BS: Mathematical Probability Query

Subject: RE: BS: Mathematical Probability Query
From: GUEST,Murray MacLeod
Date: 12 Oct 00 - 06:49 PM

Fionn, my brain is starting to hurt again. My understanding of Marion's problem depends on the fact that the house DOES have prior knowledge, and is never going to turn over the cup containing the prize. AM I missing something ?

Murray

Subject: RE: BS: Mathematical Probability Query
From: Wolfgang
Date: 13 Oct 00 - 04:57 AM

Murray,
I didn't understand the Homer Simpson reference (lack of cultural background) so there's no need at all for an apology.
For everybody else:
Murray's dead right on Marion's problem and Fionn is wrong. If the house knows where the prize is and alwaysalways offers the claimant to switch then the response to the question what is the probability of winning if I switch is 2/3.
If, however, the house always offers a switch and always turns over a cup which (a) was not chosen in the first choice but (b) may contain the prize (e.g., if the house doesn't know where the prize is or knows where the prize is, but nevertheless opens the prize cup in 1/3 of all cases) then the response to the same question is 1/2.

This puzzle is considered technically identical to a number of teasers in conditioned probability, including the prisoners' problem. All of these problems have in common that
(a) they are problems in conditioned probability in which the question is whether one information given, which itself is not conclusive yet, should lead me to adjust my prior probabilities of an outcome,
(b) they have different solutions depending upon the exact wording of the problem (more technically: they have different event spaces, since some events are excluded due to one wording and not to the other),
(c) one of these solutions is intuitively clear and therefore is chosen by nearly all persons encountering this problem for the first time (even if the problem is clearly presented in a way that calls for the other response), and
(d) for didactical purposes instructors chose to present the problem in a wording that leads nearly everyone to be wrong (well, it's just less fun to present the Monty Hall dilemma in a wording that leads close to all subjects to get the in this case correct 50/50 response).

Other problems which are mathematically nearly identical are

the prisoners problem (see above). Mostly chosen response: my chances are higher to be executed, counterintuitive but correct response (in some wordings, but not in Marions wording): my chances to be executed remain as they were, but the chances of the third person increase

the three card problem (see above): obvious response: 1/2, counterintuitive response: 2/3

the father and son problem: a man walks to me and says: "I have exactly two children, one of them is a son". What is the probability that the other child is a son as well? Obvious response: 1/2 counterintuitive response: 2/3

the two dice problem: I throw two dice and tell you truthfully (after I had a look without you looking) "at least one of these two dice shows a six". What is the probability that the other also shows a six? Obvious response: 1/6, very counterintuitive response: 1/11

All these problems can get much more complicated if you allow for unequal prior probabilities (e.g., one prisoner gets executed with prob 1/2, the next with prob 1/3 and the last with prob 1/6) or if you allow for biases (e.g., the father of a mixed pair of kids usually tells about his son, but sometimes about his daughter).

If you are really interested I can only advise you to read :
Nickerson, R. S. (1996). Ambiguities and unstated assumptions in probabilistic reasoning. Psychological Bulletin, 120, 410-433.

Wolfgang

Subject: RE: BS: Mathematical Probability Query
From: Wolfgang
Date: 13 Oct 00 - 05:15 AM

More than 100 posts!!

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