[math-fun] best published explanation of the Monty Hall Paradox?
What book or magazine article or website has the best (in your opinion) written explanation of the Monty Hall Paradox for the lay reader? (Obviously the best way to explain something mathematical to anyone is through a dialogue in which the teacher gets information from the student about the student's state of understanding and changes tactics accordingly; but I'm looking for the best canned explanation that you've seen.) Did Martin Gardner ever try his hand at explaining the paradox? Jim
On Wed, Oct 28, 2009 at 3:17 PM, James Propp <jpropp@cs.uml.edu> wrote:
Did Martin Gardner ever try his hand at explaining the paradox?
Yes, but not in the form of the Monty Hall game. One of his columns, anthologized in one of his books, has the puzzle in the form of 3 prisoners, one of whom is to be pardoned, but by order of the governor, they're not to be told whether they are the one to be pardoned or not. A gets told that B is to be executed, and then tells this to C. The question is what A and C should now consider their chances to be. Same problem, except without the possibility of "switching". Part of his explanation is the thing I always find most effective in convincing people; play the game with them with 52 doors. I shuffle a deck of cards, and give you one (don't look yet!). The winning card is the Ace of spades. I then say I'm going to turn up 50 non-Ace-of-Spades cards. I look at my 51 cards, and say "Let's see, I think I'll turn up all but....*this* one!". Now all the people who insisted the answer is 1/2 in the original problem want to switch their card for *that* one. Andy
My favorite write up was by Monty Hall and John Tierney in 1991. http://www.mathpuzzle.com/MontyHall.txt --Ed --- On Wed, 10/28/09, James Propp <jpropp@cs.uml.edu> wrote: From: James Propp <jpropp@cs.uml.edu> Subject: [math-fun] best published explanation of the Monty Hall Paradox? To: math-fun@mailman.xmission.com Date: Wednesday, October 28, 2009, 2:17 PM What book or magazine article or website has the best (in your opinion) written explanation of the Monty Hall Paradox for the lay reader? (Obviously the best way to explain something mathematical to anyone is through a dialogue in which the teacher gets information from the student about the student's state of understanding and changes tactics accordingly; but I'm looking for the best canned explanation that you've seen.) Did Martin Gardner ever try his hand at explaining the paradox? Jim _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 28 Oct 2009 at 15:17, James Propp wrote:
What book or magazine article or website has the best (in your opinion) written explanation of the Monty Hall Paradox for the lay reader?
Would a video do? <http://www.youtube.com/watch?v=koPBkK_Ra-k> this one is good, too: <http://www.youtube.com/watch?v=mhlc7peGlGg> /Bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--
In August 2001, I argued stridently on the fledgling Wikipedia that switching was irrelevant, and wrote code to prove it, only to be proven wrong by my own code. I posted the code as an "empirical example" subpage off the main article. Unfortunately, the current article no longer includes an executable demonstration. !/usr/bin/perl # Here's a Perl script with which to empirically test # the Monty Hall problem. # I'll admit that the results completely blew me away # the first time I ran the program, and I still wonder # if there isn't some odd bug which just happens to # make the numbers look the way they do... # 9 Aug 2001 # -J # Empirical test of the Monty Hall problem # Run with -v to see each game. $iterations = 10; # How many games to play @items = ("goat1", "goat2", "car"); #We'll number the items 0-2 $verbosity = 1 if $ARGV[0] eq "-v"; sub verbose { if($verbosity == 1) { print($_[0]); } } for(1..$iterations) { $door[0] = int rand 3; # What's behind door #1? if($door[0] == 0) { $door[1] = 1 + (int rand 2); # Door #2 gets 1 or 2 if($door[1] == 1) { $door[2] = 2; } else { $door[2] = 1; } } if($door[0] == 1) { $door[1] = 2 * (int rand 2); # Door #2 gets 0 or 2 if($door[1] == 0) { $door[2] = 2; } else { $door[2] = 0; } } if($door[0] == 2) { $door[1] = int rand 2; # Door #2 gets 0 or 1 if($door[1] == 0) { $door[2] = 1; } else { $door[2] = 0; } } verbose( "Door 1: $items[$door[0]] Door 2: $items[$door[1]] Door 3: $items[$door[2]] " ); $contestant = int rand 3; verbose( "Contestant chooses door " . ($contestant+1) . ". " ); # If the contestant picked the car, Monty picks either other door if($items[$door[$contestant]] eq "car") { $coinflip = int rand 2; if($contestant == 0) { $monty = ($coinflip ? 