BS"D |

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**The Nations Sample
**

By Doron Witztum

Introduction

I. The Hierarchy of Stories

II. The Great "Proof" of Deception

III. Is the Significance They Report Accurate?

IV. Did Their Principal Story Really Succeed?

V. Without Any Stories

Appendix to Part II

- The list of 136 prefixes
- Do BMS know how to contrive a story?
- Concerning RPWL as the preferred measuring device
- BMS' main story
- The best quartet

Bibliography and Notes

In this part of the article we will examine their assertions one at a time:

1. Are all explanatory stories really equal?

2. Does the success of our prefixes indeed indicate deception?

3. Is the significance they claim to have achieved accurate?

4. Did their principal story really succeed?

In

I. The Hierarchy of Stories:

BMS claim that for any four words one could contrive a likely story, and that all explanatory stories are of

BMS claim to have found a tremendous proof that WRR cheated. What is their "proof"? They claim that three out of four of the prefixes we selected were the most successful of the 136 prefixes they examined. Their assertion is erroneous on two accounts:

1. The data that BMS present to the reader is misleading. The measures that are relevant here are the ranks of P

2. Note that BMS's "proof" is

This is the height of absurdity! They label a success predicted by the research hypothesis

BMS report a significance of

Using the RPWL test the results for BMS's four prefixes (using 100,000,000 permutations) are:

BMS found their four prefixes through acknowledged optimization using the original permutation test. In Chapter V we will see what happens when optimization is conducted using the RPWL test.

IV. Did Their Principal Story Really Succeed?

No! As it turns out, the story they concocted for the four most successful prefixes in

Therefore, there is no story; there is only an arbitrary selection of prefixes, nothing more. It turns out to be not as simple as BMS imagined to contrive a story that works, even if one permits himself to consider all stories of equal status and to ignore the fundamental issue of hierarchy.

1. As we mentioned above, BMS published a list of 136 prefixes. One reader of their article, Yosef Beremez, proposed investigating how the "best quartet" out of this list succeeds in Genesis, versus the "best quartet" in

In order to compile the "best quartet" for any text T, one must rank all the prefixes in BMS's list twice: once according to the value P

As it happens, the P

םע , בתכ , בכר , תודש ("nation of," "script of," "chariot of," and "fields of").

The best quartet in

The significance of the statistic P

For the best quartet in Genesis:

Whereas for the best quartet in

Details on the measurements referred to in this section can be found in the Appendix, sec. 5.

2. BMS made experiments (using the permutation test) [4], in order to compare the distribution of the 136 prefixes in Genesis, with their distribution in 10 similar texts. It is clear from their data that the results for Genesis are significantly exceptional. Details can be found in the Appendix, sec. 3.1-2.

1. The list of 136 prefixes:

Here is the list of 136 prefixes as published by BMS in Table 2 of their article. The prefixes are arranged alphabetically:

