21. PHIL 158d - Critical Thinking: The Monty Hall Problem CriticalThinking Philosophy 158D 2000 monty hall problem, The MontyHall Problem. You're on a TV game show. In front of you are three http://qsilver.queensu.ca/~phil158d/intro/montyh3.htm

C ritical T hinking Philosophy 158D 2000 Monty Hall Problem The Problem One Question Three Answers Eighteen ... The Demo The Monty Hall Problem You're on a TV game show. In front of you are three doors: there's a great prize behind one door, and nothing behind the other two. You choose a door. Then the host (Monty Hall) opens one of the two doors you didn't choose to show that there is nothing behind that door. It would be bad for the TV ratings if he opened the prize door: you'd know you had lost and the game would be over; so Monty knows where the prize is, and he always opens a door that doesn't have a prize behind it (Monty is Canadian, so you know you can trust him). You're now facing two unopened doors, the one you originally picked and the other one, and the host gives you a chance to change your mind: do you want to stick with the door you originally chose, or do you want to switch to what's behind the other door? The Question What should you do? The Answers There are only three answers: a) stick with the door you originally chose b) switch to the other door, or

22. Monty Hall Problem THE monty hall problem. Throughout Links to websites devoted to. The MontyHall Problem . *art work courtesy of http//cartalk.cars.com/, 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." 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:

23. 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

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). Winning Probability pick stick pick switch The problem can be generalized to four doors as follows. Let one door conceal the car, with goats behind the other three. Pick a door

24. THE MONTY HALL PROBLEM (FRONT PAGE) The monty hall problem gets its name from the TV game show, Let's MakeA Deal, hosted by Monty Hall. RUN THE monty hall problem APPLET. 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.

25. THE MONTY HALL PROBLEM (APPLET) SEE THE DESCRIPTION OF THE monty hall problem. SEE THE RESULT (AND PROOF OF THERESULT) OF THE monty hall problem. RETURN TO THE TLE APPLET SELECTION PAGE. 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

26. The Monty Hall Problem - A State-space Explanation The monty hall problem a state-space explanation. Here is yet another explanationof the answer. Notes A friend called to ask about the monty hall problem. http://www.vendian.org/mncharity/dir2/montyhall/

The Monty Hall Problem - a state-space explanation Here is yet another explanation of the answer. It requires tables and color Some other pages The WWW Tackles The Monty Hall Problem Monty Hall Problem Three Door Puzzle Monty Hall Problem There are 3 doors one of which has a Prize. So the universe is in one of 3 possible states P P P Now, you choose a door Looking inside, we see P in just one of the 3 possible states. P P P If the universe is in state1, we get the P. But if it is in or , we don't. Our chances are 1/3 for "P", and 2/3 for "-" Now "Monty" takes the other two doors P P P and opens one without the prize P P P Now which door do you want, red or green P P P In green , we get two P's! If the universe is in or , we get the P. Only if it is in state1, do we lose. So our chances are 2/3 for "P", and 1/3 for "-" As you know, a 2/3 chance of getting the Prize is better than a 1/3 chance so we tell "Monty" we've switched our choice to green Comments encouraged. Mitchell N Charity mcharity@lcs.mit.edu Notes: A friend called to ask about the Monty Hall Problem. I liked this explanation, but it didn't work over the phone. Not seeing it already online, I wrote this. This presentation is similar to this Answer to the Monty Hall Problem , but sufficiently different that I went ahead with this one anyway. Doables: History: 1998.Oct.15 Created.

