Why the Monty Hall puzzle is categorically 1/2 and not 2/3rds

[DonAntonio]

... and are equally likely ...

Of course, but since this is assumed to be a "choose in a random way something" out of several possibilites, this is

a rather straightforward assumption.

DonAntonio

 

[haruspex]

Of course, but since this is assumed to be a "choose in a random way something" out of several possibilites, this is a rather straightforward assumption.
DonAntonio

Yet that's where DaveC went wrong.

 

[alewisGB]

There are loads of proofs, I think a simple one is the initial probability if 1/3 yes and 2/3 no. The initial probabilery can't change at all so the chance of winning if you stick is always 1/3

 

[haruspex]

There are loads of proofs, I think a simple one is the initial probability if 1/3 yes and 2/3 no. The initial probabilery can't change at all so the chance of winning if you stick is always 1/3

Quite so, but the tricky part is understanding why the probability doesn't change. It is key that MH is always able to open another door without revealing the prize, so the fact that he does so tells you nothing about whether your choice was correct. This breaks down if MH chooses one of the other two randomly, or only knows that a particular door is wrong so can't open a door if you pick that one first.

 

[arildno]

Quite so, but the tricky part is understanding why the probability doesn't change. It is key that MH is always able to open another door without revealing the prize, so the fact that he does so tells you nothing about whether your choice was correct. This breaks down if MH chooses one of the other two randomly, or only knows that a particular door is wrong so can't open a door if you pick that one first.

Prior to choosing the door, we KNOW that at least one of the doors we do NOT open will contain..a goat.
Knowing as well that Monty knows exactly which door, and then opens, adds us no new information.
Had Monty been ignorant, our prediction would become sufficiently skewed towards the "two goats unopened"-scenario (i.e, that we picked the CORRECT door to begin with!) that it would become irrelevant whether we switch or not.

 

[haruspex]

Prior to choosing the door, we KNOW that at least one of the doors we do NOT open will contain..a goat.
Knowing as well that Monty knows exactly which door, and then opens, adds us no new information.
Had Monty been ignorant, our prediction would become sufficiently skewed towards the "two goats unopened"-scenario (i.e, that we picked the CORRECT door to begin with!) that it would become irrelevant whether we switch or not.

Yes. Did I appear to be saying something different?

 

[Thecla]

One way of thinking adout it:
If you have chosen a door and Mr Hall opened a different door, so there are now two unopenned doors. If at this moment a stranger comes in off the street in the middle of the contest and sees 2 unopenned doors, the stranger's odds of picking the correct door is 50% But you as a contestant have more information than the stranger who came from the street. You know more- what door you picked and which door Mr. Hall openned- than the stranger. This helps a lot.

 

[DonAntonio]

One way of thinking adout it:
If you have chosen a door and Mr Hall opened a different door, so there are now two unopenned doors. If at this moment a stranger comes in off the street in the middle of the contest and sees 2 unopenned doors, the stranger's odds of picking the correct door is 50% But you as a contestant have more information than the stranger who came from the street. You know more- what door you picked and which door Mr. Hall openned- than the stranger. This helps a lot.

Well, ALSO the stranger knows what door you picked and what door M.H. opened as he sees two doors. The really important

point is that the contestant knows that a third door exists which was opened and was empty . This is what changes

the odds for the contestant in cpmparison with the stranger.

DonAntonio

 

[arildno]

Yes. Did I appear to be saying something different?

No.
Sorry for not making clear I made my own reformulation of the content of your post, inspired by it.