1. הקירה תמודיקה – the empty prefix. 2. בא - father of, progenitor of . 3. יבא – father of, progenitor of. 4. תמדא – land of . 5. לכוא – food of . 6. תמוא - nation of 7. יפוא – character of 8. תזחא - + estate of, property of. 9. יא – isle of, see Genesis 10-5. 10. ייא – isles of, see Genesis 10-5 . 11. ליא – lord of, magnate of. 12. יליא – plural of ליא . 13. שיא – person of, man of 14. לכא – food of. 15. לא – god of. 16. הלא – god of, celestial power of . 17. יהלא – plural of הלא . 18. ףולא – head of, leader of. 19. לילא – idol of, false god of. 20. ףלא – thousand of, family of, part of the tribe of. 21. יפלא – plural of ףלא . 22. תמא - + nation of. 23. ישנא – people of , men of. 24. ץרא – country of. 25. םשא – guilt of. 26. תמשא – guilt of. 27. תשא - + woman of. 28. דגב – cloth of, dress of. 29. ידגב – plural of דגב . 30. תריב – capital of. 31. תיב - + house of, family of dynasty of. 32. ןב - + son of, native of. 33. תונב – + plural of תב . 34. ינב – ןב plural of. 35. תב - daughter of, female native of + 36. יתב – plural of תיב . 37. לובג – border of. 38. רובג – hero of. 39. רבג – man of, hero of. 40. ירבג - plural of רבג . 41. דודג – battalion of. 42. יוג – nation of. 43. עזג – race of. 44. רג – convert of. 45. ירג – רג plural of. 46. רובד – speech of. 47. לגד – flag of. 48. ילגד – plural of לגד . 49. ןיד - justice of. 50. יניד – plural of ןיד . 51. םד – + blood of 52. ימד – plural of םד . 53. ךרד – way of. 54. יכרד – plural of ךרד . 55. תד – religion of 56. יתד – plural of תד . 57. בהז – gold of. 58. תוכז – privilege of. 59. ןמז – time of, period of, fate of. 60. רמז - song of, melody of. 61. תרמז – variation of רמז . 62. ערז – seed of, offspring of. 63. לבח - region of, district of. 64. אטח – sin of. 65. יאטח - plural of אטח . 66. ליח - army of, corps of. 67. יליח – plural of ליח . 68. קח - law of. 69. יקח – plural of קח . 70. ברח – sword of. 71. עבט – nature of, character of. 72. ןהכ - priest of. 73. ינהכ - plural of ןהכ . 74. חוכ – alternative spelling of חכ . 75. חכ - force of, vigour of. 76 . תוחכ - plural of חכ . 77. ףסכ – money of. 78. יפסכ – plural of ףסכ . 79. בתכ – script of. 80. ביתכ .- orthography of 81. םאל - nation of. 82. שובל - dress of, clothing of 83. ןושל – dialect of. 84. ןשל – dialect of 85. לכאמ – food of, diet of. 86. בשומ – place of, seat of. 87. לזמ – luck, destiny, constellation of. 88. עבטמ – coin of. 89. בתכמ – letter of, writing of. 90. ךאלמ – angel of. 91. ךלמ – king of. 92. יכלמ – plural of ךלמ . 93. תכלמ – queen of. 94. ללמ – speech of. 95. לשממ – rule of, government of. 96. גהנמ – manner of, custom of. 97. סמ – tax of, levy of. 98. יסמ – plural of סמ . 99. תועמ – money of 100. םוקמ – place of. 101. הארמ – appearance of, view of. 102. סומנ - + manners of, custom of, laws of. 103. סנ – flag of, banner of, miracle of. 104. יסנ – plural of סנ . 105. תושנ – + plural of תשא . 106. אישנ – president of. 107. ףוס – end of, conclusion of. 108. למס – symbol of, emblem of. 109. ילמס – plural of למס . 110. רפס - book of. 111. ןווע – alternative spelling of ןוע . 112. ןוע – sin of. 113. לש ז"ע – idolatry of . 114. ריע – city of, angel of. 115. םע – nation of. 116. ירע – towns of. 117. עשפ – crime of. 118. יעשפ - plural of עשפ . 119. אצאצ – child of. 120. אבצ – army of, troops of. 121. שאר - head of, leader of. 122. ישאר – plural of שאר . 123. חור – spirit of. 124. בכר – chariot of 125. דוקר – dance of 126. הדש – + field of 127. תודש – + plural of הדש 128. ריש – song of, hymn of, music of. 129. יריש – plural of ריש . 130. םש – + name of, fame of 131. תומש - + plural of םש . 132. תפש – language of. 133. רש – ruler of, head of, minister of, prince of. 134. ירש – plural of רש . 135. שרש – root of 136. בשות – resident of.

Note – I have reproduced BMS's list here verbatim (including the mistakes).

In their article concerning the Nations Sample BMS demonstrate their impressive acrobatic ability. They managed to do precisely the reverse of what they did in the article mentioned above. The experiment on the Nations Sample really

They claim [8] that we could not have used the permutation test for our alleged optimization because we did not have sufficient computing facilities. They support this claim by noting that in our original article on the Nations Sample we write that we were unable to perform the billion permutations necessary for calculating the ranking of P

One cannot deny that this story is very cute, but does it reflect reality? Unfortunately it does not. They did indeed quote us accurately, but they left out a minor detail that BMS should be made aware of [9]: The program for calculating the rankings of P

As I mentioned earlier, when it comes to concocting stories, Scheherazade did not come up to their ankles.

