Re: [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?
Question: --------- Behind one of three doors - A, B, and C -- is a car; behind each of the other two is a goat. The player, P, gets to keep what's behind the door he chooses to open. First P picks one door that remains closed at first, say door A. Now one of the other doors is opened to show a goat -- say door B -- and P gets to make their final choice: open the original choice of door A, or the other door -- call it C -- that's still closed. Which is more likely to get the car? ------------------------------------ Answer: ------- The player should pick door C for the best chance of getting the car. Explanation: ------------ Player P has 1/3 chance of guessing the car with their original choice of door A. Showing the goat sheds no light on whether the initial choice of door A is correct, so its probability remains 1/3. Hence the probability the car is not behind door A must be 2/3. Since door B has been excluded from hiding the car, that 2/3 probability belongs to door C. QED --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
On Wed, Oct 28, 2009 at 3:44 PM, Dan Asimov <dasimov@earthlink.net> 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?
My impression was that Jim was looking for a published version that could be cited, rather than asking for an explanation. In my experience, most versions of this I've seen present the problem in an ambiguous way...
Question: ---------
Behind one of three doors - A, B, and C -- is a car; behind each of the other two is a goat. The player, P, gets to keep what's behind the door he chooses to open.
First P picks one door that remains closed at first, say door A.
Now one of the other doors is opened to show a goat -- say door B -- and P gets to make their final choice: open the original choice of door A, or the other door -- call it C -- that's still closed.
Which is more likely to get the car?
Including this one! The answer to this one is indeterminate, because it doesn't indicate how the door is chosen to be opened. Consider the following possibilities: 1. Door is opened by someone who has no idea what is behind which door, and happens on this occasion to reveal a goat (Probabilities of A and C having the prize are now 1/2 - 1/2) 2. Door is opened by someone who knows what is behind the doors, and always opens a door with a goat behind it, choosing randomly if the prize is behind A (probabilities are now 1/3 - 2/3) 3. Door is opened by someone who knows what is behind the doors, and always opens a door with a goat behind it, choosing B if the prize is behind A (probabilities are now 1/2 - 1/2) 4. Door is opened by someone who knows what is behind the doors, and always opens a door with a goat behind it, choosing C if the prize is behind A (probabilities are now 0 - 1) 5. Door is opened by someone who knows what is behind the doors, and wants A to get the prize: If prize is behind A, he just reveals that A has won; if A has not gotten the prize, he reveals a goat, and offers to let you switch (probabilities are 0 - 1) 6. Door is opened by someone who knows what is behind the doors, and wants A not to get the prize: If prize is behind A, he reveals a door at random, and offers to let A switch. If the prize is behind B or C, he doesn't open a door, just reveals that A has lost (probabilities are 1 - 0) The "Monty Hall problem" is case 2, but a correct statement needs to make it clear that this is the intended case.
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Answer: -------
The player should pick door C for the best chance of getting the car.
Explanation: ------------
Player P has 1/3 chance of guessing the car with their original choice of door A.
Showing the goat sheds no light on whether the initial choice of door A is correct, so its probability remains 1/3.
Well that's the question, isn't it? In cases 1, 3, 4, 5, and 6, showing the goat does shed light. I don't think there's enough of an argument here to persuade someone who thinks that it also sheds light in case 2 that it doesn't. Andy Latto andy.latto@pobox.com
participants (2)
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Andy Latto -
Dan Asimov