27. The Monty Hall Problem Explained The monty hall problem. A game show host offers you the chance to choose one ofthree doors. Simulate the monty hall problem. A brief history of the problem. http://www.engin.umich.edu/soc/informs/Games/MontyHall/

The Monty Hall Problem A game show host offers you the chance to choose one of three doors. You know that there is a nice price (a new car) behind one door, and a silly prize (historically, a goat, but conceivably a gift from a sponsor such as Rice-a-Roni) behind the other two. Once you have made your choice, the game show host opens a different door, to reveal a goat. He then offers you a choice: you may keep the first door you chose, or you may switch to the other closed door. Should you keep the first door, or switch to the other door? Simulate the Monty Hall problem. A brief history of the problem The Monty Hall problem apparently captured the public's attention when it was reported in Parade magazine, in Marily vos Savant's column. Before that, it was reportedly discussed in American Statistician in 1976, and in the Journal of the American Mathematical Society about ten years after that. An earlier version, the Three Prisoner Problem, was analyzed in 1959 by Martin Gardner in Scientific American . He called it "a wonderfully confusing little problem...in no other branch of mathematics is it so easy for experts to blunder as in probability theory." The answer...should you switch?

28. Monty Hall Problem - Wikipedia Other languages Deutsch. monty hall problem. From Wikipedia, the freeencyclopedia. The monty hall problem is a riddle in elementary http://www.wikipedia.org/wiki/Monty_Hall_problem

Main Page Recent changes Edit this page Older versions Special pages Set my user preferences My watchlist Recently updated pages Upload image files Image list Registered users Site statistics Random article Orphaned articles Orphaned images Popular articles Most wanted articles Short articles Long articles Newly created articles All pages by title Blocked IP addresses Maintenance page External book sources Printable version Talk Log in Help Other languages: Deutsch Monty Hall problem From Wikipedia, the free encyclopedia. The Monty Hall problem is a riddle in elementary probability that arose from the American game show "Let's make a deal" with host Monty Hall . The problem's main claim to fame is that after its solution was discussed in Marylin vos Savant's "Ask Marylin" question-and-answer column of Parade magazine in 1990, many readers including several math professors wrote in to declare that her solution was wrong, thereby making public fools of themselves. The problem is as follows: at the end of the show, a player is shown three doors. Behind one of them, there's a car for him to keep, behind the other two there are goats. Of course, the player does not know where the car is, but Monty knows. The player chooses one door. Before that door is opened however, Monty opens one of the two other doors with a goat behind it. He then offers the player the option to switch to the other closed door. Should the player switch? The classical answer is yes . Suppose you are a "sticker". This means you point to a door, ignore what Monty does, and stick with your choice. When you chose, all three doors were equally likely, so you win the car with probability 1/3. Now consider a switcher. He chooses a door, waits for Monty to expose a goat and then switches to the other remaining door. The switcher wins if and only if his initial door had a goat behind it (think it through). How likely is it that his initial choice had a goat? Two thirds, of course. So switchers are twice as likely as stickers to win the car.

29. Monty Hall Problem/Empirical Proof - Wikipedia monty hall problem/Empirical Proof. !/usr/bin/perl Here's a Perlscript with which to empirically test the monty hall problem. http://www.wikipedia.org/wiki/Monty_Hall_problem/Empirical_Proof

Main Page Recent changes Edit this page Older versions Special pages Set my user preferences My watchlist Recently updated pages Upload image files Image list Registered users Site statistics Random article Orphaned articles Orphaned images Popular articles Most wanted articles Short articles Long articles Newly created articles All pages by title Blocked IP addresses Maintenance page External book sources Printable version Talk Log in Help Monty Hall problem/Empirical Proof From Wikipedia, the free encyclopedia. See also : Monty Hall problem Edit this page Discuss this page Older versions ... Recent changes It was last modified 15:51 Feb 25, 2002. All text is available under the terms of the GNU Free Documentation License Main Page Recent changes Random page ... Bug reports

30. The Monty Hall Problem The monty hall problem. The following problem is taken from a quiz showon TV that really existed (or still exists). It is an interesting http://www.remote.org/frederik/projects/ziege/