"To investigate whether non-uniformity and exaggerated extremes are the norm, we ran the same tests on 10 control texts. We generated each control text by randomly permuting the order of the words within each verse of Genesis except for the verses in Chapter 10. Chapter 10, where all nation names are found in a specific fixed order, was left alone in order to make the comparison with Genesis and WGP more meaningful." To ensure independence of the results, ELSs lying entirely inside the section of the text, containing Chapter 10 of Genesis were ignored.

The results for

Bin | G | Text 0 | Text 1 | Text 2 | Text 3 | Text 4 | Text 5 | Text 6 | Text 7 | Text 8 | Text 9 |

0 | 23 | 15 | 19 | 22 | 23 | 18 | 17 | 14 | 13 | 19 | 19 |

1 | 17 | 13 | 13 | 8 | 15 | 12 | 6 | 13 | 18 | 13 | 18 |

2 | 9 | 17 | 13 | 16 | 15 | 8 | 18 | 13 | 15 | 11 | 13 |

3 | 13 | 13 | 14 | 7 | 9 | 7 | 16 | 18 | 10 | 12 | 11 |

4 | 11 | 9 | 7 | 13 | 7 | 10 | 12 | 9 | 13 | 7 | 7 |

5 | 12 | 16 | 11 | 13 | 12 | 16 | 5 | 12 | 8 | 10 | 8 |

6 | 10 | 10 | 7 | 9 | 22 | 22 | 10 | 8 | 13 | 18 | 10 |

7 | 14 | 9 | 13 | 20 | 10 | 9 | 13 | 11 | 15 | 16 | 10 |

8 | 16 | 13 | 15 | 15 | 14 | 14 | 23 | 15 | 13 | 11 | 13 |

9 | 11 | 21 | 24 | 13 | 9 | 20 | 16 | 22 | 18 | 19 | 27 |

min | 493 | 5314 | 3991 | 1584 | 608 | 874 | 4979 | 180 | 6184 | 6247 | 2481 |

mean | 463125 | 504905 | 517197 | 496493 | 458654 | 534142 | 519963 | 522828 | 505910 | 512851 | 514246 |

max | 990921 | 999807 | 999923 | 999141 | 997583 | 999864 | 999849 | 999229 | 997025 | 991129 | 998778 |

"The most obvious characteristic of these distributions is their inconsistency. A few are near-uniform, but others are skewed markedly in the positive or negative direction. Both the minima and maxima appear exaggerated.

Several things should be clear from these results. Firstly, WRR's assumption of uniformity in the rank orders is unfounded, as many of the texts give profoundly non-uniform distributions. More importantly, the non-uniformity may be more pronounced at the extremes of the distribution where WRR measure their 'significance levels'."

2. Comment on BMS' Data:

Looking at the same table and checking the tendency to low ranks at the extremes, we can't escape the conclusion that the data for Genesis are exceptional. It is pronounced in each of the following parameters:

- The occupation number of bin 0 is the highest (there is another
such number) of all 11 texts.

- The occupation number of bin 9 is second to the lowest of all 11
texts.

- The minimum is second to the lowest of all 11 texts.

- The maximum is the lowest of all 11 texts.

3. Data concerning the RPWL test:

Bin | Text 0 | Text 1 | Text 2 | Text 3 | Text 4 | Text 5 | Text 6 | Text 7 | Text 8 | Text 9 |

0 | 13 | 14 | 8 | 5 | 13 | 12 | 15 | 14 | 14 | 9 |

9 | 10 | 16 | 20 | 8 | 11 | 13 | 12 | 11 | 16 | 13 |

min | 2177 | 16145 | 3064 | 34204 | 14822 | 2867 | 5334 | 2984 | 7771 | 13605 |

max | 985212 | 998915 | 992263 | 977197 | 998763 | 996302 | 997474 | 980781 | 997363 | 994129 |

The results from a similar experiment done for

Bin | Text 0 | Text 1 | Text 2 | Text 3 | Text 4 | Text 5 | Text 6 | Text 7 | Text 8 | Text 9 |

0 | 14 | 17 | 14 | 9 | 12 | 15 | 12 | 10 | 10 | 11 |

9 | 12 | 24 | 15 | 7 | 14 | 13 | 13 | 12 | 13 | 15 |

min | 5112 | 17038 | 7705 | 21471 | 11176 | 6775 | 1076 | 1659 | 10225 | 10626 |

max | 984823 | 996099 | 998372 | 999790 | 999204 | 997275 | 995230 | 989584 | 995961 | 998581 |

- no accumulations at the extremes, and

- no exaggerated minima or maxima

4. BMS's main story:

BMS's main story, " Celestial Guardians" [10] is based on Nachmanides commentary to Chapter 18 of Leviticus. There, as BMS put it, "he discusses the celestial beings who represent and supervise the Nations of the world".

