1. The Infamous Monty Hall Problem Short introduction for beginners. http://www.comedia.com/hot/monty.html

The Infamous Monty Hall Problem The Setup you are presented with 3 doors (A, B, C) only one of which has something valuable to you behind it (the others are bogus) you do not know what is behind any of the doors You choose a door Monty then counters by showing you what is behind one of the other doors (which is a bogus prize), and asks you if you would like to stick with the door you have, or switch to the other unknown door The question is should you switch? Another question is Does it matter? The answer lies behind this link Don't look until you've decided upon your answer. Home Broadcatch Technologies CoMedia Consulting Words ... Hot List This page maintained by CoMedia Consulting webmaster@CoMedia.com Last modified: Fri Aug 30 13:51:58 PDT 1996

2. Monty Hall Problem Web Sites The WWW Tackles The monty hall problem. Discourse on the monty hall problem Simulationsof the monty hall problem Three Door Puzzle A great site! http://math.rice.edu/~ddonovan/montyurl.html

The WWW Tackles The Monty Hall Problem Discourse on the Monty Hall Problem: The following sites all pose the problem and discuss the solution and the conflict the solution has with our intuition. Monty Hall, Let's Make a Deal, Problem http://www.nadn.navy.mil/MathDept/courses/sm230/montyhal.htm The Infamous Monty Hall Problem http://www.comedia.com/Hot/monty.html ... http://www.wiskit.com/marilyn.gameshow.html Simulations of the Monty Hall Problem: Three Door Puzzle A great site! The problem is meticulously posed, the graphics are excellent and it is fun to "play". The is a full explaination of the solution and nice generalization of the problem for n doors. http://www.intergalact.com/threedoor/threedoor.html Education, Mathematics, Fun, Monty Hall Dilemma Another great site. Details of the "Ask Marilyn" controversy and a cool simulation. http://www.cut-the-knot.com/hall.html

3. The Monty Hall Problem The monty hall problem. The Statement. 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. http://astro.uchicago.edu/rranch/vkashyap/Misc/mh.html

The Monty Hall Problem The Statement 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? The Solution 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

4. 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

A new approach to the Monty Hall problem 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.

5. Monty Hall Problem THE monty hall problem. Throughout the many years of Let's Make A Deal's popularity, mathematicians and a mathematical urban legend has developed surrounding "The monty hall problem http://www.letsmakeadeal.com/problem.htm

THE MONTY HALL PROBLEM Throughout the many years of Let's Make A Deal 's popularity, mathematicians have been fascinated with the possibilities presented by the "Three Doors" ... and a mathematical urban legend has developed surrounding "The Monty Hall Problem." A heated debate began when Marilyn Savant published a puzzle in her Parade Magazine column. One of her readers posed the following question: “Suppose you’re on a game show, and you’re given a choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the other doors, opens another door, say No. 3, which has a goat. He then says to you, ‘Do you want to pick door No. 2?’ Is it to your advantage to take the switch?” Ms. Savant, who’s listed in the Guinness Book of World Records Hall of Fame for “Highest IQ” (228), answered “Yes.” Because of the estimated 10,000 letters she received in response, she published a second article on the subject. Due to the fervor created by Ms. Savant’s two columns, the New York Times published a large front page article in a 1991 Sunday issue which declared:

6. Andrewgraham.co.uk.games Simulation of the 3door problem in Flash, along with a brief discussion. http://www.andrewgraham.co.uk/maths.html

Picture the scene. You've made it through to the final round of a game show. There are three doors in front of you. Behind one of the doors is the star prize - a brand new car. There is a goat behind each of the other two doors. You make your choice, hoping to select the star prize. The game show host (who knows where the car is hidden) opens a different door to reveal a goat. The choice is now down to two doors. He asks whether you'd like to stick with your original choice or whether you'd like to switch doors... do you switch? yes no it doesn't matter I don't care, take me to the index ... e-mail

7. 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

Navigation Panel: Go backward to Interactive Activities and Games Go forward to Questions and Discussion Switch to text-only version (no graphics) Go to University of Toronto Mathematics Network Home Page Problems and Puzzles You can select any of the items below: International Mathematical Talent Search Problems Try your hand at these problems, and mail in your answers! Interesting Mathematical Games 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. The Math Forum A good, comprehensive source of many mathematical materials. MAT 007 I News 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

8. The Monty Hall Problem The monty hall problem. The monty hall problem gets its name from the TV game show, "Let's Make If the monty hall problem ended with the selection of the first http://www.io.com/~kmellis/monty.html

9. 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

Tom Luthman's webpage has moved! The new webpage is: PlanetTom.home.mindspring.com

10. Education, Mathematics, Fun, Monty Hall Dilemma Monty Hall Dilemma. The Monty Hall Dilemma was discussed in the popular "Ask Marylin" questionand-answer column of the The WWW Tackles The monty hall problem. Win a car http://www.cut-the-knot.com/hall.html

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Monty Hall Dilemma The Monty Hall Dilemma was discussed in the popular "Ask Marylin" question-and-answer column of the Parade magazine. Details can also be found in the "Power of Logical Thinking" by Marylin vos Savant, St. Martin's Press, 1996. Marylin received the following question: Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors? Craig. F. Whitaker Columbia, MD Marylin's response caused an avalanche of correspondence, mostly from people who would not accept her solution. Several iterations of correspondence ensued. Eventually, she issued a call to Math teachers among her readers to organize experiments and send her the charts. Some readers with access to computers ran computer simulations. At long last, the truth was established and accepted. Below is one simulation you may try on your computer. For simplicity, I do not hide goats behind the doors. There is only one 'abstract' prize. You may either hit on the right door or miss it. You make your selection by pressing small round buttons below input controls that substitute for the doors. Down below other controls update experiment statistics even as you progress.

