The discussion centers around the Monty Hall problem, specifically debating the probabilities associated with switching doors after one has been opened. Participants explore the implications of the game's structure, assumptions about the host's behavior, and the resulting probabilities of winning a car versus a goat. The conversation includes both theoretical reasoning and personal interpretations of the problem.
Participants do not reach a consensus, with multiple competing views on the probabilities involved in the Monty Hall problem. Some maintain that switching doors yields a 2/3 probability of winning, while others argue for a 1/2 probability based on their interpretations of the game's mechanics.
Participants express uncertainty regarding the assumptions made about the host's behavior and the implications of those assumptions on the probabilities. There are also questions about the application of conditional probabilities in different scenarios presented during the discussion.
haruspex said:... and are equally likely ...
Yet that's where DaveC went wrong.DonAntonio said: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
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.alewisGB said: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 said: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.
Yes. Did I appear to be saying something different?arildno said: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.
Thecla said: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.
haruspex said:Yes. Did I appear to be saying something different?