Talmud for Numbers 3:40

“And Moses was presiding.” ³⁵⁹Tanhuma Bemidbar 21, Tanhuma Buber Bemidbar 25, Num. rabba 4(9), shortened Babli 17a. Moses found a way to settle things. When the Holy One, praise to Him, told him, count all male firstborns among the Children of Israel³⁶⁰Num. 3:40. The reference should have been to vv. 46,47 where Moses was commanded to collect 5 šeqel per person from 273 of the 22’273 firstborn of the 11 tribes. The other 22’000 were redeemed by the service of 22’000 Levites. Since he was instructed to take the 5 šeqel from 273 persons, he could not simply collect 1’365 šeqel from 22’273 persons., he said, who will consent voluntarily to give five sheqel per head? What did he do? He took 22’000 tickets³⁶¹Greek πιττάκιον “ticket, label, etc.” and wrote on them “Levite” and 273 on which he wrote “five sheqel” and put all of them in an urn³⁶²Greek κάλπη, “urn”.. He told them, come and take your ticket. To any one who drew a ticket saying “Levite” he said, a Levite already redeemed you. To anybody who drew a slip saying “five sheqel” he said, what can I do, it is from Heaven; Rebbi Jehudah³⁶³The Midrash sources make it clear that the preceding was R. Jehudah’s opinion; the following is R. Nehemiah’s. The Babli only mentions R. Jehudah’s opinion.. But Rebbi Nehemiah the Tanna objected to the colleagues: If you had given me the possibility of a Levite, it would have come up for me.³⁶⁴Since there were only 22’273 tickets, it was a mathematical necessity that 273 people would have to pay. The distribution of these was a pure matter of probabilities; there is no reason to invoke Heaven in the matter. But the following is what he did. He took 22’000 tickets³⁶⁵The parallel sources, and the paragraph after the next, make it clear that one has to read: 22’273 tickets reading “Levite”. and wrote on them “Levite” and two hundred seventy three on which he wrote “five sheqel” and put all of them in an urn. He told them, come and take your ticket. To any one who drew a slip saying “Levite” he said, a Levite already redeemed you. The anybody who drew a ticket saying “five sheqel” he said, what can I do, it is from Heaven. The Tanna objected to the colleagues: Think of it, if all came up as “Levite”³⁶⁶Assuming that the tickets in the urn are well mixed and all tickets have the same probability to be drawn, the probability that in 22’273 drawings of 22’546 tickets no slip of “5 šeqel” or that all of the 273 extra tickets would be drawn is practically zero. (The first probability starts with a string of 117 zeroes after the decimal point, the second with 639.) If all 273 extra tickets were actually drawn, it was a clear sign from Heaven.? They told him, it was miraculous and they came up alternatingly. Rebbi Samuel said, in the opinion of the second Tanna, it was a miracle. In the opinion of the first Tanna, it was not a miracle³⁶⁷While in the first scenario by necessity 273 tickets of “5 šeqel” were drawn, these slips were drawn with approximately constant frequency, about 122.6 per 10’000 draws, against the probabilistic frequency of 121.1 per 10’000. One cannot say that alternatingly “Levite” and “5 šeqel” were drawn since then the drawing would have ended after 546 draws.. They told him, it was a miracle in any case since they came up alternatingly.