- BMS claim that in the course of this discussion Nachmanides uses
the descriptive terms: רש (prince), יכלמ (kings of), םיהלא (gods), and
ןיריע (angels).

- BMS contrived a set of "guidelines," which they claim
leads to the selection of these four prefixes: 1. יכלמ , 2. רש , 3. ריע , 4. יהלא .

- Nachmanides actually used a much broader range of descriptive terms,
of which BMS arbitrarily selected four.

- Their supposed "guidelines" are a nothing more than a joke.

אמטתש ץראה רובעב ,תוירעב בותכה רימחה - "ץראה איקתו
הילע הנוע דוקפאו ץראה אמטתו"

םירבד) בותכב רבדה דוס לבא ,ץראב תויולת ןניאו ףוגה
תבוח תוירעה הנהו ,תושועה תושפנה איקתו ןהב

םשה יכ ןינעהו ."'וגו ומע 'ה קלח יכ 'וגו םימע תולובג
בצי םדא ינב ודירפהב םיוג ןוילע לחנהב" (ט-ח ,בל

רשאכ עודי __לזמ__ו __בכוכ__ םהייוגל םתוצראב םעו
םע לכ לע ןתנו ,םינוילעב םינותחתה חכ םשו ,לכה ארב דבכנה

םלוכל קלח יכ ,"םימעה לכל םתוא ךיהלא 'ה קלח רשא" (טי
,ד םירבד) רמאנש והזו .תונינגטצאב עדונ

(גי קוספ ,םש) ביתכדכ __םיכלמ__ םיארקנו ,"אב ןוי
__רש__ הנהו" (כ קוספ ,םש) ביתכו ,ידגנל דמוע סרפ __תוכלמ__

לבא ,םלועה לכל __םינודא__ה ינודאו __םיהלא__ה
יהלא אוה דבכנה םשה הנהו ."סרפ __יכלמ__ לצא םש יתרתונ ינאו"

יל יכ םימעה לכמ הלוגס יל םתייהו" (ה ,טי תומש) רמאש
והזו ,ויבהוא ערז ומש דחימה ומעל התוא וליחנהב

ןלהל) רמא ךכלו ,ומשל םתויהל תוצמה יבורבו תוירעה תשודקב
וצראב בשויה םעה שדיק הנהו .ללכ םירחא

קוספ ,םש) ביתכו ,"ץראה םכתא איקת אלו םתוא םתישעו יטפשמ
לכ תאו יתוקוח לכ תא םתרמשו" (בכ ,כ

יתלדבה רשא םכיהלא 'ה ינא התוא תשרל םכל הננתא ינאו
םתמדא תא ושרית םתא םכל רמאו" (דכ

תא ונל ותתב ,םירחא __םיהלא__ו __םירש__ םהילע
ןתנ רשא םימעה לכמ ונתוא לידבה יכ רמאי "םימעה ןמ םכתא

לכ איקת דבכנה םשה תלחנ איהש ץראה הנהו .ומשל םידחוימ
היהנו םיהלאל ונל ךרבתי אוה היהיש ץראה

םע הרז הדובע לולכל ךלומה הריכזה תאזה השרפהו .תוירע
םילגמו ז"ע ידבוע לובסת אלו התוא אמטמ

ןכו ,"היבשוי תא ץראה איקתו ,םיוגה ואמטנ הלא לכב יכ
הלא לכב ואמטת לא" רמא םלוכ לעו .תוירעה ןורכז

ךכלו ,הרז הדובע רוסיא רמוח אוהש ,"יל תויהל םימעה ןמ
םכתא לידבאו" (וכ ,כ ןלהל) הינשה השרפב רמא

םתא םכל רמואו" (דכ קוספ ,םש) רמאנש ,ץראה םהל ןתנ ןכ
רובעב ומשל םידחוימ םהש ינפמ יכ רמא

