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/>
