
On how to pick goats from cars: The Monty Hall Problem - prakash
http://garry.posterous.com/on-how-to-pick-goats-from-cars
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llimllib
This never made _intuitive_ sense to me until I heard it rephrased this way:

In front of you are 1002 doors. You pick a door, and then Monty Hall opens
1000 doors to reveal goats. You have the option to pick the remaining door or
stick with your original choice. Which should you do?

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vitaminj
I've always thought the combining doors way of explaining this made the most
sense -
[http://en.wikipedia.org/wiki/Monty_Hall_problem#Combining_do...](http://en.wikipedia.org/wiki/Monty_Hall_problem#Combining_doors)

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hardik
After a _lot_ of brain twisting it sinks in.. and I do probability for living!

For doubters: Before the door is opened by the host, you are less likely to
have chosen the right one(with prob 1/3). Once he opens the door, he
eliminates one "bad door". Now, given that you were less likely to have chosen
the right one in the first place and one wrong one has been removed, it is
highly likely that the residual door is the right one.

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cconstantine
Right, so the first choice is door 1, and the host reveals that door 3 is a
goat.

The decision to switch or not is really a decision between door 1 and door 2.
At this point isn't your first guess completely irrelevant?

I've heard this problem and the reasoning before, and for some reason I can't
get myself to believe that switching is better.

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emmett
There are two possibilities when you pick door 1: either you've picked a door
with the prize behind it (33%) or you've picked a door with a goat behind it
(67%).

Assume you stick. In that case, you have a 33% chance you've picked the right
door, as before. Opening another door doesn't change whether you've picked the
right door or not originally.

Assume you switch. In the case you've picked a door with a goat originally
(67%), you will necessarily switch to the door with the prize. In the case
you've picked a door with the car originally (33%), you will necessarily
switch to a door with a goat. Therefore switching gives you a 67% chance of
picking the door with the prize.

Another interesting strategy to consider is "randomize" - what if you picked
randomly between the two remaining doors? Then you'd have a 50% of getting the
prize, since you're selecting randomly between two doors, one of which you
know contains the prize.

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superkarn
Imagine a similar but slightly different scenario: We now have two
contestants: Mr A and Mr B. Mr A is the contestant in the original story. He
gets to pick a door. The host then opens a door with a goat (and removes it
from the stage). Then Mr A gets to choose whether to keep his door or open the
third.

Now the second contestant, Mr B, joins the show. He has not seen anything
prior and has no knowledge of Mr A or the other door previously removed from
stage. All he sees is two doors. He has the option of opening one of the two.

Mr A has 2/3 chance of switching door and coming out with a car. Whereas Mr B
has 1/2 chance (from his perspective) of picking either door and coming out
with a car.

I think the main problem why so many people have a hard time grasping the
solution is that they're looking at the probability as Mr B and ignoring the
extra information they do have as Mr A.

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vinutheraj
Seriously, I had one of my friends ask me this question and I was stymied too
!! Neway, the 1002 thing is really intuitive and I definitely agree that we
think from the perspective of Mr.B and the point of the matter is that the
host removes all the "bad" doors !

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bitdiddle
classic teaser. If you extend the problem to n doors and you're shown all but
one, I think it helps the intuition.

Years ago I recall a lunch conversation with a group of mathematicians, some
with PhDs in statistics, that was quite lively.

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dcurtis
This is also explained well by Kevin Spacey in the movie 21.

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timmorgan
Not really. That was just enough to get me to read the Wikipedia article to
figure out what the heck they were talking about in that movie. It made the
student sound super smart, but most people I know didn't have a clue what they
were saying (including myself).

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dangrover
Yeah. And what was that whole business about the Newton-Rhapson method?

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qqq
you pick door 1. the host does his thing. now the situation is:

if it was behind door 1, you win by not switching.

if it was behind door 2 or 3, switching would win.

seems intuitive to me.