הנהו ."םימעה ןמ םכתא יתלדבה רשא םכיהלא 'ה ינא התוא
תשרל םכל הננתא ינאו םתמדא תא ושרית

םימעהו הילע __םילשומ__ה __םיתרשמ__ה רובעב ,המילש
הב הרהט ןיא ,דבכנה םשל לכהש פ"עא ,ץראל הצוחב

יהלא אוה יכ ,"ארקי ץראה לכ יהלא" (ה ,דנ היעשי) בותכה
רמאי ךכלו .םתוא םג דובעל __םהירש__ ירחא םיעות

סורהלו __םינוילע__ה תלשממ ריסהל "םורמב םורמה אבצ
לע" ףוסב דוקפי אוהו לכה לע לשומה __םיהלא__ה

(די ,ד לאינד) רמאש בותכה ןינע והזו ."המדאב המדאה יכלמ"
לע דוקפי ןכ ירחאו ,__םיתרשמ__ה תכרעמ

וררועתי ןתוליצאמ יכ __ןיריע__ וארקיו ,ךכ תושעל
ןהמ ןילצאנה תוחוכה לע ורזגש אתלאש __ןישידק__ רמאמו אמגתפ

ודוג רמא ןכו ליחב ארק תיחנ אימש ןמ __שידק__ו __ריע__
ולאו" (אי י ,םש ,םש) רמאש ומכ ,תולועפה לכב תוחכה

והזו ,ןכ תושעהל ורזג ןכ ירחאו וילע ןוילעה ןוצרה המ
ולאשש רמולכ ,אתלאש __ןישידק__ רמאמו ."'וגו אנליא

יהלא ךרבתי דבכנה םשה הנהו .ךרבתי ותאמ לכה יכ ,"איה
האליע תרזגו" (אכ ,םש) לאינד ול רמאש

,אל םירבד) "ץראה רכנ __יהלא__ ירחא הנזו" םעט והזו
,'ה תלחנ איהש לארשי ץרא יהלאו םלועה לכב __םיהלא__ה

ץראה יהלא טפשמ תא ועדי אל" (וכ ,זי ב"מ) רמאנש והזו
,ותלחנבו םשה ץראב םירכנ __תוהולא__ה יכ ,(זט

אל םייתוכה הנהו ."ץראה יהלא טפשמ תא םיעדוי םניא רשאכ
םתוא םיתיממ םנהו תויראה תא םב חלשיו

םהישעמכ םש ושעו םשה ץראב םאובבו ,תויראה תא םהב חלשל
םהיהלא תא םדבעב םצראב םישנענ ויה

'וגו םכתא ץראה איקת אלו" (די ,אי םישודק) ארפסב ונש
ןכו .םתוא םיתיממה תויראה םהב חלש םינושארה

רכנ __לא__ ומע ןיאו" (וטש וניזאה) ירפסבו ."הריבע
ירבוע תמייקמ הניא ,תוצרא ראשכ הניא לארשי ץרא --

(חל ,הכ ןלהל) רמאנש ,הולא ול ןיאש ימכ המוד ץראל הצוחב
רדה לכ" (ב יק תובותכ) םרמאמ אוהו ."'וגו ןוי

רמאל 'ה תלחנב חפתסהמ םויה ינושרג יכ (טי ,וכ א"ש) רמואו
,םיהלאל םכל תויהל ןענכ ץרא תא םכל תתל

"םירחא םיהלא דובע ךל

בתכו .ךיניע וב יתחקפש רחא םהב שרופמ ונארתו ,םיבותכב
םיבר תומוקמב אוה הזה ןינעהו
"

הנשי אל םשהו ,םילבקמהמ אבי יונישהו דחא םשה יכ ונעדי"
(זט ,אל םירבד) ךליו תשרפב א"ר

טפשמ תא בותכ ןכ לע םוקמה יפכ לובקה חכ רומשל םשה תדובעמו
,המכחב םה םלוכ יכ וישעמ

ליכשמהו ראש םהש תוירעב קבדה םוקמה ךפהו ,רכנה __יהלא__
תא וריסה בקעיב רמאו ,ץראה יהלא

שקבל תרשמ רש אוה יכ (אכ ,י לאינד) "םכרש לאכימ" קוספמ
ילע בישת לאו .ל"ז וירבד ולא ".ןיבי

