
The Mosteller Hall Puzzle - gpresot
http://jonathanweisberg.org/post/Teaching%20Monty%20Hall/
======
BoppreH
I don't understand why the given solution doesn't fall prey to the same
problem of "her logic would make it unnecessary to consult the guard".

By this solution, if the guard says "B" (as in the original story), the
probably of A being condemned goes up. But if the guard says "C", the probably
also goes up by the same amount, because the graph is symmetric!

Going by this logic, if the guard is not allowed to say "A", then their answer
doesn't matter. So why isn't the final probability still 1/3?

~~~
shardo
I think it's the author's fault for switching between probabilities and odds
and the readers for not recognizing the switch.

"Thus A’s chance of being condemned remains twice that of being pardoned."

When he says this, he is talking of odds of 2:1 and therefore, the probability
of not being condemned is 1/1+2 = 1/3

Also when he says "Because, unlike in Monty Hall, the intuitive judgment is
the correct one in Mosteller’s puzzle." He means the intuitive judgment that
1/2 is incorrect.

I was extremely confused the first 2-3 times I read it and kept trying to
understand the author's viewpoint because everything other than these two
statement seemed to make sense. At least I wasn't the only one.

------
kcanini
I think a correct analysis of this problem requires you to know the guard's
exact policy for what to say in every possible situation. The author has
assumed that in the case that A is pardoned, the guard randomly chooses
whether to answer "B" or "C" with equal probability, but we have no evidence
for this.

Consider, for example, how you would analyze the situation where the guard
says to A, completely unsolicited, "Hey, just so you know, B is still
condemned to die."

On the one hand, it seems that the author's exact same analysis could be used
in this situation: the guard's policy here might be exactly the same as it was
in the original situation (where A asks the guard to name one of the other
prisoners). On the other hand, perhaps the guard's policy was to only offer
this information to A if A were pardoned. Or maybe the guard would have only
said this if C were pardoned. We just don't know, so we can't calculate the
probability that A is pardoned.

~~~
FabHK
> a correct analysis of this problem requires you to know the guard's exact
> policy for what to say in every possible situation.

Just as with Monty Hall, where one ought to specify that the host knows
exactly what's where, and always opens a remaining door such that it contains
a goat (choosing uniformly if there's more than one such choice).

The nice thing about the variant in the article is that it is fairly well and
intuitively specified.

(By the way, if the guard chooses B with probability x (rather than 1/2) in
case A is pardoned, then the probability that A is pardoned is x/(1+x), which
is between 0 and 1/2, but indeed is 1/3 only for x = 1/2.).

    
    
            A = A is pardoned
    	etc.
    	
    	P(A) = P(B) = P(C) = 1/3
    	
    	
    	SA = guard says A still condemned
    	etc.
    	
    	P(SB | A) = x
    	P(SB | B) = 0
    	P(SB | C) = 1
    	
    	P(SC | A) = 1-x
    	P(SC | B) = 1
    	P(SC | C) = 0
    	
    	P(SA) = 0
    	P(SB) = (1+x)/3
    	P(SC) = (2-x)/3
    	
    	
    	P(A | SB) = P(SB | A) P(A) / P(SB)
    	          = x * 1/3 / (1+x) * 3 = x/(1+x)

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dmurray
This seems identical to the Monty Hall problem, except that the typical
telling of the Monty Hall problem elides an important detail: that Monty knows
which door conceals the car and intentionally always picks one that conceals a
goat.

With that omitted (on Wikipedia [0], it's omitted in both the introduction to
the page and the "canonical" reader's letter it quotes), people correctly
conclude that their chance of winning doesn't go up by switching doors.

[0]
[https://en.m.wikipedia.org/wiki/Monty_Hall_problem](https://en.m.wikipedia.org/wiki/Monty_Hall_problem)

~~~
aptwebapps
It's not the same as the Monty Hall problem because the prisoner is not given
a choice. If you are sure you know the answer to the Monty Hall problem and
are a programmer then I suggest writing a little simulator to test your
answer.

~~~
FabHK
It is structurally the same. With the wrong conclusion that A now has 1/2
chance of being pardoned, he'd be indifferent between remaining himself or
rather being C.

With the (correct) conclusion that A still has 1/3 chance of being pardoned,
and given that B has 0 chance of being pardoned, A would now rather "switch"
with C, who has 2/3 chance of being pardoned. (That he doesn't have that
choice doesn't matter.)

~~~
aptwebapps
I didn't mean it was mathematically different, but the framing is sufficiently
different to cause most people to miss the most interesting part. I was
leaving out the correct conclusion so as not to spoil it for anyone.

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glitcher
Bit of a tangent here, but I don't get the relevance of the goat animated gif
at the bottom of the article. Perhaps it was just for fun, or is there some
deeper metaphor I'm missing?

~~~
agf
In the Monty Hall problem, the bad outcome is opening a door and finding a
goat --
[https://en.wikipedia.org/wiki/Monty_Hall_problem](https://en.wikipedia.org/wiki/Monty_Hall_problem)

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petters
This is a puzzle that everyone gets intuitively, but formulating the error
exactly is still a good learning exercise.

Maybe a good interview question?

~~~
logicallee
I found it!! The elusive source of it all! Here, ladies and gentleman, you can
see the birth of the exact thought:

"Hey I just read this really obscure article on something I've never heard of
in school or used in my professional life, and didn't have to figure out for
myself.

I bet it would make a great interview question."

~~~
pavlov
You don't deserve the downvotes because that seems to be the exact thought
process here: an interview is not about what the candidate can do, but how
[s]he responds to weird gotchas.

