🚀 go-pugleaf

In <BzqJx8.3Fv@unx.sas.com> sasjec@asimov.unx.sas.com (Jerry M. Cox) writes:

>Odds are that I could stand a review course in probability. But hear me 
>out anyway :-) If the contestant selects one door of three and that was 
>all there was to it then the odds of initially selecting the winning door 
>are one out of three. I have no problem with that, but I don't think 
>this describes the situation. If Monty is *always* going to reveal a 
>goat behind one of the doors, that door is removed from consideration. 

It is precisely because Monty ALWAYS reveals a losing door that makes it
significant.  

All you have to see is that if the initial guess was incorrect, the person
will always win when they switch.  Think about that very carefully.  The
initial guess is incorrect - of the two door, one has the prize, and the
other has the goat - Monty will reveal the goat, so the door switched to
will win.  This is only in the case when the initial guess was a goat.


Now, the probability that the initial guess was incorrect is 2/3.  That
cannot change, because a selection was made from 3 doors, only one of which
is correct.  Monty doesn't remove the door from consideration, because it
was there when the selection was made.    

Sure if he took away one of the doors, and scrambled the prizes and then
let you select randomly from the two, it would be 50/50, but the guess was
made from three doors.  Revealing a door does not change the probability
of the original guess, it can't.  It does reveal a lot of information about
the remaining door.

>The final choice then becomes a choice between one of two doors. If 
>Monty *always* removes one of the doors from consideration, then the 
>final choice (as well as the initial choice) is between a door with a 
>car behind it, and a door with the "other" (the one Monty doesn't show) 
>goat behind it. The odds of picking the car are 50/50 whether you switch 
>or not because Monty is *always* going to take away one of the losing 
>choices.

Just because it is a choice between two doors does not make it 50/50
probability. 

>If he said "You can have the door you picked or the TWO doors that you 
>didn't pick", and didn't reveal anything about the two doors, then I 
>would agree that the best thing to do would be to switch. 

Ah, but he does equivalently say this.  Look at the cases very carefully.
Any time the prize is in one of the two doors you did not pick, you win,
and any time the prize is the one that you initially chose, you lose.


But, Monty 
>changes the odds when he shows you one of the goats behind one of the 
>two doors. He removes that door from consideration completely. 

No one, not even Monty can change the probability of something that you
already selected from three choices.  That remains fixed at 1/3.  It cannot
change. 



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