Quote Originally Posted by Hiro Protagonest View Post
Actually, 1 is 1 and 2, then 2 and 3 are 3 and 4.

1. The prize is behind door A. Monty opens door B. Switch, you lose.
2. The prize is behind door 1. Monty opens door C. Switch, you lose.
Those two combined have a probability of one third - the probability that you picked the right door initially. You've split it into four possibilities, but not four possibilities of equal probability.

Quote Originally Posted by Susano-wo View Post
Thank you for the detailed explanation. But its still wrong. :P (I know, I know, it's widely accepted but I am still arguing. sorry >.<)
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The choice is between door number 1 and door number 2. was it unlikely that you picked right initially? yes. does that change the probability that you are choosing between 2 equally possible choices now? no. Each choice is independent.

Its like the difference between what your chances are to roll nat 20 twice in a row, vs your chance on either or 2 rolls. Your chance to roll 20 is always 5%, no matter how many 20s were rolled before that, but the aggregate probability that the overall occurrence could happen is much lower.

In the Hall problem, you have already made the first choice, in which you either conformed to probability (picked wrong door) or did not (picked right door.) that part is done, and now you have a new choice, with a 50 50 chance. The second door is not 2/3, because 3 is no longer really a choice, only 1 and 2 are.
Sorry if I appear belligerent, but I just cant see the logic in the proposed probabilities.
You don't appear belligerent - just mistaken.

Consider the following nine cases, all equally likely:

You picked A; prize in A. Either B or C has been opened. Switching is bad.
You picked A; prize in B. C has been opened, switching is good.
You picked A; prize in C. B has been opened, switching is good.
You picked B; prize in A. C has been opened, switching is good.
You picked B; prize in B. Either A or C has been opened, switching is bad.
You picked B; prize in C. A has been opened, switching is good.
You picked C; prize in A. B has been opened, switching is good.
You picked C; prize in B. A has been opened, switching is good.
You picked C; prize in C. Either A or B has been opened, switching is bad.

2/3 chance that switching is good, because if you picked wrong first, you were given new information.

If you won't accept the word of a Statistics instructor, then I suggest that you actually try it. Run 1,000 cases in which you choose randomly, Monty opens a worthless door, and then you switch.