Non-Random Equidistant Letter
Sequence Extensions in Ezekiel

By R. Edwin Sherman, FCAS, MAAA, and Nathan Jacobi, Ph.D.

Appendix B—A Markov Chain ELS Extension Model

This addendum provides a derivation of formula (1), which is used to determine the expected number of extended ELSs (with varying numbers of extensions) to emerge from a search process beginning with n initial ELSs. We shall denote initial ELSs by the symbol I. After a Hebrew expert examines the string of letters resulting from taking every j-th letter before and after the initial ELS with a skip of j letters, one of two results will occur in each instance that there is an opportunity to find an extension of the initial ELS. If no extension is found, this will be denoted by N and if one is found, by E.

The following chart presents the complete range of outcomes from this process up through the location of four extensions to an initial ELS. From this chart, the formula for determining the expected number of ELSs with a given number of extensions is evident.

Each outcome is represented first by a combination of the letters I, E and N, indicating the order in which these events appeared in that letter string. In the middle section of the table is a row of formulae for the expected number of ELSs for each type of outcome in that column. In each formula, n is the total number of initial ELSs, d is the probability of finding a grammatically correct Hebrew extension of the preceding ELS, and (1-d) is the probability of failing to find an extension. It was assumed that d is independent of the number of extensions that have already been discovered—even though consideration of the different factors affecting d as the number of extensions increases suggests that d most likely declines as the ELS becomes longer.

The beginning of the search process is represented by the leftmost column of the table, where there are n instances where I, an initial ELS, appears. Each column represents the range of outcomes that would result in the indicated number of final extensions.

The only way that zero extensions will be found is the situation where no extension is found either before or after the initial ELS (denoted by NIN). The probability of this occurring for any given initial ELS is (1-d)2. So the expected number of final ELSs that have no extension is n(1-d)2.

There are two ways that a final ELS can have one and only one extension: NIEN and NEIN. The expected number of final ELSs that are of the NIEN type is nd(1-d)2, as is the case for the NEIN type. So the total expected number of final ELSs with exactly one extension is 2nd(1-d)2, as shown at the bottom of the "One" column.

There are three ways that a final ELS can have exactly two extensions: NIEEN, NEIEN and NEEIN. For any given column (representing k extensions), I can appear in the 1st, 2nd… or (k+1)st position after the first N. Thus the total expected number of final ELSs with exactly k extensions will be

(1)                                                     (k+1)ndk(1-d)2.

The above model is an example of a Markov chain, since the outcome of any trial depends at most on the outcome of the immediately preceding trial and not upon any other previous outcome, and the probability of each state is clearly defined. When the state N occurs, it is an "absorbing state."

As a check on formula (1), it should be possible to show that the sum of (1) for k ranging from 0 to infinity is n, the total number of initial ELSs. That derivation is as follows:

Since n and (1-d)2 appear in each term, we can factor them out of an expression for the total number of final ELSs, to get
n(1-d)2 [Ó (k+1)dk],

where k ranges from 0 to infinity.

If we multiply each term of this power series by d in both the numerator and denominator, we obtain
{[n(1-d)2]/d} [Ó (k+1)(d{k+1}].

If we shift the value of (k+1) by one for each term, so that the series is summed from k = 1 to infinity, rather than from k = 0 to infinity, it becomes

(2)                                                 {[n(1-d)2]/d} [Ó kdk].

According to formula 40 on page 8 of Summation of Series, collected by L.B.W. Jolley:

Ó nxn = x/(1-x)2,

where x < 1, and the series is summed from n=1 to infinity. If we substitute k for n and d for x in this formula, we have
Ó k(dk) = d/(1-d)2.

By substituting the expression on the right for the power series expression in (2) above, we get

{[n(1-d)2]/d} [d/(1-d)2] = n.

