Q&A: A Question on Hizkuni’s Words
A Question on Hizkuni’s Words
Question
From Hizkuni’s comments on the selection of the seventy elders from among the tribes
“And in the writings of Rashi it is explained: he took seventy-two slips and wrote ‘elder’ on seventy of them, and left two blank. And if you should ask: surely this involves great jealousy, since the later ones could say that whoever draws first will presumably get an ‘elder’ slip, because the ‘elder’ slips outnumber the blank slips. Rather, one may answer that by this same reasoning the first seventy can say: perhaps the two blank slips will come into our hands, since we are the majority, before they reach the last two, who are the minority; and it turns out that neither side has an advantage over the other.”
His answer doesn’t sit so well with me.
Answer
I don’t know what Hizkuni had in mind. But one can do the probability calculation. The first person to draw a slip has a 70/72 chance of being an elder. For the second, the probability depends on what the first drew: if the first got an elder slip, then the second has a 69/71 chance. If the first got a blank slip, then the chance is 70/71.
If we do not know what the first drew (or we calculate independently of what he drew), then we have to compute the expectation: for the second, the probability is: 2/72X70/71+70/72X69/71=70/72. Exactly the same as for the first.
And of course, if they first conduct a random lottery to determine who draws which slip, then a priori (before the first drawing) everyone’s chances are equal.
Discussion on Answer
I completely agree. I was only trying to explain his process. In the Bible there are lotteries conducted in ways that are not logical. For example, the author of Gevurat Ari (unlike Rashi) holds that the lottery between the goats was done by means of two slips, each with something different written on it, and not by one marked slip and one blank one. Meaning, two slips had to be drawn, not one.
It is “not logical” if you view the lot merely as clarification, as establishing a fact. But if the lot is an act of consecration (or even a selection in a state of ontic ambiguity), then it is understandable that the consecration “to Azazel” does not take effect automatically—something like inherent sanctity does not simply lapse for nothing—and therefore a special slip consecrating it to Azazel is needed, not a blank slip.
[Perhaps you too understand that here the lot is obviously not only a matter of clarification, since it reveals nothing that is not already known before Heaven. Rather, in your view the inscription “Azazel” is merely a convention in order to define the lot, and one could just as well have written “Shmalke” and “Barele” instead of “for the Lord” and “for Azazel,” so why should a blank slip be any worse than the inscription “Azazel”? The truth is that this really is “not logical.”].
I looked up the words of Gevurat Ari and found that he deals with the question of how it could be that the High Priest raises one slip with his right hand and the deputy priest (or the High Priest with his left hand) raises the second slip, when the Yom Kippur service is valid only when performed with the High Priest’s right hand. To this he says that raising the slip in the High Priest’s right hand determines the entire matter—which goat is for the Lord and which is thereby for Azazel—and the second slip is in practice unnecessary. https://hebrewbooks.org/pdfpager.aspx?req=47113&st=&pgnum=192
So what were you trying to show by this? The lottery itself is logical; it’s just that an additional ritual of raising another slip is attached to it.
It means that aside from the act of the lottery there are also rituals. So it could be here as well. But that is a side point. After all, it is obvious that there is a mechanism by which one can lottery 70 out of 72 with equal probability.
Seemingly, Hizkuni is saying that just as there are more “elder” slips (70/72), so too there are more draw sequences in which there is a chance of drawing a blank slip (also 70/72), and therefore “it cancels out.” That is, the expected number of blank slips in the first 70 draws is 70 times 2/72, and therefore the chance of each of the first 70 being blank is, as usual, 2/72.
But his wording is not precise. Even if there were only 40 draws (from a box containing 70 “elder” slips and 2 blank slips), there still would be no advantage to anyone. The chance of each of the first 40 being blank would still be 2/72. Because in random sampling, the distribution in the sample matches, in the limit, the distribution in the space.
This sequential process is exactly equivalent to a situation in which all the slips are shuffled and one slip is assigned to each person all at once, in which case it is obvious that everything is fair.
Rabbi Michi,
what you suggested—that they should first conduct a random lottery to determine who draws which slip (if that were really needed here)—seems like a step backward. If there is competition over who will be among the first 70 drawers, and you found a fair way to choose who those first 70 drawers will be, then you could simply decide that this itself will be the selection of who will sit on the Sanhedrin, and no second lottery would be needed.