the weekly Torah reading by the faculty of
Lots of Luck
Prof. Ely Merzbach
Department of Mathematics
“Moses … gathered seventy of the people’s elders and stationed them around the Tent” (Num. 11:24).
In this week’s reading Moses appoints seventy elders who are to constitute the Sanhedrin, a sort of Supreme Court. The way in which they were selected is explained at length in Tractate Sanhedrin (17a):
The rabbis taught: “Two men … had remained in camp” – some say they remained in the ballot box, for when the Holy One, blessed be He, told Moses to gather seventy elders from among the Israelites, Moses said: How shall I do it? If I select six from each tribe, there will be two left over; if I select five from each tribe, I will be ten short; if I select six from one tribe and five from another, it will lead to jealousy between the tribes. What shall I do? He selected six from each tribe, and collected seventy-two notes. On seventy of them he wrote “elder” and two he left blank. He mixed them up and put them in a ballot-box. Then he told them, “Come take your notes.” To every person who drew a note saying “elder,” he said, “The heavens have already set you aside”; to whomever drew a blank note he said, “The Omnipresent does not want you; so what can I do?”
According to this description, there were seventy-two candidates, of whom seventy were elected – those on whose note was written “elder.” This process of drawing lots is described in the Jerusalem Talmud, as well (Sanhedrin 1.4), except that there the story has a continuation:
Rabbi Yode and Rabbi Nehemiah asked one another: Each of the two who drew blanks could still claim: Had I written “elder,” I too might have drawn a note saying “elder,” but since there were only seventy such notes, I did not draw one. So this is what he did. He took seventy-two notes and wrote “elder” on each of them, and another two blank notes, and put them all in the ballot-box. He then told them: come and take your notes. To those who drew a note saying “elder” he said, “You were already appointed by Heaven.” To those who drew blanks, he said, “What can I do? It is from Heaven.” One asked the other: Consider what if they all came out saying “elder”? He answered, “It was a miracle [that they did not all draw notes saying “elder”], but came out alternating.
According to the Jerusalem Talmud, so that the lottery would be perfect, seventy-four notes were put in; “elder” was written on seventy-two of them, and two were blank. Miraculously two of the seventy-two candidates selected the blanks. However, the halakhah did not rule according to the method in the Jerusalem Talmud, and when picking lots one does not rely on a miracle.
Using lots to decide legal or halakhic questions is very widespread in Jewish sources. In the Torah itself there are several examples of this practice:
1) In Genesis, after the brothers sold Joseph to the Ishmaelite caravan going down to Egypt, they decided to tell their father that Joseph had been torn apart by a wild beast. The midrash recounts that the brothers cast lots to determine which one of them would have to carry out the frightful task of telling their father, and the lot fell on Judah.
2) Lots are cast on the Day of Atonement: “and he [Aaron] shall place lots upon the two goats, one marked for the Lord and the other marked for Azazel” (Lev. 16:8). The gemara in Tractate Yoma emphasizes the importance of these lots, the halakhah ruling that casting lots is a necessary procedure: the entire Yom Kippur ritual is not valid if the goats are not selected by lot.
3) Redemption of the first-born: According to Numbers , there were 273 firstborn in excess of Levites. The firstborn were “exchanged” for Levites and the excess firstborn were redeemed for five Shekels each. How did one know which first-borns were to be redeemed by a Levite and which by a sum of money? The gemara (Sanhedrin 17b) says it was done by casting lots: Moses placed in a box twenty-two thousand slips on which “to a Levite” was written, and two hundred and seventy-three slips on which “five shekels” was written, and each first-born was redeemed according to what was written on the slip which he drew.
4) Apportionment of the land: “The land, moreover, is to be apportioned by lot” (Num. 26:56). Here, too, the gemara ( Bava Batra 122a) explains that apportionment of various regions of the land among the tribes was done by slips that were drawn from two lotteries, one specifying the regions and one, the tribes.
On the basis of these examples (and many others) the Sages in many instances ruled that the court must draw lots (for example, in dividing property among heirs or partners; see Shulhan Arukh, Hoshen Mishpat 175), and in certain cases even a private individual may make a lottery.
When the lottery method is a drawing, the people drawing slips one after the other, is the order in which they draw of any importance? The rishonim, it turns out, discussed this question with reference to selection of the seventy elders.
Responsa Radbaz (Part II, par. 779) reads:
A question arises regarding this baraitha, for Moses was trying not to incur jealousy among the tribes, yet much jealousy remained since it is not the same for the person who puts in his hand first, when the box has 70 slips saying “elder,” as it is for the one who draws last, when there are only a few slips remaining. And even though it is said that “everything which is fixed is equal fifty-fifty” (meaning that every one who draws has the same fifty-fifty chance), in any event it is clear that there is jealousy here, and each one will want to be the first to draw.
Radbaz raises several possible solutions to this difficulty, such as, for example, that the ballot box was wide and all seventy-two participants in the lottery drew their slips simultaneously. In the end, he rejects all these possibilities and writes:
It seems to me that the Israelites did not care at all about the order of the tribes and that they were not jealous of one another in this regard, for Heaven came to their aid… One might even say that the Lord was provident, seeing to it that those who were not deserving would draw blank slips among the first to draw, so that any contention would be removed; and this is correct. Likewise, with the two goats selected by lot, one always finds that the one marked for the Lord always comes out on the right, and the one marked for Azazel on the left; and this is clearly an act of Providence, and it is this way in order to avoid dispute and contention. This also follows from the Jerusalem Talmud, as said between the first and last Tanna in the discussion: the seventy-two slips that said “elder” and the two blank slips were miraculously alternated, elder and blank, thus also pleasing everyone and removing any contention.
This special method is quite surprising and not easy to understand. In terms of probability it is easy to show (using conditional probability and the formula for inverse probability) that the probability of being elected an elder is exactly the same for the first one who draws as for the last, and a simple computation shows that the probability at all stages of the lottery equals 70 divided by 72.
Indeed, Hizkuni writes on Numbers 11:26 that the order of selection makes no difference whatsoever. He says as follows:
You might think that great jealousy would be caused by this, the people who draw later conjecturing that those who draw first will most likely all draw slips saying “elder” since there are more slips with “elder” than blank slips. However, one could equally well conjecture that the first seventy might say: the two blank slips are likely to come our way since we are the majority, before they come to the last two, since they are the minority. So we see that there is no advantage for one over the other.
In my humble opinion, and thus also according to the halakhah, the order of drawing lots make no difference whatsoever, and there is no reason to insist on any specific order (also since we have found no references to the matter on the part of posekim, save for Radbaz).
From these and other examples we can deduce three rules that the halakhah requires when drawing lots:
Comprehensiveness of the lottery: the first rule is that there must be a drawing for each and every item involved. Generally there is only one item and therefore one can make do with a single ballot box; but if there are more items to be decided by lot, one must use more ballot boxes, according to the number of items.
Fairness: the second principle in the process of drawing lots is that it be fair; all participants in the lottery must have an equal chance. In other words, scientifically put, the distribution of the outcome should be uniform.
Appearance: the third and last rule demands that care be taken to have the lottery also appear fair to all those participating, without any tricks, and that the entire public be present at all stages of the lottery: preparing the slips, mixing them, etc. Great importance attaches to appearances, and the public must sense the presence of the selection by lots throughout the entire process.
We conclude by quoting the remarks of Havot
Yair, which are supposed to guide us whenever we
draw a lottery: “If a lottery is