Re: Advanced Monty Hall Problem with N door and M cars
From: Duncan Smith (buzzard_at_urubu.freeserve.co.uk)
Date: 02/23/05
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Date: Wed, 23 Feb 2005 17:33:30 -0000
"David Winsemius" <dwin$emiu$@comcast.not.rot> wrote in message
news:Xns9606692DC56A8dwinscom@216.196.97.136...
>
>
> 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
The second event in my random case description has positive prior
probability in the mind of the contestant. It shouldn't be ignored in the
construction of the (i.e. her) prior just because we (i.e. not her) know
that it will be conditioned away later. As you say, "the analysis focus
should be on the state of knowledge of the contestant". Knowing that the
opened door will not reveal the car leads to the 1/3 and 2/3 probabilities,
as per the non-random case. But in the random case the contestant does not
know a priori that this is going to happen.
The "two distinct states of information" are not equivalent, for the same
reason that setting the level of a variable is not the same as observing the
level of a variable. If I observe somebody wearing a dress I might
reasonably confidently infer that the person is female, but selecting
someone at random and sticking them in a dress doesn't actually help me
infer their gender. In each case I see somebody in a dress, but the
motivation behind the dress-wearing is very relevant to my inference.
Observing that a randomly selected door does not hide the car is not the
same as being shown that a non-randomly selected door does not hide the car.
The first is worth conditioning on (regarding the probability that the
originally chosen door hides the car), the second isn't.
Duncan
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