Appendix C – Comparison of the Number of Lengthy
ELSs Found in Ezekiel 37 with Those Expected by Chance

ELSs 46+ Letters Long       [Expected: 0.092, Actual: 6. A/E = 65.0 ]

  • (61) There is quarrel in his speeches. A living brother uttered words to them and to me. And Zubaidah turned to his sea, without then lying for a whole week. Oh, the mountain of her interior will bear a testimonial to her name.
  • (53) The island was restful, elevated, and it happened. Where is Libya? And you have disrupted the nation. She changed a word. He answered them with combat. Why the navy and the smell of the bottom of the sea?
  • (52) For where has God consumed from you? And in it are stones of substantial sickness for us. You will indeed delay their diagnosis, because of His own reflection in the one who solves.
  • (52) The trouble of the new-born one is vigilant and honest because of the ruin. Get out as if Iraq had been sent out. The majority is aware that, rest in peace, you will come -- the villainy with light. You will understand the heart of granite.
  • (48) Let the oppressed be congratulated, saturated from Him at 2001. And let them be guarded by the echo of the Father's Son, supported by the U.S. I will see but He has the knowledge.
  • (47) Rabbi, behold! The temple mount is dormant. And He will deliver the fallen, as well as my mother, as she will see. She will lead a dried out enemy with her guilt offering.

ELSs 39-45 letters long    [ Expected: 0.32, Actual: 5. A/E = 15.6 ]

  • (45) Imprison your burden there, to immerse her distinct feature, and in my waters to anthrax, king of all. Embrace it in the sea.
  • (41) From the salt of betrayal and from fire, a sand dune provided the foundation for a peace treaty. Yah -- indeed God -- came to the heights of the mountain.
  • (41) Hussein is a vapor. Like a guarded lamb, God is keeping Jews and Levites whole. And the cell inside your dwelling will become a torture chamber.
  • (40) God delivers the joyous. I will indeed inflict pain upon the rock. I am the God who strikes. They wiped out the nakedness.
  • (40) Third temple, the fullness of the illness will take place. And a land will emerge from a tight place. God is lofty, and it is time for a prince.

ELSs 32-38 letters long    [ Expected: 1.36, Actual: 6. A/E = 4.41 ]

  • (34) He will suffer pain and restlessness. But the halo is His, and for her the monument will be Baghdad.
  • (34) The joyful God is ruling. What is in it for me? And the red heifer is crying from the fire and the guilt offering.
  • (33) The rest of my terminal illness is spreading, Saddam, as if from a missile made for you. Where is he? Or who is the tyrant?
  • (33) You were harnessed. Contemplate my might. Let Zubaidah be kept in the limelight. Will they strike the sea?
  • (32) Hussein, his shoulder is dead. And it is appropriate that he prevented home-building everywhere, and is echoing in me.
  • (32) The major dryness of hand in her is in your image. While setting the stage, become Russia.

ELSs 25-31 letters long    [ Expected: 5.6, Actual: 16. A/E = 2.86 ]

  • (30) He loves me that much. Watch in me the rehabilitation since Sept. 11, when his sea was rendered bitter.
  • (30) N. Korea, exalt, rejoice, see that my creed is the truth. Here, their congregation is in her.
  • (29) And from Armageddon God has screamed to us. And who has ambushed whose shadow?
  • (29) She has her army cells, which will be prepared and precise. The inside of Haman is water.
  • (29) We will get the foreigner, Tom Brokaw. Something smells. Consider the flag valuable.
  • (28) The temple mount is poor in my faith. God and Father as a friend of the mountain. And he stopped from giving birth.
  • (27) Armageddon, relieving my heart, will waste away a quarter dead. And God is my refuge.
  • (27) Bear the end of the fool's gift, 2001-2; show the substance of the contemplation.
  • (27) Carry the mountain. Zubaidah will tell something of value as a gift as the monument of the sect is finished.
  • (26) An idol to reflect his lot, Taliban. I will thus rise eastward.
  • (26) From a daughter of his. Where are they from? It is my season of the third temple – please.
  • (26) His people have been easily established, and the Iran you loved has died by me.
  • (26) To her and to the secret he rendered at the nose of the beast, anti-semitism is dead.
  • (25) He was saved in the sea of Hamburg, without any commandment in them for the rabbi's lamb.
  • (25) If she is a whore to the six, N. Korea is God's gift in proliferation.
  • (25) The farmer of Rome came. Bin Laden is dead. The deep throat burned everything.
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