Re: Advanced Monty Hall Problem with N door and M cars

From: David Winsemius (dwin$emiu$_at_comcast.not.rot)
Date: 02/23/05 

Date: Wed, 23 Feb 2005 09:20:18 -0600

snipped prior text but will insert here the OP's positing of the problem:

> Assuming that the player made an initial selection (she picked a door
> D1), now.... Monty open a door that has a $1 bill behind .... now he
> asks her, if she want to stay with her initial choice or if she
> wants to switch?

Duncan Smith wrote in news:cvi13u$aet$1@newsg2.svr.pol.co.uk:

> In the non-random case the contestant knows a priori that a car will
> not be revealed, so there are only two possible events,
>
> contestant chooses door hiding car, Monty reveals no prize, p = 1/3
> contestant does not choose door hiding car, Monty reveals no prize, p
> = 2/3
>
> Subsequent conditioning doesn't exclude either of these events (and
> the posterior probability of having chosen the correct door is equal
> to the prior, 1/3).
>
> In the random case the contestant does *not* know a priori that a car
> will not be revealed, and there are three possible events,
>
> contestant chooses door hiding car, Monty reveals no prize, p = 1/3
> contestant does not choose door hiding car, Monty reveals a prize, p =
> 1/3
> contestant does not choose door hiding car, Monty reveals no
> prize, p = 1/3
>
> Subsequent conditioning excludes the second event and results in
> probabilities of 1/2 for the remaining events.
>
> The prior knowledge of the contestant is different in each case, so
> despite the evidence being the same, the posteriors turn out to be
> different.
>
The problem posed by the OP stated that a non-car door was opened after
the contestant choice had been made. That means your second event in your
random case description was excluded by the manner in which the problem
was posed. The prior probability that the contestant chose the car was
still 1/3, so the probability that the car is behind the other door
becomes 2/3 after non-car-door-opening by whatever mechanism, and the
logical choice remains to switch.

My sole point is that the mechanism by which the non-car-door-opening
occurs is immaterial. If the problem is restricted to analyzing only
those situations where a non-prize door is opened, the probabilities and
knowledge of the contestant at the two distinct states of information
remain equivalent. The analysis focus should be on the state of knowledge
of the contestant and her logical choices given the evolution of his
information. The contestant does not need to know that Monty always opens
a losing door, only that on this occasion that a losing door was
disclosed AFTER the first decision (which set the priors). 

-- 
David Winsemius

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