does it make a difference. Most people's first reaction is that it can't matter. How can it? The car has a one in three chance of being behind each of the doors.
U Of T Mathematics Network -- Problems And Puzzles
Includes interactive games, problems and puzzles including the monty hall problem and the Tower of Hanoi and questions pages with answers and discussion.
http://www.math.toronto.edu/mathnet/probpuzz.html

Extractions: Go to University of Toronto Mathematics Network Home Page You can select any of the items below: Try your hand at these problems, and mail in your answers! If your interest is in recreational mathematics, try playing these games, then figuring out the mathematics behind them. The following sites are not part of the University of Toronto Mathematics Network, but since there are already many good traditional-style problems available on the Internet, we decided we'd just point you to them, while we spend more time developing the interactive projects and activities unique to this site. A good, comprehensive source of many mathematical materials. Not a problem collection, but a newsletter chock full of puzzles, trivia, humour, and even some real mathematics. Published by undergraduate mathematicians at the University of Toronto. This page last updated: September 27, 1999
Marilyn Vos Savant's Monty Hall Problem
Simulator. Uses buttons as labels and controls. Counts tries and provides percentages. Can be Reset without page refresh.
http://www.mindspring.com/~tluthman/vossavant.htm
Education, Mathematics, Fun, Monty Hall Dilemma
Subject The monty hall problem. Date Wed, 18 Apr 2001 122306 +0200. From PeterStikker. Dear madam/sir,. Your sincerely, PH Stikker. The monty hall problem.
http://www.cut-the-knot.com/peter.shtml

Extractions: Subject: The monty hall problem Date: Wed, 18 Apr 2001 12:23:06 +0200 From: Peter Stikker Dear madam/sir, As a fourth year student at the Educational faculty of Amsterdam (The Netherlands), studying math-teacher, I found your website very usefull over the years. I recently took some interest in the Monty Hall problem (a.k.a. the goat problem) and found no satisfactory explenations of the answer. Sure they are all correct but "non-mathematical" friends of mine weren't really convinced by these "proofs". So I started to try to proof it myself. I went back into my very elemantery books on probability and decided to draw a probability tree well the results are attached as a word-2000 document. If you find this usefull enough to put on your website please feel free to use it and correct my English (I am dutch). Hope to hear from you soon. Your sincerely, P.H. Stikker The Monty Hall Problem When I read about this problem for the first time, I could not believe it (like many of us). I started looking for some proofs and finally found some on the internet and in some books, but none where really easy or not satisfactory (? Sorry for the bad English, I'm Dutch). As I am studying to become a math-teacher I wanted an easy to understand proof and started making a probability tree. This gave the following result: But what happened? If I add up the "Wins" and add up the "Looses" they are both 6. Then I remembered that I forgot to put in the chances. So I added those with the following result:
Monty Hall
The monty hall problem. (This is similar to the routine on the TV game showLet's Make a Deal, hosted by Monty Hall, hence the name of the problem.)
http://www.hofstra.edu/~matsrc/MontyHall/MontyHall.html

Extractions: Suppose you're on a game show, and you're given the choice of three doors: Behind one door is the Grand Prize; behind the others, Booby Prizes. You pick a door, say Door A, and the host, who knows what is behind each door, opens another door, say Door B, revealing a Booby Prize. The host then offers you the opportunity to change your selection to Door C. Should you stick with your original choice or switch? Does it make any difference? (This is similar to the routine on the TV game show Let's Make a Deal , hosted by Monty Hall, hence the name of the problem.) Assuming that the host always chooses to open a door with a Booby Prize, and would never reveal the Grand Prize, the possibly surprising answer is that you should switch to the third door, which is now twice as likely as your original choice to be hiding the Grand Prize. This problem can be analyzed using Bayes' theorem or trees (see "You're the Expert" at the end of Chapter 7 of Finite Mathematics , Second Edition ), but here is an intuitive argument. When you chose Door A, the probability that you chose the Grand Prize was 1/3 and the probability that it was behind one of the other doors was 2/3. By showing you which of Doors B and C does not hide the Grand Prize (Door B, say), the host is giving you quite a bit of information about those two doors. The probability is still 2/3 that one of them hides the Grand Prize, but now you know which of the two it would be: Door C. So, the probability is still only 1/3 that the Grand Prize is behind Door A, but 2/3 that it is behind Door C.
Answer To The Monty Hall Problem
Answer to the monty hall problem. Hold on to your hats you *double*your chances by switching. This is, at first look, way counter
http://www.comedia.com/hot/monty-answer.html