11. Interactive Mathematics Miscellany And Puzzles, Probability 1 2 3. If this computer simulation of random selection seems sufficiently credible,you may want to continue and tackle the monty hall problem. Remark. http://www.cut-the-knot.org/probability.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Probability In a world as crazy as this one, it ought to be easy to find something that happens solely by chance. It isn't. Kevin McKeen The Orderly Pursuit of Pure Disorder. Discover, January, 1981 American Heritage Dictionary defines the Probability Theory as the branch of Mathematics that studies the likelihood of occurrence of random events in order to predict the behavior of defined systems. Starting with this definition, it would (probably :-) be right to conclude that the Probability Theory, being a branch of Mathematics, is an exact, deductive science that studies uncertain quantities related to random events. This might seem to be a strange marriage of mathematical certainty and uncertainty of randomness. On a second thought, though, most people will agree that a newly conceived baby has a 50-50 chance (exact but, likely, inaccurate estimate) to be, for example, a girl or a boy, for that matter. Interestingly, a recent book by Marilyn vos Savant dealing with people's perception of probability and statistics is titled "The Power of Logical Thinking". My first problems will be drawn from this book.

12. 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

Answer to the Monty Hall Problem Hold on to your hats... you *double* your chances by switching 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

13. 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

The Monty Hall Problem 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.

14. The Monty Hall Problem The monty hall problem 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? Math Forum, a Classic Problem from the Ask Dr. http://rdre1.inktomi.com/click?u=http://mathforum.org/dr.math/faq/faq.monty.hall

15. 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

THE MONTY HALL PROBLEM 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.

16. The Monty Hall Problem Sorry, this program require Javascript, it will not work for you. ? ? ? Loading,Please wait Keep choice 0 times Wins 0, cars, (0%). Losses 0, goats, (0%). http://www.grand-illusions.com/simulator/montysim.htm

Sorry, this program require Javascript, it will not work for you Loading, Please wait... Keep choice: times Wins: cars Losses: goats Change choice: times Wins: cars Losses: goats Run times and keep change the choice: Cheat

17. 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

Ask Dr. Math: FAQ The Monty Hall Problem Dr. Math FAQ Classic Problems Formulas Search Dr. Math ... Dr. Math Home 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

18. Monty Hall Problem - Wikipedia, The Free Encyclopedia monty hall problem. From Wikipedia, the free encyclopedia. This articlehas been Enlarge The monty hall problem. The monty hall problem http://en.wikipedia.org/wiki/Monty_Hall_problem

Monty Hall problem From Wikipedia, the free encyclopedia. This article has been nominated on Wikipedia:Featured article candidates . Please refer to that page if you wish to second or contest the nomination. The Monty Hall problem The Monty Hall problem is a puzzle in probability that is loosely based on the American game show Let's Make a Deal ; the name comes from the show's host Monty Hall . In this puzzle a contestant is shown three closed doors; behind one is a car, and behind each of the others is a goat. The contestant chooses one door and will be allowed to keep what is behind it. Before the door is opened, however, the host opens one of the other doors and shows that there is a goat behind it. Should the contestant stick with the original choice or change to the remaining door; or does it make no difference? The question has generated heated debate. As the standard answer appears to contradict elementary ideas of probability, it may be regarded as a paradox . As the answer relies on assumptions that are not in the statement of the puzzle and are not obvious, it may also be considered a trick question. Table of contents 1 Problem and solution 1.1 The problem

19. 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 don’t agree with an answer. http://www.coastaltech.com/monty.htm

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.

20. Monty Hall Problem -- From MathWorld monty hall problem. The monty hall problem is named for its similarityto the Let s Make a Deal television game show hosted by Monty Hall. http://mathworld.wolfram.com/MontyHallProblem.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Applied Mathematics Game Theory Monty Hall Problem The Monty Hall problem is named for its similarity to the Let's Make a Deal television game show hosted by Monty Hall. The problem is stated as follows. Assume that a room is equipped with three doors. Behind two are goats, and behind the third is a shiny new car. You are asked to pick a door, and will win whatever is behind it. Let's say you pick door 1. Before the door is opened, however, someone who knows what's behind the doors (Monty Hall) opens one of the other two doors, revealing a goat, and asks you if you wish to change your selection to the third door (i.e., the door which neither you picked nor he opened). The Monty Hall problem is deciding whether you do. The correct answer is that you do want to switch. If you do not switch, you have the expected 1/3 chance of winning the car, since no matter whether you initially picked the correct door, Monty will show you a door with a goat. But after Monty has eliminated one of the doors for you, you obviously do not improve your chances of winning to better than 1/3 by sticking with your original choice. If you now switch doors, however, there is a 2/3 chance you will win the car (counterintuitive though it seems).

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