[Avodah] Is it better to have one person do a vadai mitzva...

[saul mashbaum]

REReich
> We see that out of the 120 there are 76 cases (120-44), i.e a 
> 63%+ possibility that will be at least one Yibum and 30 cases 
> (1 + 10 +19), i.e. a 25% possibility that there be more than one.

RCL
>>
I have now had an opportunity to brush up on my probability theory
(which is a much simpler way of doing this kind of analysis) and which
gives the probability of getting five actual yibums at 1/5 x 1/5 x 1/5 x
1/5 x 1/5 = 0.00032 while of getting five chalitzas at 4/5 x 4/5 x 4/5 x
4/5 x 4/5 = 0.32768.  That means that the probability of getting one or
more cases of yibum (ie of equalling or bettering the situation where
you have one brother marry all the women) is 1 - 0.32768 = 0.67232.
>>
As far as the probability of getting 5 yibumim goes, RCL has gotten the math wrong. The possible combinations of the yevamim and yevamot is,  5 factorial (5*4*3*2*1) = 120, not five to the fifth power (=3125).Thus the chance that there will be 5 yebumim is, as REReich correctly points out, 1/120, an exponentially :) larger chance than RCL's  1/3125  This is because the five yibumim are not independent of one another. If , for example the first yibum is a "match", the chance that the second one is also a match is 1/4, not 1/5, since there are only 4 yevamot the second yavam may marry, one of whom is "his" yevama. In the probability examples we are probably familiar with, this illustrates the idea of "with replacement" and "without replacement". If five people blindly choose a ball from a box which contains one white ball and 4 black balls, there is a fundamental difference between if the ball chosen is replaced in the box after each choice or not. In the former case the number of possible choices is 5 to the fifth power, since each choice has no influence on any other; in the latter 5 factorial, since they do. In our case,obviously the yevamot are married "without replacement"
RCL wrote about my suggestion that we should take into account that in any event there are mitzvot chalitza being done.
>>
Thinking about this some more (and doing some research), I now wonder
can we indeed say this at all?  Is not the general principle that - even
though chalitza is a mitzva, b'mkom yibum aino mitzva.  >>
 I believe that this is not to be taken entirely literally. "Eino  mitzva" be "eino mitzva kol kach"; yibum is preferable to chalitza. The monei hamitzvot count chalitza as one of the taryag mitzvot, and I am unaware that there is a qualification "*only* if mitzvat yibum for some reason yibum cannot be done".
Clearly. sometimes mitzvat chalitza is *preferable* even if yibum could be done, even according to the opinion that mitzvat yibum kodemet. According to the Torah, the yavam is visited by beit din who talk to him about his obligation. The gemara says that they advise him as to the option he should exercise. If the yavam is very old, and the yevama is very young, yibum is considered unadvisable, and beit din may endeavor to persuade the yavam to do chalitza, rather than enter into an unhappy and problematic marriage.  If the yavam listens to this sage advice and does chalitza, he has definitely been makayem a mitzva from the Torah.  I am convinced that even in other than this extreme case, every yavam who does chalitza is mekayem a mitzva min haTorah.
>>Because if the yevama is forbidden to the yavam by way of a lav, yibum can still be
performed, on the basis that aseh docheh lo ta'ase, and we do *not* say
that since it is possible to do chalitza and it is not as though he will
not do a mitzvah at all, since he will perform chalitza, and it is
merely not a mitzva min hamuvchar, we should be mevatel the mitzvah of
yibum in favour of the mitzvah of chalitza  <<
I do not think that we can deduce from asei doche lo taaseh by yibum that chalitza is not considered a mitzva if yibum could possibly be done. Asei doche lo taaseh even if one could somehow both do a mitzva and avoid the lo taaseh. Tzitzit is doche k'laim, and wollen tzitzit can be put on a linen garment, even if linen tzitzit (without t'chelet) could possibly be used, both performing a mitzva (in perhaps a lesser way) and avoiding the lav. Despite the wording of the principle, I think it's fair to say that the lo taaseh is hutra, not merely d'chuya, by the asei.
Shabbat Shalom and KvCT
Saul Mashbaum
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