Extractions: This is, at first look, way counter-intuitive, so here's an attempt at an explanation: Take a look at this matrix of possibilities: Door ~~~~ case A B C ~~~~ 1 bad bad good 2 bad good bad 3 good bad bad Let's assume you choose door A you have a 1/3 chance of a good prize. But (this is key) Monty knows what is behind each door , and shows a bad one. In cases 1 and 2, he eliminates doors B and C respectively (which happen to be the only remaining bad door) so a good door is left: SWITCH! Only in case 3 (you lucked out in your original 1 in 3 chances) does switching hurt you. So, your probability goes up from 1/3 to 2/3 if you switch after being shown a bad door. Caveat: of course, this only works if Monty is guaranteed to show you a bad door every time after you choose a door, something that was not assured in the original game show. Home Broadcatch Technologies CoMedia Consulting Monty ... Hot List This page maintained by CoMedia Consulting webmaster@CoMedia.com
Cheap Monty Hall
Fully HTMLbased simulator for the problem. All on one page.Category Science Math Recreations Famous Problems Monty HallTHE monty hall problem. The following is a simple simulation of themonty hall problem. It took a total of 1.5 hours to create. Adding
http://www.utstat.toronto.edu/david/MH.html
Welcome To Monty Hall!
Contains a introduction to the problem, a hint to the solution and the solution. The author attempts Category Science Math Recreations Famous Problems Monty Hall The socalled monty hall problem is an ancient net.chestnut which, every time itappears on the net ignites mega-flame wars and consumes enormous bandwidth as
http://www.sover.net/~nichael/puzzles/monty/

Extractions: The so-called Monty Hall Problem is an ancient net.chestnut which, every time it appears on the net ignites mega-flame wars and consumes enormous bandwidth as folks wrangle (once again) over the problem and its solution. The following is an attempt to supply an introduction to the problem (in case you haven't seen it before)
The Monty Hall Problem, Part 2
The monty hall problem, Part 2. We problems. If you have an axe to grindtry a newsgroup such as rec.puzzles. monty hall problem Links.
http://www.icdc.com/~samba/marlright/monty1.htm

Extractions: We are proceeding on the basis that you have no advance knowledge of which door the automobile is behind and that Monty offers the switch whether you have chosen the correct door or not. The popular but incorrect answer is that the probability of winning is 1/2 whether you switch or not. The correct answer is that you should always switch and if you do your probability of winning is 2/3. That is the answer Marilyn gave, and it inspired a flood of indignant mail telling her that she was wrong. She took delight in printing the most ostentatious of these, ones on university stationery signed by the heads of mathematics departments. On the other hand, her biggest supporters were classes of school children who didn't try to figure it out but ran simulations instead. Here is the way Monty explains it. When the contestant made his first choice his probability of being right was 1/3. When Monty opened the second door the contestant would think his chance of being right had gone up to 1/2. It hadn't, though, it was still 1/3; and since the only other place the auto could be was behind the third door, the probability of it's being there was 2/3. This is the way Marilyn first explained it: "Suppose there were 100 doors and Monty opened 98 ot them. You'd switch pretty fast then, wouldn't you?"
Math Forum: Ask Dr. Math FAQ: The Monty Hall Problem
The monty hall problem Let's Make a Deal Francois Bergeron The monty hall problem- Keith M. Ellis Marilyn is tricked by a game show host - Herb Weiner.
http://mathforum.org/dr.math/faq/faq.monty.hall.html

Extractions: For a review of basic concepts, see Introduction to Probability and Permutations and Combinations. Let's Make a Deal! Imagine that the set of Monty Hall's game show Let's Make a Deal has three closed doors. Behind one of these doors is a car; behind the other two are goats. The contestant does not know where the car is, but Monty Hall does. The contestant picks a door and Monty opens one of the remaining doors, one he knows doesn't hide the car. If the contestant has already chosen the correct door, Monty is equally likely to open either of the two remaining doors. After Monty has shown a goat behind the door that he opens, the contestant is always given the option to switch doors. What is the probability of winning the car if she stays with her first choice? What if she decides to switch? One way to think about this problem is to consider the sample space, which Monty alters by opening one of the doors that has a goat behind it. In doing so, he effectively removes one of the two losing doors from the sample space. We will assume that there is a winning door and that the two remaining doors, A and B, both have goats behind them. There are
About "The Monty Hall Problem"
The monty hall problem. Visit this site http//mathforum.org/dr.math/faq/faq.monty.hall.html.Author Math Forum, a Classic Problem from the Ask Dr. Math FAQ.
http://mathforum.org/library/view/7488.html

