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

[saul mashbaum]

RCL:
>
While I would seem to have gotten the maths wrong, because as you say,
the yevamot choices to have an influence on the next choice, I am
struggling a bit to see how this case matches the white/black ball case
either.  If five people choose a ball from a box with one white ball and
five black balls, then in the end *somebody* gets the white ball.
However in our case, it is quite possible that nobody gets *his* yevama.
>>
My example of 4 black balls and one white one indeed did not completely correspond  to the case of yevamot, and was merely designed to illustrate the difference between selecting with replacement and without replacement, to show why the number of possible selections is five factorial and not five to the fifth power.   A complete match to the yevamot case is the follows: A box has 5 balls of the following colors: black, white, brown green and grey. 5 people, who happen to be named Mr. White, Mr. Black, Mr Brown, Mr. Green, and Mr Grey choose balls, one after another without replacenment, from the box. A "match" is when someone chooses a ball from the box whose color corresponds to his name (this is like a yavam marrying his yevama). A perfect match is all five doing so, which will happen on average in only 1/120 of the cases. Clearly it is possible that there will be no matches, or one or 2 or 3 (only four matches is impossible, since if the first 4  match, the fifth one must be a match as well). I have been unable to devise a general formula for calulating the likelyhood of a certain number of matches for any given number of brothers; empirically, greatly aided by REReich, I conclude that the mishna's method will on average produce fewer yibumim: for 5 yevamim the mishna's method is better (ie multiple yibumim) in 30 cases out of 120, against 44 non-yibumim out of 120. It seems that even taking into account the additional matches in the mutiple yibumim, the total number of yibumim out of the total number of marrisges by the mishna's procedure will be less that the 1/5 we get by the alternative procedure.
RCL
>>
That means, as I understand it, that while there may be a whole debate about Reish
Lakish's principle and whether it is d'orisa or d'rabbanan, elsewhere,
that debate does not start to apply to yibum/chalitza, because chalitza
is already discounted as a mitzvah when one is discussing yibum, so the
whole Resh Lakish principle does not apply if the option is chalitza as
opposed to yibum. 
>>
Reish Lakish makes his statement (or more accurately *could* make his statement - "amar l'cha Reish Lakish") that "chalitza b'makom yibum lav mitzva hi" in order to save his whole principle, which the gemara suggests is refuted (leima tehavei t'yuvta d'Riesh Lakish) by the case of chayvei lavim and yibum. To present this tentative suggestion that Reish Lakish could conceivably make to avoid being entirely refuted as a fundamental principle universally accepted and applicable to our case is IMO something of a stretch. I am happy with my statement that "lav mitzva" means "lav mitzva kol kach" even according to RL. 
My proof from the case where when appropriate the yavam is persuaded to do chalitza instead of yibum is still valid; the case of safek is merely another case where chalitza is a proper procedure, and one who does so is m'kayem a mitzva.
Saul Mashbaum
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