[remote] [frederik] [projects] [monty hall] The Monty Hall Problem The following problem is taken from a quiz show on TV that really existed (or still exists). It is an interesting topic to discuss in almost any group of people because even the most intelligent often get into trouble, and is (in other languages) referred to as the Goat Problem. The Scenario The quiz show candidate has mastered all the questions. Now it's all or nothing for one last time: He is lead to a room with three doors. Behind one of them there's an expensive sports car; behind the other two there's a goat. (Don't ask me why it's a goat. That's just the way it is.) The candidate chooses one of the doors. But it is not opened; the host (who knows the location of the sports car) opens one of the other doors instead and shows a goat. The candidate is now asked if he wants to stick with the door he chose originally or if he prefers to switch to the other remaining closed door. His goal is the sports car, of course! The Question The question now is: Is the candidate better off if he sticks with his original choice

31. Faisal.com: The Monty Hall Problem The monty hall problem. The monty hall problem is a classic exampleof the nonintuitive nature of statistics You are playing a game. http://www.faisal.com/docs/monty.html

The Monty Hall Problem News Archive Quotes Documents ... Contact Search: Note: for the time being, this page requires a "4.0 browser" in order to display colors properly. I apologize to those of you with older browsers, and also to anyone who has trouble with the page due to red/green color-blindness. The "Monty Hall Problem" is a classic example of the non-intuitive nature of statistics: You are playing a game. In the game, there are three doors with prizes hidden behind them. One door hides a new car while the other two doors hide goats. If you pick the door with the car behind it, you win the car. You pick a door. At this point the game host opens another of the doors, revealing a goat. Now the game host offers you the opportunity to change your pick to the other door. Should you switch? Most people believe that there is no advantage to changing their pick: of the two remaining doors, one of them has the car, so they have a 50% chance of winning. In fact, this is incorrect: while 1 of the doors will have the car behind it, 2/3 of the time it will be the other door. It is generally to your advantage to switch.

32. The Monty Hall Problem The monty hall problem. This problem has rapidly become part of themathematical folklore. The American Mathematical Monthly, in http://db.uwaterloo.ca/~alopez-o/math-faq/mathtext/node32.html

Next: Master Mind Up: Mathematical Games Previous: Mathematical Games The Monty Hall problem This problem has rapidly become part of the mathematical folklore. The American Mathematical Monthly, in its issue of January 1992, explains this problem carefully. The following are excerpted from that article. Problem: A TV host shows you three numbered doors (all three equally likely), one hiding a car and the other two hiding goats. You get to pick a door, winning whatever is behind it. Regardless of the door you choose, the host, who knows where the car is, then opens one of the other two doors to reveal a goat, and invites you to switch your choice if you so wish. Does switching increases your chances of winning the car? If the host always opens one of the two other doors, you should switch. Notice that of the time you choose the right door (i.e. the one with the car) and switching is wrong, while of the time you choose the wrong door and switching gets you the car. Thus the expected return of switching is which improves over your original expected gain of Even if the hosts offers you to switch only part of the time, it pays to switch. Only in the case where we assume a malicious host (i.e. a host who entices you to switch based in the knowledge that you have the right door) would it pay not to switch.

33. The Monty Hall Problem Previous Mathematical Games. The monty hall problem. This problemhas rapidly become part of the mathematical folklore. The American http://db.uwaterloo.ca/~alopez-o/math-faq/node65.html

Next: Master Mind Up: Mathematical Games Previous: Mathematical Games The Monty Hall problem This problem has rapidly become part of the mathematical folklore. The American Mathematical Monthly, in its issue of January 1992, explains this problem carefully. The following are excerpted from that article. Problem: A TV host shows you three numbered doors (all three equally likely), one hiding a car and the other two hiding goats. You get to pick a door, winning whatever is behind it. Regardless of the door you choose, the host, who knows where the car is, then opens one of the other two doors to reveal a goat, and invites you to switch your choice if you so wish. Does switching increases your chances of winning the car? If the host always opens one of the two other doors, you should switch. Notice that 1/3 of the time you choose the right door (i.e. the one with the car) and switching is wrong, while 2/3 of the time you choose the wrong door and switching gets you the car. Thus the expected return of switching is 2/3 which improves over your original expected gain of 1/3. Even if the hosts offers you to switch only part of the time, it pays to switch. Only in the case where we assume a malicious host (i.e. a host who entices you to switch based in the knowledge that you have the right door) would it pay not to switch.

