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The Monty Hall Problem

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Pamela Conley

on 4 August 2011

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Transcript of The Monty Hall Problem

Door #1 Door #2 Door #3 The Monty Hall Problem You are on a game show. On this game show, the objective is to win a car as a prize. The host shows you three doors. He says that there is a car behind one of the doors and goats behind the other two doors. The host asks you to pick a door. The host opens one of the doors you didn’t pick to reveal a goat (because he knows what is behind the doors). You have one final chance to change your mind before the doors are opened and you get a car or a goat. The host asks you if you want to stay with your original choice or change your mind and pick the unopened door instead. What should you do? You pick a door but it is not opened. pam Conley 11A The Show The Problem The Controversy The Equation The Solution It's original host was Monty Hall, hence the name of the famous paradox. The Monty Hall problem is a probability puzzle loosely based on the television show "Let's Make a Deal." You are asked to pick a door Chosen Door Contents: GOAT Chosen Door Contents: GOAT Chosen Door Contents: CAR You Stay You Stay You Stay You Switch You Switch You Switch You Get:
GOAT You Get:
GOAT You Get:
GOAT You Get:
CAR You Get:
CAR You Get:
CAR Here's an easier way to think about it: Let the doors be called X, Y, and Z.
Let C(x) be the event that the car is behind door X, etc.
Let H(x) be the event that the host opens door X, etc. Supposing that you choose door X, the probability that you win a car if you then switch your choice is given by: Probability of getting a car if you stay: 1/3 Probability of getting a car if you switch: 2/3 In 1991, a reader of Marilyn vos Savant's Sunday Parade column wrote in and asked whether it was to the advantage of the contestant to stay with or switch from their original choice. Marilyn's response was that the contestant should always switch doors in order to have a higher probability of winning the car. She received nearly 10,000 responses from readers, most disagreeing with her. Several were from mathematicians and scientists whose responses ranged from hostility to disappointment at the nation's apparent lack of mathematical skills. Those that disagreed believed that it did not matter if the contestant stayed or switched; the probability of winning the car would be 1/2. Although this is the intuitive answer, it is a false one; it is only true if a slight change to the problem is made: that the host does not know the contents of each of the doors. The impact Sources Therefore, it is to the advantage of the contestant to always switch doors, as there is a greater probability that this action will yield the winning prize of a car. The Monty Hall Problem remains one of the finest examples of a mathematical veridical paradox, in which a result that appears absurd is demonstrated to be true. The Monty Hall Problem and similar paradoxes continue to be addressed and analyzed in academic journals, published works and academia, as well as television, film, and other popular media around the world. http://reason.com/blog/2009/11/24/lets-make-a-deal-health-care-r http://www.istockphoto.com/file_thumbview_approve/333541/2/Goat_head_above.jpg http://t0.gstatic.com/images?q=tbn:ANd9GcSW8XbGycX_mhg_R6vMf07A87_JZEYb2nvPNSMrfMwsjyhGrux Haddon, Mark. The Curious Incident of the Dog in the Night-time. New York: Doubleday, 2003. Print. Created for Mrs. Tallman's
Pre-Calculus Class
MMSTC 11A 2011 www.youtube.com
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