21. NPR Math Guy The Birthday Problem The Two Envelopes Paradox. The monty hall problem http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

22. Monty Hall Problem - Wikipedia, The Free Encyclopedia monty hall problem http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

23. Monty Hall Problem - Wikipedia, The Free Encyclopedia Talkmonty hall problem Wikipedia, the free encyclopediaI ve moved the existing talk page to Talkmonty hall problem/Archive2, I m going to show that the monty hall problem is a specific case of a more http://en.wikipedia.org/wiki/Monty_Hall_problem

Monty Hall problem From Wikipedia, the free encyclopedia. In search of a new car, the player picks door 3. The game host then opens door 1 to reveal a goat and offers to let the player pick door 2 instead of door 3. The Monty Hall problem is a puzzle in game theory involving 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 player is shown three closed doors; behind one is a car , and behind each of the other two is a goat . The player is allowed to open one door, and will win whatever is behind the door. However, after the player selects a door but before opening it, the game host (who knows what's behind the doors) must open another door, revealing a goat. The host then must offer the player an option to switch to the other closed door. Does switching improve the player's chance of winning the car? The answer is yes — switching results in the chances of winning the car improving from 1/3 to 2/3. The problem is also called the Monty Hall paradox , in the sense that the solution is counterintuitive, although the problem does not yield a logical contradiction.

24. Math Forum: Ask Dr. Math FAQ: The Monty Hall Problem In the threedoor problem, once you ve seen what s behind the second door is itbetter to stay with your first choice or switch doors? 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

25. Welcome To 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 http://www.sover.net/~nichael/puzzles/monty/

Welcome to Monty Hall 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) and provide a reasonable and clear explanation of the answer. The Problem A Hint The Solution Return to Nichael Cramer's HomePage ... nichael@sover.net

26. 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, 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

27. Monty Hall Problem Web Sites Discourse on the monty hall problem. The following sites all pose the problem Simulations of the monty hall problem. Three Door Puzzle A great site! http://math.rice.edu/~pcmi/mathlinks/montyurl.html

28. Monty Hall Problem The CBS drama series NUMB3RS featured the monty hall problem in the final monty hall problem, Explanation of the problem with a Java applet to run a 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." The CBS drama series NUMB3RS featured the Monty Hall Problem in the final episode of its 2004-2005 season. The show's mathematician offered his own, very definite solution to the problem involving hidden cars and goats. The London FINANCIAL TIMES published a column about the Monty Hall Problem on August 16, 2005, declaring positively that "the answer is, indeed, yes: you should change." However, the columnist, John Kay, notes that "Paul Erdos, the great mathematician, reputedly died still musing on the Monty Hall problem." The column resulted in several letters published on the "Leaders and Letters" page of the FINANCIAL TIMES on August 18 and 22 - and two follow-up columns by Mr. Kay on August 23 ( So you think you know the odds ) and August 31 ( The Monty Hall problem - a summing up ) in which he acknowledges that he received "a large correspondence on Monty Hall."

29. 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.

30. Monte Hall, Let's Make A Deal, Problem The monty hall problem. Monty Hall was the host of a game show called Analysis of the monty hall problem. It does not make a difference which door the http://www.usna.edu/MathDept/courses/pre97/sm230/MONTYHAL.HTM

Monty Hall (Let's Make a Deal) Problem Gary Fowler, revision date: January 22, 1996. Statement of problem and background. Analysis of problem. Simulation Red or black? Another challenging problem. Return to the SM230 home page. The Monty Hall Problem Monty Hall was the host of a game show called "Let's Make a Deal." This was a very popular show due in part to the finale. The stage was set with three doors. Behind each door was a prize. One prize was very desirable and valuable, e.g. , two week, all expense paid trip for two to Hawaii. There was a much less desirable prize, e.g. , living room furniture. The remaining prize was undesirable. The undesirable prize is traditionally called a "goat," but since this is the Naval Academy we will call it a "mule." After the contestant selected a door, another door was opened to show the prize and the contestant was given the choice between the already selected door or the other door that had not been opened. A few years ago, the popular press contained several articles debating whether the contestant should switch doors. This debate was sparked by an analysis of the given by Marilyn vos Savant in Parade Magazine in which she concluded that the contestant should switch. She received many letters objecting to her analysis and conclusion. Several of these letter were from college professors who teach statistics. The debate spread to professional journals including