Extractions: Visit this site: http://mathforum.org/dr.math/faq/faq.monty.hall.html Author: Math Forum, a Classic Problem from the Ask Dr. Math FAQ Description: Let's Make a Deal: If you're shown a goat behind the second of three doors, should you stay with your first choice or switch? Levels: Elementary Middle School (6-8) High School (9-12) Languages: English Resource Types: Problems/Puzzles Math Topics: Probability
Untitled
They called it the monty hall problem the title of an analysis in the journalAmerican Statistician in 1976 or sometimes Monty's Dilemma or the Monty
http://www.dartmouth.edu/~chance/course/topics/Monty_Hall.html
GRAND ILLUSIONS
THE monty hall problem. This story is true, and comes from an Americantv game show. Here is the situation. Finalists in a tv game
http://www.grand-illusions.com/monty.htm

Extractions: This story is true, and comes from an American tv game show. Here is the situation. Finalists in a tv game show are invited up onto the stage, where there are three closed doors. The host explains that behind one of the doors is the star prize - a car. Behind each of the other two doors is just a goat. Obviously the contestant wants to win the car, but does not know which door conceals the car. The host invites the contestant to choose one of the three doors. Let us suppose that our contestant chooses door number 3. Now, the host does not initially open the door chosen by the contestant. Instead he opens one of the other doors - let us say it is door number 1. The door that the host opens will always reveal a goat. Remember the host knows what is behind every door! The contestant is now asked if they want to stick with their original choice, or if they want to change their mind, and choose the other remaining door that has not yet been opened. In this case number 2. The studio audience shout suggestions. What is the best strategy for the contestant? Does it make any difference whether they change their mind or stick with the original choice? The answer to this question is not intuitive. Basically, the theory says that if the contestant changes their mind, the odds of them winning the car double. And over many episodes of the tv show, the facts supported the theory - those people that changed their mind had double the chance of winning the car.
Torrez.net : Let's Make A Deal
It's sometimes referred to as the monty hall problem because of the gameshow Let's Make a Deal where contestants were faced with this very game.
http://www.torrez.net/archives/lets_make_a_deal.php

Extractions: A few weeks back I read this account of a brain teaser. One of those logic puzzles that, no matter how many times someone explains it to you, your brain keeps saying, "That can't be right." The gist of it is this: You are on a gameshow where you are put before three doors. You are asked to choose a door you think the prize is behind. The announcer reveals one door that has a booby prize behind it, such as a goat. You are then asked, "Would you like to change your answer?" The experts (most of them, anyway) say you should. It's sometimes referred to as the "Monty Hall Problem" because of the gameshow "Let's Make a Deal" where contestants were faced with this very game. I decided to whip up an example of this logic puzzle so you could try it for yourself. Before you start check out the related links below for more background if you're not clear on anything. Try it out here . Don't want to wait? Let Java prove it Artwork by Goopy
The Monty Hall Problem
The monty hall problem, This problem goes back a number of years and is used todemonstrate how angry people can get when they dont agree with an answer.
http://www.coastaltech.com/monty.htm

Extractions: The Monty Hall Problem Now you are facing the 2 remaining doors. The one you originally chose and the remaining closed door. You are now asked whether you want to keep door 1, the choice you originally made or switch to door 3, the other closed door. Do you maximize your chances of winning by switching doors, staying with your first choice, or does it not make any difference? Answer : Switching to door 3 increases the probability of winning the prize from 1/3 to 2/3. If you think that the problem really involves 2 doors and 1 prize then the odds must logically be 50-50. But opening a door with full knowledge of what is behind it does not add any information to the problem and the probabilities do not change. When all three doors were closed, there was a one out of three (1/3) probability of the prize being behind the door you chose. There was a two out of three (2/3) probability that the prize was behind one of the other two doors. Now door 2 is opened, and the probabilities do not change. Since you obviously won't choose the open door, the odds are in your favor to choose door 3. Another way to view the problem is to imagine another person entering the room and seeing two closed doors and one open door. If this person is asked about the odds of finding the prize the chances are 50-50. But if the person is allowed to ask you one question, they will ask which door you chose first. That one clearly had a 1/3 probability of being correct and they will select the other door.

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