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Topic: Maths question
Replies: 354   Last Post: May 31, 2007 8:04 PM

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 Dave Seaman Posts: 3,624 Registered: 12/8/04
Re: Maths question - totally correct actually
Posted: May 25, 2007 3:54 PM

On Sat, 26 May 2007 00:29:35 +1000, Richard Kelly wrote:
><BigMama> wrote in message news:4656a9b2\$1@dnews.tpgi.com.au...
>> "Richard Kelly" <blamesociety@yahoo.com> wrote in message
>> news:1180073767_91@qrz.fr...

>>> Assuming the host will always open a door containing a goat after you
>>> have selected, and you know he will do this, you will double your chances
>>> of getting the car by switching.

>> Ah, but you don't know that the host will do that, so your explanation is
>> moot.

> Did it just get dumber in here?

> Even without the above assumptions, my explanation is correct.

> Assume the contestant is completely in the dark about what is going to
> happen (with respect to the host opening a door after they have made their
> initial selection and offering the chance to re-select). The contestant
> selects a door. The host then (surprisingly) opens a different door and
> reveals a goat. He then asks if you would like to pick again from one of the
> remaining two doors or stick with your original selection. As the
> contestant, I will change my selection, because suddenly the odds of getting
> the car are twice what they were when I started the game (it has increased
> from 50% to 67%).

You have to be careful here. It actually does make a difference that the
host knows where the car is and always reveals a goat. If the host
merely opens an unchosen door at random, the outcome is drastically
altered.

Scenario A (the original). Host always opens an unchosen door and
reveals a goat. Contestant always switches. The contestant wins a car
whenever the original choice was a goat (2/3 of the time), and the
contestant wins a goat whenever the original choice was the car (1/3 of
the time).

Scenario B (the modified version). Host opens an unchosen door at
random, which means that 1/3 of the games end prematurely with the car
being revealed by the host. Outcome: the contestant still loses wheneer
the original choice is the car (1/3) of the time. The results are split
in the 2/3 of the cases where the contestant originally chooses a goat,
because in 1/2 of those (1/3 of the total) the contestant wins the car,
but in the other 1/2 of those cases (1/3 of the total), the contestant
never gets the opportunity to switch. Hence, in scenario B:

probability that contestant wins a car: 1/3
probability that contestant wins a goat: 1/3
probability that contestant wins nothing (car revealed early): 1/3

Now we can see exactly why the host's behavior in scenario A is
beneficial to the contestant. If the host is *required* to reveal a
goat, then all those type-3 scenarios (car revealed early) are converted
into type 1 (contestant wins car).

--
Dave Seaman
Oral Arguments in Mumia Abu-Jamal Case heard May 17
U.S. Court of Appeals, Third Circuit
<http://www.abu-jamal-news.com/>

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