31. Untitled Document Here is a page that reproduces the 1991 New York Times article discussing themonty hall problem, and provides links to several online simulations. http://www.dartmouth.edu/~chance/course/topics/Monty_Hall.html

Here is a page that reproduces the 1991 New York Times article discussing the Monty Hall problem, and provides links to several online simulations.

32. Graphical Proof Of The Monty Hall Problem THE monty hall problem An original graphic by John de Pillis, from his book, (For more on the solution to the monty hall problem, see Curiouser, http://math.ucr.edu/~jdp/Monty_Hall/Monty_Hall.html

T HE M ONTY H ALL P ROBLEM An original graphic by John de Pillis from his book, "777 Mathematical Conversation Starters (citing Jay Leno)" REVIEWS: amazon.com. 5-star reader reviews; [2] Laugh and Learn with John (with cartoons) by Prof. Philip J. Davis, Brown University. [3] Dull and No Life by Susan Palmer Slattery. ( For more on the solution to the Monty Hall problem, see Curiouser, a site of paradoxical puzzles, illusions, etc. RULES of the GAME: There are three inverted cups, one of which hides a valuable diamond. A contestant chooses one of the three cups at random (Move One). At this point, the probability of success, i.e., choosing the diamond, is 1/3. Monty Hall, who knows where the diamond is, must eliminate one of the empty, unchosen cups, leaving only two cups on the table (Move Two). If the contestant always switches cups (Move Three), then the chance of winning will double - from the original 1/3, to 2/3. Using the graphic on the left, we review these points. Move One: One of the three cups is chosen. The graphic shows all three possibilities. The first column shows the case when the diamond is chosen, and the last two columns show when an empty cup is chosen.

33. Monty Hall I think the reason the monty hall problem raises people s ire is because a basic In the case of the monty hall problem, however, the outcome is that a http://www.maa.org/devlin/devlin_07_03.html

Search MAA Online MAA Home Devlin's Angle July-August 2003 Monty Hall A few weeks ago I did one of my occasional "Math Guy" segments on NPR's Weekend Edition. The topic that I discussed with host Scott Simon was probability. [Click here to listen to the interview.] Among the examples we discussed was the famous - or should I say infamous - Monty Hall Problem. Predictably, our discussion generated a mountain of email, both to me and to the producer, as listeners wrote to say that the answer I gave was wrong. (It wasn't.) The following week, I went back on the show to provide a further explanation. But as I knew from having written about this puzzler in newspapers and books on a number of occasions, and having used it as an example for many years in university probability classes, no amount of explanation can convince someone who has just met the problem for the first time and is sure that they are right - and hence that you are wrong - that it is in fact the other way round. Here, for the benefit of readers who have not previously encountered this puzzler, is what the fuss is all about.

34. The Monty Hall Trap This puzzle (also called the monty hall problem) has become fairly famous.If you ve never seen it, you won t believe the answer that I will give below. http://www.jimloy.com/puzz/monty.htm

Return to my Puzzle pages Go to my home page The Monty Hall Trap This puzzle (also called the Monty Hall problem) has become fairly famous. If you've never seen it, you won't believe the answer that I will give below. Suppose that Monty Hall (on TV's Let's Make a Deal ) asks you to choose between three doors: #1, #2, and #3. Behind a random door is a new Rolls Royce. Behind each of the other two doors is a goat. Let's assume that you would prefer a Rolls Royce to a goat. You choose a door. Now, Monty, who knows which door hides the Rolls Royce, shows you a goat behind one of the two doors that you did not choose. He then gives you the opportunity to change your choice. Assume that Monty always does this, regardless of your guess. Should you change your choice? There is a fairly simple solution to this. But, people don't believe it. There has been much heated debate over it. I'll show my reasoning below. And then, see if you agree. Originally, your chances of choosing the correct door are one in three. Monty, who knows where the Rolls Royce is, shows you a goat behind another door. It will turn out that he did not change your odds. You still have one chance in three of being right. So you should switch to the remaining door, which now has odds of 2/3. You probably don't believe that Monty did not change the odds of your first guess being right. He showed you a door, seemingly at random. So, aren't the odds now 50-50 for the two unseen doors? Of course, Monty's choice of doors was not at random. But, what difference does that make?