1 : 2); } if($contestant == 1) { $monty = ($coinflip ? 0 : 2); } if($contestant == 2) { $monty = ($coinflip ? 0 : 1); } } # Otherwise, he picks the other goat if($items[$door[$contestant]] eq "goat1") { if($contestant == 0) { $monty = ($items[$door[1]] eq "goat2" ? 1:2); } if($contestant == 1) { $monty = ($items[$door[0]] eq "goat2" ? 0:2); } if($contestant == 2) { $monty = ($items[$door[0]] eq "goat2" ? 0:1); } } if($items[$door[$contestant]] eq "goat2") { if($contestant == 0) { $monty = ($items[$door[1]] eq "goat1" ? 1:2); } if($contestant == 1) { $monty = ($items[$door[0]] eq "goat1" ? 0:2); } if($contestant == 2) { $monty = ($items[$door[0]] eq "goat1" ? 0:1); } } verbose( "Monty opens door " . ($monty+1) . ". " ); $sticker = ($items[$door[$contestant]] eq "car") ? "wins" : "loses"; $switcher = ($items[$door[$contestant]] eq "car") ? "loses" : "wins"; verbose( "Sticker $sticker. Switcher $switcher. " ); if($sticker eq "wins") { $stickers++ } if($switcher eq "wins") { $switchers++ } } print " Grand totals: Stickers: $stickers Switchers: $switchers ";
Jason wrote:
In August 2001, I argued stridently on the fledgling Wikipedia that switching was irrelevant, and wrote code to prove it, only to be proven wrong by my own code.
Yes, I've heard more than one person say they were persuaded by their own code, even without running it, even without finishing the program. I think that's because writing the program forces one to confront the ambiguity in the statement of it. If it were set up as usual, up until the door selection, but then proceeded with the announcer saying "Is that your final door?" and the contestant replying, "I'm not sure yet; what I want you to do is to show me a goat that is different from the one that I'm 2/3 sure is behind this door," then it wouldn't sound very mysterious. A new puzzle category: the every so slightly ill-posed problem. Hilarie
This reminds me of the classic puzzle genre where one person knows the sum of two integers and the other knows the product; the two each confess their ignorance about the values of the numbers, alternating confessions until one of them says "Now I know!". The actual puzzle isn't important. However: somehow this particular mini-genre is horribly vulnerable to tiny mistakes in wording which completely invalidate the intended solution. I've seen it happen more than once. On Wed, Oct 28, 2009 at 5:58 PM, Hilarie Orman <ho@alum.mit.edu> wrote:
Jason wrote:
In August 2001, I argued stridently on the fledgling Wikipedia that switching was irrelevant, and wrote code to prove it, only to be proven wrong by my own code.
Yes, I've heard more than one person say they were persuaded by their own code, even without running it, even without finishing the program. I think that's because writing the program forces one to confront the ambiguity in the statement of it.
If it were set up as usual, up until the door selection, but then proceeded with the announcer saying "Is that your final door?" and the contestant replying, "I'm not sure yet; what I want you to do is to show me a goat that is different from the one that I'm 2/3 sure is behind this door," then it wouldn't sound very mysterious.
A new puzzle category: the every so slightly ill-posed problem.
Hilarie
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On Wednesday 28 October 2009 19:17:24 James Propp wrote:
What book or magazine article or website has the best (in your opinion) written explanation of the Monty Hall Paradox for the lay reader?
Jason Rosenhouse has written a whole book on the subject: http://www.oup.com/us/catalog/general/subject/Mathematics/ProbabilityStatist... I haven't read it and have no idea how good it is, either in general or for your particular purpose. I've seen a couple of positive reviews, but I saw them because the author posted them on his blog so there might just be a bit of selection bias there. -- g
participants (8)
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Allan Wechsler -
Andy Latto -
Bernie Cosell -
Ed Pegg Jr -
Gareth McCaughan -
Hilarie Orman -
James Propp -
Jason