34. Mudd Math Fun Facts: Monty Hall Problem 19992003 Francis Edward Su. From the Fun Fact files, here is a FunFact at the Medium level monty hall problem. Figure 1 Figure 1. http://www.math.hmc.edu/funfacts/ffiles/20002.6.shtml

hosted by the Harvey Mudd College Math Department Francis Su Any Easy Medium Advanced Search Tips List All Fun Facts Fun Facts Home About Math Fun Facts ... Other Fun Facts Features 887384 Fun Facts viewed since 20 July 1999. Francis Edward Su From the Fun Fact files, here is a Fun Fact at the Medium level: Monty Hall Problem Figure 1 Here's a problem that makes the round every few years, and each time, it is hotly debated. You are on a game show. You are presented with a choice of 3 doors: behind one is a luxury car, and behind the other two are nothing. The host asks you pick one of the doors. After you do this, as part of the game he opens one unpicked doors which he knows is empty. There are now only the door you picked and one remaining door which are unopened. You are asked if you would like to switch your choice. Should you switch? Presentation Suggestions: Another version of this problem uses cards in a game called "three-card monte", often played by scam artists on the streets of New York who prey on easily-duped tourists. The Math Behind the Fact: The answer is YES, you should switch, because the probability that you will find the car by doing so is 2/3. This is because the probability that you picked the correct door in the first place does

35. A Simulation Of The Monty Hall Problem A simulation of the monty hall problem. The monty hall problem can bestated as follows A gameshow host displays three closed doors. http://www.ram.org/computing/monty_hall/monty_hall.html

A simulation of the Monty Hall problem The Monty Hall problem can be stated as follows: A gameshow host displays three closed doors. Behind one of the doors is a car. The other two doors have goats behind them. You are then asked to choose a door. After you have made your choice, one of the remaining two doors is then opened by the host (who knows what's behind the doors), revealing a goat. Will switching your initial guess to the remaining door increase your chances of guessing the door with the car? The answer is yes. I did a simulation of this problem in the following manner: Randomly assign the car to be behind one of the three doors. The other two doors will be assigned goats. Randomly pick one of the three doors. Open one of the two remaining doors to reveal a goat: if the choice in (2) was the one behind which the car was hidden, then randomly choose between the remaining two doors. if the choice in (2) was not the one behind which the car was hidden, then one of the two remaining doors contains a car and the other contains a goat. The one with the goat is always chosen. Pick whether to switch or stay with the choice made in (2) randomly.

36. Sci.math FAQ: Monty Hall Problem sci.math FAQ monty hall problem. Newsgroups sci.math,sci.answers,news.answersFrom alopezo@neumann.uwaterloo.ca (Alex Lopez-Ortiz http://www.faqs.org/faqs/sci-math-faq/montyhall/

sci.math FAQ: Monty Hall problem Newsgroups: sci.math sci.answers news.answers From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Subject: sci.math DI76M3.KAF@undergrad.math.uwaterloo.ca alopez-o@neumann.uwaterloo.ca Organization: University of Waterloo Followup-To: sci.math alopez-o@barrow.uwaterloo.ca Tue Apr 04 17:26:57 EDT 1995 By Archive-name By Author By Category By Newsgroup ... Help Send corrections/additions to the FAQ Maintainer: alopez-o@neumann.uwaterloo.ca Last Update March 05 2003 @ 01:20 AM

37. The Monty Hall Problem . Welcome to the probability problemthat has vexed many a person over many a year. Rumours about......The monty hall problem. http://people.bu.edu/trachten/java_stuff/monty.html

The Monty Hall Problem Description Welcome to the probability problem that has vexed many a person over many a year. Rumours about this problem have flown far and wide, the most egregious being that the wrong answer was once published in a journal. When the woman with the highest known IQ (at the time) wrote in with a correction, her correction was refused because it is quite counter-intuitive. I, myself, have seen professors of mathematics run computer simulations because they did not believe the theoretical result. How to play The question is simple: there are three doors, with the prize of your dreams behind one of them (use your imagination). You select one door, and the host decides he's going to be nice to you; he shows you which of the other doors definitely does not contain the prize. Your task now is to decide: do you stick with your choice, or do you change your mind and opt for the one remaining closed door. This java applet was originally developed for use by the CS 273 class at the University of Illinois. Back to Ari's

