Monty Hall Strikes Again
Subject: Probability Question from OZ! Date: Wed, 2 Nov 1994 17:13:05 +1100 (EST) From: "Sean Pryor" Hi... Basically the problem goes like this. There are three cups, one of which is covering a coin. I know the whereabouts of the coin, but you don't. You pick a cup, and I take one of the remaining cups, one which DOESN'T contain a coin. Both you and I know the cup I pick doesn't contain a coin. You then have the option to swap your cup with the third, remaining cup, or keep your first choice. What is the probability of the coin being in the cup if you keep your first choice, or if you decide to swap them? To summarise: Three cups with a coin under one. You pick one, I pick one that DOESN'T have the coin. You then either stay with your choice, or swap it with the remaining cup. What is the probability of getting the coin, either way? BTW the answer I get is 50% either way - though it has been suggested that you have a 2/3 chance if you swap cups.... I disagree with this, but I don't think my maths is capable of giving a definitive answer (which is why you're reading this!) (This problem has placed a cool Australian $50 on the line here in a bet - so I need an answer proving me right! :-) My mate would never let me forget it if he was right...) Thanxx Sean Pryor
Date: Thu, 3 Nov 1994 08:20:34 -0500 From: Phil Spector Sean- This question is a famous brain teaser that is usually described in terms of the game show Let's Make a Deal. Indeed, you do have a 2/3 chance of winning if you swap cups. The answer is anti-intuitive, and my efforts at explaining it usually fall a little flatter than others' explanations, so I'm going to let other Dr. Math's try to provide an answer, and I'm going to work on a productive explanation. Basically, the solution is based as follows: Let's say it was under cup 1. Option 1: You originally choose cup 1. Then, you get shown one of the empty cups (let's say cup 2 or 3) at which point if you CHANGE (to the remaining empty cup, either cup 2 or 3), you LOSE, but if you REMAIN with cup one, you WIN. Option 2: You originally choose cup 2. Then, you get shown the remaining empty cup (cup 3), at which point if you CHANGE (to cup 1), you WIN, but if you REMAIN with cup 2, you LOSE Option 3: You originally choose cup 3. Then, you get shown the remaining empty cup (cup 2), at which point if you CHANGE (to cup 1), you WIN, but if you REMAIN with cup 3, you LOSE. Each of the three options has an equal chance of occuring, based upon your original random pick. If you change 1/3 of the time, you lose (in the case of option 1, which has a 1/3 probability of occuring), while if you change 2/3 of the time (with options 2 or 3, which have a 2/3 probability of occuring), you win. Therefore, the proper choice is always to change! Sorry about that 50 dollars. :( I think the place where things skew to the anti-intuitive is the fact that you will -always- get shown an EMPTY cup by the game-show host/tricker, not necessarily a random cup. Hope this helped. Phil, a Dr. Math who knows only statistics revolving around game shows...
Subject: Probability Question from OZ! From: "Sean Pryor" Hey, it wasn't my $50! It was a mate you was silly enough to bet on a maths problem!! :-) I think I understand. My query is that perhaps you have a 2/3 of getting the coin when changing, taking into account the whole process. If you isolate the last choice and say - now it's either in this cup or that and I can choose this or that, its 1/2. Is that a reasonable method of explaining the anti-intuitive bit? Or have I missed the point here...
X-Sender: email@example.com Date: Fri, 4 Nov 1994 18:46:18 -0500 Exactly. Nope, you explained it better than I did... Phil, a Dr. Math who knows only statistics revolving around game shows... >What better place to start - my favourite was always the Wheel of Fortune >- spinning wheels, angular momentum, friction, force, acceleration, sectors, >degrees, probability - that show had it all! :-) True, but I always thought Pat Sajak was a dweeb. :) Glad Dr. Math could help, and of course, write back with any other problems you have... Phil
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