35. Thoughts Arguments And Rants: Monty Hall Problem monty hall problem. Via Justin Leiber, here s a playable version of the MontyHall Problem. It s simultaneously a lesson in decision theory and in the http://tar.weatherson.net/archives/002705.html

Thoughts Arguments and Rants Donate to Oxfam: Australian link New Zealand link UK link US link ... Main July 20, 2004 Monty Hall Problem Via Justin Leiber , here's a playable version of the Monty Hall Problem . It's simultaneously a lesson in decision theory and in the perils of small sample sizes - my first two plays I lost the car by switching. TrackBack Comments Post a comment Name: Email Address (won't be displayed): Remember personal info? Yes No Comments (note no links to .com sites allowed): Search Search this site: Archives September 2005 August 2005 July 2005 June 2005 ... March 2002 My sites Home page Crooked Timber Papers Blog Syndicate this site (XML) This weblog is licensed under a Creative Commons License Powered by Movable Type 2.661

36. Monty Hall CONCLUSION As for the correct solution to the monty hall problem, The problemwith The monty hall problem is that the possibilities are not http://barryispuzzled.com/zmonty.htm

A classic probability teaser In September 1990, Marilyn vos Savant, puzzle columnist for the U.S. magazine Parade, was sent a probability teaser by a reader. Its publication in her "Ask Marilyn" column together with her solution has produced much debate amongst mathematicians and laymen alike ever since. Apart from it's challenge to common intuition, one reason it has attracted so much interest is that Ms von Savant is listed in the Guinness Book of World Records as having the highest recorded score in an IQ test (228 : based on a score attained at 10 years old) and that proving her wrong, some respondents no doubt reasoned, might indicate that they too were blessed with such uncommon prowess. To date, Ms vos Savant has received over 10,000 letters on the puzzle, mostly disagreeable, and articles are still being written claiming new insights that show where Ms vos Savant got it wrong. The present article argues that the solution really depends on one's interpretation of the stated problem. One of the earliest known appearances of the problem was in Joseph Bertrand's Calcul des probabilites (1889) where it was known as Bertrand's Box Paradox. It later reappeared in Martin Gardner's 1961 book

37. The Monty Hall Problem And Monte Carlo Simulations monty hall problem I love this problem as an example of a very straightforwardbut nonintuitive result. I once presented this problem to a group of young http://haacked.com/archive/2004/07/21/836.aspx

you've been HAACKED ...and you like it. Home Archives Contact Login ... Resume posts - 744, comments - 2101, trackbacks - 303 The Monty Hall Problem and Monte Carlo simulations Ian Griffiths blogs about the Monty Hall problem The problem, named after the host of a game show on which it sometimes appeared, is as follows: There are three doors, behind one of which is a valuable prize, but you don't know which door. Choose a door. You are not told straight away whether you've made the right choice. Instead, the host of the game will then open one of the doors you did not pick, showing you that there is no prize behind it. You are now offered the chance to change your mind. This effectively narrows down your choice - the prize is behind one of two doors, either the one you picked, or the door that neither you nor the host picked. What should you do to maximize the probability of winning the prize? Should you stick with your first choice, or switch to the other door? Or does it not matter? I love this problem as an example of a very straightforward but non-intuitive result. I once presented this problem to a group of young kids who were in a summer math and science enrichment program. They were floored by the result. I demonstrated the proof to the kids via both the logical proof (as Ian does) as well as by running a Monte Carlo simulation. I had two teams play the game over and over, one choosing to switch every time, and one choosing to stay. Like mathematical magic, over a series of 20 or so trials it becomes quite clear that always switching is indeed the better strategy. Argue with me as they did, they could not argue with their own eyes.

