Monty Hall Paradox (let's make a deal) 2.5. ==> monty.hall <== [decision] Three doors; one has a prize. If you pick the door with the prize, you get the prize. You pick a door, Monty Hall opens a different door that does not contain the prize and allows you the right to switch you choice to the remaining unopened door. Should you switch? The answer is that it is advantageous to switch: your probability of winning the prize is 2/3 if you do so. The probability that your first guess is wrong is obviously 2/3, and switching doors will gain you the prize if and only if your first guess was wrong. The solution assumes that Monty knows which door contains the prize and that his decision to offer you the switch is independent of whether you originally chose the correct door. These assumptions are implicitly made because they were the way the game was conducted by Monty Hall on "Let's Make A Deal" -- if you don't consider these assumptions to be natural then your conclusions may be different. References: Math. Gazette 75 (Okt 1991) 275-277 Math. Teacher 85:4 (Apr 1992) 250-252-256, 85:6 (Sep 1992) 409 Math. & Comp. Education 26:2 (1992) 153-156 Der Spiegel 34/1991 212-213 Drei T\"uren Problem Der Spiegel 36/1991 12-13 (Leserbriefe zum Drei T\"uren Problem) Math. Magazine 64 (1991) 359 [Reviews], -> New York Times, Focus Die Zeit 30/1991, 33/1991, 34/1991 p58 (Leserbriefe) - Herb Bailey; Monty Hall Uses a Mixed Strategy, Mathematics Magazine, 73:2 (April 1999) 135-141 - Ed Barbeau; Marilyn's Problem, Prisoner's Paradox, Two Children, and Three Cards, (Fallacies, flaws and flimflam) The College Math. Journal 22:4 (1991) 308 - Ed Barbeau; The Problem of the Car and Goats (survey) (Fallacies, flaws and flimflam) The College Math. Journal 24:2 (Mar. 1993) 149-154 (163-165??) - Ed Barbeau; Probability Problems References (Update) (Fallacies, flaws and flimflam) The College Math. Journal 26:2 (1995) 132-134 - Alex Bogomolny; Monty Hall Dilemma, http://www.cut-the-knot.com/hall.html - M. Borovcnik; Stochastik in der Schule, 11:3 (1991) 46-51 Problemecke - D. Donovan; The WWW Tackles the Monty Hall Problem, http://math.rice.edu/~ddonovan/montyurl.html - Keith M. Ellise; The Monty Hall Problem, http://www.io.com/~kmellis/monty.html - E. Engel, A. Ventoulias; Monty Hall's probability puzzle, Chance 4:2 (1991) 6-9 - Luis Fernandez, Robert Piron; Should She Switch? A Game-Theoretic Analysis of the Monty Hall Problem, Mathematics Magazine, 72:3 (June 1999) 214-217 - D. Friedman; Monty Hall's three doors: construction and deconstruction of a choice anomaly, The American Economic Review, 88 (1998) 933-946 - Leonhard Gillman; The 'car' and the 'goats', American Mathematical Monthly 99:1 (Jan 1992) 3-7 - D. Granberg, T. Brown; The Monty Hall Dilemma, Personality and Social Psychology Bulletin, 21 (1995) 711-723 - R. Grothmann; Das Ziegenproblem, http://mathsrv.ku-eichstaett.de/MGF/homes/grothmann/Projekte/ZiegenProblem/ - Robert Koch; Das Ziegenproblem, http://www.mathematik.uni-osnabrueck.de/staff/phpages/koch/ziegen/ziegen.html - J. Paradís, P. Viader, L. Bibiloni; A Mathematical Excursion: From the Three-Door Problem to a Cantor-Type Set, American Mathematical Monthly 106 (1999), 241-251. - Jochen Paulus; Das Rätsel der drei Türen, DIE ZEIT 18.11.2004 Nr.48 http://www.zeit.de/2004/48/N-Ziegenproblem - John de Pillis; 777 Mathematical Conversation Starters, Mathematical Association of America, 2003, ISBN 0-88385-540-2 pp. 143-146 http://math.ucr.edu/~jdp/Monty_Hall/Monty_Hall.html - Gero von Randow; Das Ziegenproblem, Denken in Wahrscheinlichkeiten, Rowohlt, rororo science 19337, 1992 (2. Auflage 2004) - Gero von Randow; Geisterfahrer des Journalismus oder wie kommt Mathematik in die Zeitung? DMV-Mitteilungen, (Dez 1998) 30-35 - Gero von Randow; Ziegen wählen statt Schafe zählen, DIE ZEIT 09.12.2004 Nr.51 http://www.zeit.de/2004/51/N-Ziegenproblem_neu - Marilyn vos Savant; Ask Marylin, Parade Magazine, Sept. 9, 1990 - Marilyn vos Savant; The Power of Logical Thinking, St. Martin's Press, 1996 (German: Brainpower - Die Kraft des logischen Denkens, Rowohlt, Hamburg, 2000, ISBN 3-498-06334-0, - Das Monty-Hall-Dilemma 28-42 - Anhang (Donald Granberg): Das Monty-Hall-Dilemma - revidieren oder nicht revidieren 241-278) "A very nice book about the Monty Hall Dilemma and statistical inference" (says Alexander Bogomolny) - Steve Selvin; A Problem in Probability, (Letters to the Editor) American Statistician 29:1 (Feb 1975), 67 American Statistician 29:3 (Aug 1975), Comments http://www.io.com/~kmellis/AS.HTML - Ian Stewart; Mathematische Unterhaltungen, Spektrum der Wissenschaft, 11/1991 12-16 - Marc C. Steinbach Autos, Ziegen und Streith\"ahne, Mathematische Semesterberichte 47 (2000) p 107-117. - John Tierney; Behind Monty Hall's doors: puzzle, debate and answer, The New York Times (July 21, 1991), 1 - H. Winter; intuitive Aufkl\"arung prob. Paradoxien JMD 13:1 (1992) 23-53, (Nachsatz 52-53) The 5-Step Probability Solver, Pi Mu Epsilon J. 9 (1992) 445-447 Simulation..., Stochastik in der Schule 12:3 (1992) 2-25 Generalizing Monty's Dilemma, Quantum 5:4 (Mar Apr 1995) 17-21 & 59-60 and 40-41 Let's make a deal http://math.ucsd.edu/~crypto/Monty/montybg.html The Monty Hall Problem http://www.crosswinds.net/~marlright/monty.htm http://www.crosswinds.net/~marlright/monty1.htm - Frederik Ramm, http://www.remote.org/frederik/projects/ziege/ (german) http://www.remote.org/frederik/projects/ziege/index.html.de (german) http://www.remote.org/frederik/projects/ziege/index.html.en (english) - Daren Scot Wilson; The Goat and the Car: A Probability Puzzler http://www.newcolor.com/goat/ http://www.newcolor.com/goat/refs.html (online references) - rec.puzzles FAQ 2.5. ==> monty.hall <== [decision] -- http://www.mathematik.uni-bielefeld.de/~sillke/ mailto:Torsten.Sillke@uni-bielefeld.de