The Tribes of Israel

Eliyahu Rips

Draft, 22/06/1999

© All rights reserved

The experiment described in this paper is based on a list of birth dates of the sons of Yaakov given in Midrash Tadshe. I learned about this midrash from Doron Witztum, who used it in his paper "Personalities of Genesis and their dates of birth". However, the experiment described here, while being essentially related, is not identical to that of Witztum's. It reflects my own choices and decisions. In a sense, it can be viewed as complimentary to the paper "Personalities of Genesis."

Midrash Tadshe is quoted by Rabenu Bachya in his Commentary to the Torah, Exodus 1:6. It is also brought in Yalkut Shimoni, in Otzar Midrashim (Eisenstein), in Shalsheleth Hakabalah, in Torah Shlemah (Kasher) etc. On Fig.1 we see the corresponding passage from one of the editions of Rabenu Bachya Commentary on the Torah:

Fig.1

Thus, Midrash Tadshe gives a list of birth dates of the sons of Yaakov:

However, it turns out that there are differences between the versions of this list appearing in different sources. So, for example, the old edition of Yalkut Shimoni gives the birth date of Shimon as 28th Teveth, and it does not bring the birth date of Yosef. In the new edition of Yalkut Shimoni (ed. Mosad Harav Kook) the list of the birth dates is identical with that of Rabenu Bachya.

We strived to make the design of the experiment dependent on as few choicesand decisions as possible. The decisions and choices we have made are stated below.
1.Given that there are several versions of the list of birth dates given by different sources, we decided that each source "as it is" will serve as a basis for a separate experiment.
2.We decided to restrict ourselves only to the sources which bring the list of birth dates as a single whole. The sources which bring it scattered over other topics like Seder Hadorot or Yalkut Meam Loez are thus not considered.
3.We decided to maintain the definitions and the setting from the previous work. In case it will into conflict with the "as is" principle, we have to check both alternatives.

In this section we specify the practical consequences of the decisions described above.

In our research we have studied two different, though related phenomena, namely
(A) close meetings between ELSs, and (B) close meetings of ELSs with substrings of the text, that is with expressions with skip 1 or -1.
Each of theselternatives can be used for the design of an experiment. Let me say at once that the alternative (A) failed. We will not return to it more in our discussion.

The values c(w,w') were defined as in WRR for the case (A); their definition is modified in WRR2 to be used in the case (B). We use the same randomization scheme as in WRR. However, we supplement it with a control experiment of a new kind based on random permutation of letters (see the details below).

In order to compose the samples, we have to decide how to write the dates and the names.
We have two alternatives for the dates: to write them precisely as is the source or to use 3 forms of date we used in WRR (including the additional forms for 15th and 16th). In the spelling of ten of the manes there is no ambiguity. We have only to decide how to write Zevulun and Binyamin. Here we also have two alternatives. One is to take the names precisely as in the source. The other one is to take the names as they appear in the Torah, because we used this rule in WRR. (I originally overlooked this alternative. It was pointed out byDr. McKay after my talk at the Rationality Center, Hebrew University, January 1999).

The following table lists the data obtained from the source shown on Fig.1.

From the list of the dates "as is" only 5 appear in Genesis as ELSs (recall that we consider only expressions from 5 to 8 letters):

Taking the dates in 3 forms with the names "as is" gives a sample with 27 actually appearing pairs (of type 1-2). For this sample P1=2.285x10e-4 and P2=1.1x10e-2.

The randomization test: for 1000000 random permutations, P1 had rank 8 and P2 had rank 1186. For statistic P1 we have also conducted a test on all 12! permutations and obtained rank 3567.

For the names taken as in the Torah (sample 2-2) we consider two possibilities: with בן אוני and without בן אוני. The corrsponding samples have 39 and 36 actually appearing pairs respectively. Here are the results:
With בן אוני P1=6.992x10e-4, P2=7.5x10e-3; the ranks are 472 and 6082 respectively.
Without בן אוני P1=7.042x10e-3, P2=5.9x10e-2; the ranks are 1457 and 22615 respectively.

In the randomization test, the correct matching between the names and the dates (that is the identical permutation) is compared with a large number of wrong matchings (non-identical permutations). In order to verify, to which extentthe outcome of the randomization test (that is the ranking of the correct matching) reflects the probability level, we designed a new kind of control experiment. In this control experiment we perform a long series of randomization tests. In each one (except for the initial one) the expressions for the dates are substituted by some randomly chosen permutation of their letters. We inquire, how often the ranking of the identical permutation in such "letter-permuted" randomization tests is equal or better than the ranking obtained in the original randomization test (that is, with correct expressions for the dates)?

For the sample of type 1-2 and statistic P1, in a series of 5000000 randomization tests with letter permutations of the dates, in 27 cases better rank was obtained, with no ties (we planned to run a series of 10000000 randomization tests, but the run was interrupted shortly after the middle).

All the rest is still in progress.

[WRR] Witztum, Rips, and Rosenberg, Equidistant letter sequences in the Book of Genesis.
[WRR2] ibid. II: The relation to the text.
[W] Witztum, Personalities of Genesis and their dates of births, at Doron Witztum's web site www.torahcodes.co.il.