Thursday, May 27, 2010

Need Help on Kavua & The Monty Hall Problem

it struck me yesterday that perhaps kol kavua k'mechtza al mechtza dami has something to do with the "Monty Hall Problem" - but I can't articulate it. Can you help me out by either explaining why I'm right or why I'm wrong?

This may help (siman 46 above):



  1. You mean siman 47, right?

  2. As you might recall, I built a whole philosophy about qavu'ah based on R' Aqiva Eiger. It includes eidus, migo and terei kemei'ah as tolados of the same av, as well as chazaqa demei'ikara as a first cousin. (Whereas rov and chazaqa disvara belong to a different family of birur.) Literally a philosophy, as I use RAE's chaqira to prove the correctness of hashkafos that see mitzvos in terms of cognitive changes -- whether as deveiqus or a definition of sheleimus.

    So the mehalakh you're proposing isn't likely to appeal -- it contradicts my "baby".

    Leshitas RAE, qavua isn't statistical because it's resolving a safeiq in din. Rov only works when pasqening where the safeiq is in the metzi'us.

    The Monty Hall problem is only kemechtza al mechta because it involves three doors. Parallel problems could be made for four or more doors (n doors), where the chance of winning is 1/n if you stay but 1/n-1 if you switch.


  3. That should be read: In the MJP for n doors, the chance of winning if you stay is 1/n, but the chance of winning if you switch goes up to 1/(n-1).

    Forgot the parenthesis last time.

  4. 47, right.

    My thoughts are running to the ephemeral nature of probability, and asserting that a probability rov is not a valid halachic rov unless it is a rov b'teva - which is really something else.

  5. No connection whatsoever. From a probabilistic standpoint, the Monty Hall paradox is just an application of Bayes' Theorem. Nothing could be further from Chazal's decision methods than Bayes' Theorem, since they never relate to probabilities as numerical values.

    Kol kavua is a principle according to which a set is treated as a single mixed entity, so that an individual member of the set inherits a mixed identity from the group rather than being assigned a pure identity.

  6. Ben,

    Interestingly, I would have said the reverse of your last paragraph.

    Kol deparish is a notion that there is a single mixture, the state of the mixture is rov X, and therefore the state of each item is rov X.

    Whereas with qavu'ah, the item is treated as an independent unknown, and thus the majority of the set is irrelevant.

    From a different perspective...
    There might be a Bayesian explanation to qavu'ah. Qavu'ah is where we once knew more of the givens than we know now.

    IOW, we were once able to say P(X | C) -- the probability of X given C. Such as the probability this meat is kosher given that we know it came from a trustworthy butcher-shop. However, we forgot which items were P(X | C) and which were P(Y | not-C). I.e. which one came from the kosher shop, and which didn't.

    Whereas WRT kol deparish, we never had the set divided into C and not-C.

    So, the rule of qavuah is saying that an item can never revert from P(X|C) back to P(X).


  7. My premise is that *no* rov in Halacha is based on probability. Rather, wherever bittul b'rov does not work, thus leaving the minority as a factor, only kol d'parish works, because then we can assume that the minority remained behind.

  8. You lost me with that comment, since I don't know how you would define a ruba deleisa leqaman other than probability.

    Since it's leisa leqaman, how do you invoke bitul?

    Isn't there also a Brisker chiluq between rov as birur and bitul berov? Part of it -- "acharei rabim lehatos" is bitul, since the decision becomes that of the beis din as a whole. Unfortunately, I don't recall the rest.


  9. Rubba d'leisa kamman is not probability. It really only works when it is a rov b'teva. It certainly is not a probability based on statistics.

    I'm posting The Brisk thing to which you refer in a re-edit of this post. Let me know what you want to propose on this basis.

  10. My first assumption was that by "beteva" you meant "because of a law of teva". If so, and if the rov is beteva, and not just statistically, it would be a chazaqah, no? For example, "ein adam chotei velo lo" is such a rov beteva.

    Or perhaps you mean to say that a ruba deleisa leqaman is a physical numerical rov that happens not to be in front of you. But that is itself the frequentist interpretation of statistics.

    As for the R' Chaim... that will have to wait until after I have time to learn it. (I'm still on yesterday's daf in Kelaim.)


  11. A rubba d'leisa kamman that is effective (=rov b'teva) is a form of chazakah - but not a chazaah d'mei'ikarah, which is the very narrow definition of chazakah derived from tzora'as. Rather, it is a logical construct based not on a statistical rov, but on psychological (chazakah ein adam porei'a toch zmano) or biological (rov beheimos kesheiros) principles, etc.

  12. I fail to understand your clarification. You appear to agree with my statement that a ruba deleisa leqaman that has a reason to be true, whether a law of psychology or a law of nature, is a chazaqah disvara. (The Sheiv Shemaatsa's term for chazaqah that isn't a chazaqah demei'iqara.)

    But then when the gemara or shu"t refer to a ruba deleisa leqaman, they must mean where there is no teva forcing the rov. It just happens to be true. E.g. rov metzuyim eitzel shechitah mumchin heim. Doesn't have the same authority as a chazaqah.

    And so I return to the question of how one defines ruba deleisa leqaman other than probability. There is no set to count, that would be an isa leqaman.


  13. can someone explain to a novice what exactly "kol kavua k'mechtza al mechtza dami" means and give an example?

  14. Translation: Whatever was established is like half vs. half.

    The textbook case of Kol Qavuah is where someone lives in a town with ten butcher shops, of which 9 sell kosher. He buys a piece of meat, but doesn't recall which of the 10 it was bought from. There, he can not rely on rov (majority), rather he has to treat the doubt as though it were balanced.

    Whereas if he owned three pieces of fat, two of which are permissible and one is cheilev (e.g. it was surrounding the kidneys) and thus prohibited, and they get mixed up -- he MAY rely on majority. This is kol deparish meiruba parish -- whatever separated itself [from the group] was separated from the majority. In this case, the piece of fat he uses is presumed to come from the 2/3 which are kosher.

    There are MANY theories as to why these two types of doubt have different rules. Our host is suggesting that the Monty Hall Problem would provide such an explanation. I don't see it, for reasons given above. Now that I defined the terms, perhaps you can reread the original blog entry and explain the line of reasoning to me. RYGB seems to have given up.

    On the other hand, I have a whole theory built out of Rabbi Akiva Eiger's explanation which feeds into my hashkafah (which is the same as our host's on all relevent points) as to what we're trying to accomplish with mitzvos. So, I'm heavily emotionally invested in a different theory, and my objection may not come from enough objectivity.

    In short, R' Aqiva Eiger's position is that qavu'ah is a case where the halakhah was established, and now doubt arose as to what it was. It did come out of a store, when it had halachic state. The case of majority is where we don't know the facts on the ground to even start establishing the halakhah to begin with. Doubt about fact can use majority, doubt about halakhah cannot.

    I believe what underlies R' Akiva Eiger's position is the idea that halakhah is established based on how people experience to an object, not objective reality. Much like our ignoring things people can only experience indirectly, via tools -- microscopic mites and whatnot. Therefore, if we can not establish objective reality (and not due to criminal neglect), the doubt is itself the "reality" about which we pasqen.

    This does relate to the Monty Hall Problem, since there too how we relate to the unknown changes based on information revealed after the decision -- and the natural commonsensical answer doesn't actually correspond to what are the actual probabilities. How we relate and objective reality differ.

    But I'm not sure how to use it to explain kavu'ah differently.