Subject: Monty Hall dilemma
Date: Tue 03.11.98 11:49
From: Robert Muller
The answer to Monty's Dilemma is real simple and is keyed to the one fact I keep seeing repeated in all the 1/3 probability "answers."
Once the Host has opened the (non-winning) door, it is removed from the probability. No longer is it 3 doors, but 2, with one of which being the "winner" or desired outcome.
The open door is now "known." Moreover, it cannot even be selected. It's not like the contestant, having picked, say door number 2, can change his mind and pick the Host-opened door number 3 with the corn dog.
What remains are 2 doors each with a 50-50 chance.
However, as I wrote years ago in my (unpublished) response to Marilyn vos Savant, the other element she neglected to consider was the Host's mindset. Is the Host malevolent (ie wants you to lose), benevolent (ie want's you to win), or indifferent (ie could care less)? I would suggest that the latter one of the series is irrelevant and can be left out of the equation, since if he is indifferent, he is either indifferent good or indifferent bad, there would be a 50-50 chance of either, and each would equate to the malevolent or benevolent Host.
There has been discussions of calculating the show's TV ratings tat cloud the issue. Where in the specific case, those ratings (or even the producer's finances) may sway the outcome, without evidence of those factors presented in the problem they must be ignored, or at best considered 50-50 themselves.
I have read some responses that claim ratings drive the malevolent host, but that is best answered by the simple fact that Las Vegas didn't get big by never letting people win.
There have also been analyses comparing this problem to the 3 Card Monty or Shell Game. These games are crooked on their face, inasmuch tha slight of hand removes the pea from all the shells or cards are switched to prevent winning. Ergo, those analyses are defective as well.
So, in the specific problem, once the Host has opened a door, it boils down to the contestant having two doors with one prize and one host with 2 mindsets, each representing a 50-50 probability.
Therefor, the answer is that your odds DO improve with him opening one of the doors (ie 1:3 to 1:2), but 1:2 is 50-50 so why bother to change? Each remaining door carries the same chance of winning.
Any analysis that includes the opened door in the consideration is flawed.
The New York Times devoted a full page article, replete with an interview with Monty Hall himself, that addressed both these issues and confirmed that once the one door is opened, the probability reverts to 1 chance in 2 doors, and a 50-50 chance of a helpful host.