38. THE MONTY HALL PROBLEM (FRONT PAGE) The monty hall problem gets its name from the TV game show, Let s Make A Deal, hosted by Monty Hall. The problem is stated below. http://members.shaw.ca/ron.blond/TLE/MONTY.APPLET.FRONTEND/

The Monty Hall Problem gets its name from the TV game show, "Let's Make A Deal," hosted by Monty Hall. The problem is stated below. There are three closed doors, behind one of which is a prize (the remaining doors contain "joke" prizes). Monty Hall, the game show host, asks you to pick one of the three doors. You pick a door (which remains unopened). Monty opens a door that has a joke prize. Monty then gives you the choice of either keeping your original choice, or switching to the remaining unopened door. QUESTION : To maximize the chances of winning a real prize, should you keep your choice or switch (or does it matter)? NOTE : This isn't really how the actual game show worked. This problem gets its name from the show, because it inspired the problem. The problem is interesting because most people believe that after Monty shows a losing door, the two remaining unopened doors (whether chosen or not) each have a fifty-fifty chance of being a winning door. One with a real prize, the other with a joke prize. Most people are surprised that this is not the case. At this time you can choose one of the links below. The applet is a simulation of this problem.

39. THE MONTY HALL PROBLEM (APPLET) You select a door with a mouse click and Monty then opens a door with a JOKE prize. SEE THE RESULT (AND PROOF OF THE RESULT) OF THE monty hall problem http://members.shaw.ca/ron.blond/TLE/MONTY.APPLET.FRONTEND/MONTY.APPLET/

CONCEPT BY RICK SEPT HOW TO USE THIS APPLET Basic operation blue background. When all of the doors are opened, a listing of events for the current experiment are displayed in the top portion of the applet. When the mouse button is clicked, the results will be recorded in the chart and in the graph. Advanced operation feel for what is happening, and give a greater reinforcement to believing the result. Notes SEE THE DESCRIPTION OF THE MONTY HALL PROBLEM SEE THE RESULT (AND PROOF OF THE RESULT) OF THE MONTY HALL PROBLEM RETURN TO THE TLE APPLET SELECTION PAGE

40. Miscellanea - Blog Archive » Monty Hall Problem I’d like to refer you to the monty hall problem. It’s an interesting paradox witha very counterintuitive solution. Here’sa quick summary http://elver.cellosoft.com/2005/07/24/monty-hall-problem/

@import url(http://elver.cellosoft.com/wp-content/themes/redbar/style.css); Profile Elver Loho is (among many, many other things) a computer science student at The University of Tartu Pages Home Profile Projects Biblefind People X-G Bob Hentges Janey Peep Pullerits ... Simon of Space Archives September 2005 August 2005 July 2005 June 2005 Categories Art Books Bullshit Business ... Tutorials Powered By Proudly powered by the excellent WordPress Creative Commons Attribution, NonCommercial, NoDerivs 2.5 License 24th July 2005 Monty Hall Problem Categories: Mathematics Code Monty Hall problem monty.c The output looks something like this: wins by switching of all wins: 6684 / 10041 (66%) wins by staying of all wins: 3357 / 10041 (33%) total combined wins of all games 10041 / 20000 (50%) wins by switching of all related games 6684 / 10000 (66%) wins by staying of all related games 3357 / 10000 (33%) Jolle says: July 24th, 2005 at 10:50 pm Jolle says: July 24th, 2005 at 10:53 pm elver says: July 25th, 2005 at 12:15 am But you don’t really need a program if you think it all through. Seriously, even in high school instead of doing my maths homework like everyone else, I usually wrote a program first. Doing my homework then was easier, because I already knew the answer.

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