38. No. 1577: The Monty Hall Problem No. 1577 THE monty hall problem. by John H. Lienhard. I've been running intothe monty hall problem lately. I suspect that many of you know about it. http://www.uh.edu/engines/epi1577.htm

No. 1577: THE MONTY HALL PROBLEM by John H. Lienhard Click here for audio of Episode 1577. Today, we learn not to turn our back on information. The University of Houston's College of Engineering presents this series about the machines that make our civilization run, and the people whose ingenuity created them. I' ve been running into the Monty Hall Problem lately. I suspect that many of you know about it. It came to my attention the other day when I ran into a colleague from the math department. She told me about it and left me scoffing in disbelief. I should've known about the problem, since it goes back to the old TV show, Let's Make a Deal . Host Monty Hall would offer a contestant three doors. One had a prize behind it. If the contestant guessed the correct door, he would win the prize. But, before the door that he chose was opened, Monty Hall (who knew where the prize was) would say, "Of the two remaining doors, I'll open this one, which has no prize behind it." Then Hall would add, "Now, would you like to change your guess?" The contestant could either decide that the first guess was correct or switch to the other unopened door. "The contestant should switch," said my mathematician friend. "Why?" I asked. "Because the probability of getting the prize will rise from one chance in three to two out of three."

39. The Monty Hall Problem Half's home Creations Puzzles The monty hall problem. The monty hall problem. NextSome Explanations. Half's home Creations Puzzles The monty hall problem. http://www.rdrop.com/~half/Creations/Puzzles/LetsMakeADeal/

MLI Home Creations Puzzles The Monty Hall Problem The Monty Hall Problem An American game show left an unexpected legacy: many arguments, and more than a few Web pages. Some people even learned some probability theory. We'll leave out the theory here to concentrate on different ways to understand the problem's solution. The Problem The game show Let's Make A Deal , hosted by Monty Hall, ended each show the same way. There were three closed doors. Behind one was a prize, while the other two concealed booby prizes. Monty asked the contestant to choose a door. Then Monty opened one of the remaining doors, revealing a booby prize. Monty then offered the contestant the option to stay with the originally chosen door or switch to the other unopened door. The contestant received whatever was behind the chosen door. Now for the big question: is it better to stay , better to switch , or does it make no difference The Importance of Empiricism If you think it doesn't make a difference, stop . Right now, and I mean right now , you're going to play the game. You'll need:

40. The Monty Hall Problem: Some Explanations The monty hall problem Some Explanations. If you don't know what theMonty Hall Let's Make a Deal problem is, start here. The Answer. http://www.rdrop.com/~half/Creations/Puzzles/LetsMakeADeal/explanations.html

MLI Home Creations Puzzles The Monty Hall Problem Some Explanations The Monty Hall Problem: Some Explanations If you don't know what the Monty Hall "Let's Make a Deal" problem is, start here. The Answer You'll do better on average if you switch. There is a 2/3 chance that the prize is behind the door you switched to, and only 1/3 that prize is behind your original door. Before I try to explain why, there are a few things will remain true throughout the following discussion: The host and assistants never lie or cheat. The prize is placed behind one door and not moved. You do not know where the prize is. Random numbers are truly random. Dice are fair. Two Common Misconceptions There seem to be two reasons for thinking the chance of picking the door with the prize is 50%. After one door has been revealed, there are two doors left, and the prize is behind one of them. Hence there must be a 50/50 chance of it being behind either unopened door. There are four possible outcomes (to be shown later). Two are winning outcomes if you stay, two if you switch. So there are 2 out of 4 winning cases if you either stay or switch. That means there is a 50/50 chance of winning with either strategy. I'll tackle these misconceptions in order.

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