(גי ,ה עשוהי) וחיריב עשוהיל הארנה אבצ רש היה ןכו
.הלשממו __תוכלמ__ __רש__ אל ,לארשי לע םימחר

ונתויהב הז היהש םגו ,(הל ,טי ב"מ) והיקזחב ןינעכ םהיתומחלמ
םוחלל וחלש םשה יכ ול הארה

"
ץראל הצוחב

1. בכוכ star. 2. לזמ constellation. 3. תולזמ constellations . 4. יכאלמ the angels of. 5. םירש lords. 6. תוכלמ רש the prince of the kingdom of (a prince exercising royalty. 7. רש the prince of. 8. םיכלמ kings . 9. יכלמ the kings of. 10. םיהלא gods. 11. םינודא lords . 12. םיכאלמ angels. 13. ןיצק chief. 14. רטוש observer . 15. לשומ ruler . 16. םיתרשמ servants. 17. םילשומ rulers . 18. םינוילע celestial powers. 19. ןיריע the wakeful ones. 20. ןישידק the holy ones. 21. ריע a wakeful one. 22. שידק a holy one. 23. יהלא gods of the. 24. תוהולא gods. 25. לא the god of. 26. ירש the princes of.

(The translation of the 26 terms was taken from the same source used by BMS.)

2. Contriving guidelines:

*They only selected terms that are used explicitly by Nachmanides.*

In fact, as we have seen, Nachmanides actually uses explicitly many more terms than the ones they mention.

*"When a word appears in the quotation in both singular and plural form, we always prefer the (shorter) singular form". However,*

*"When it appears only in plural form, we can't know if it is reasonable to use it in singular form and hence we keep it in plural form".*

- First of all, let us see what prefixes we receive when we consistently
select only the singular form (we will ignore the term תוכלמ רש
because it is too long):

- As it turns out, not only does this entire
list fail in the permutation test in
*WGP*, but the set of four prefixes, 4, 5, 12, and 14, BMS's arbitrarily chosen quartet, fails as well.

- This is why BMS had to come up with "guideline" number
three. Using this guideline we receive the set:

1. בכוכ . 2. לזמ . 3.__יכאלמ__. 4. רש . 5.__יכלמ__. 6. ןודא . 7. ןיצק. 8. רטוש .9. לשומ . 10. תרשמ . 11. ןוילע . 12. ריע . 13. שידק. 14.__יהלא__. 15. תוהולא .16. לא.

- Unfortunately this set also fails. The
third guideline only helps with regard to the four prefixes, 4, 5, 12
and 14, for which BMS are trying to contrive their story. By replacing
the singular form ךלמ with the plural יכלמ and (according to [11]) by
replacing הלא with יהלא , we finally receive the desired set –
i.e., the one that succeeds.

- However, the arbitrary choice of prefixes 4, 5, 12, and 14, out
of a total of 15 prefixes, is made away from the eyes of the reader and
no explanation is given for it.

- Guideline number three is ridiculous enough in its own right. For
example, there is absolutely no doubt that ךלמ ("king of") is a perfectly
acceptable prefix. Since according to Nachmanides םירש = םיכלמ , it follows
that רש = ךלמ . Indeed the word ךלמ used as a prefix appears in Scripture
– סרפ ךלמ ("king of Persia"). In fact it appears in the
very same chapter of Daniel from which Nachmanides took the expression
סרפ יכלמ ("kings of Persia"). Therefore when BMS say that, "
*we can't know*" whether the singular form ךלמ is acceptable as a prefix they are speaking nonsense.

b. Every prefix

c. When we calculate the values of

d. We then calculate the value of P

e. We rank the sets

f. The "Best Quartet", Q(T), is defined as the union of the four sets:

g. To measure the significance of the statistic P

- Using this procedure on the book of Genesis we receive a P
_{1}value for the quartet

Q(G) of:**P**._{1}(G)=1.65X10^{-9}

- Using the randomization PRWL with 10,000,000,000 permutations we
receive a significance of:
**r**._{1}(G)=4.0X10^{-10}

- Regarding the text
*WGP*the value of P_{1}for the quartet Q(WGP) was:

**P**._{1}(WGP)=4.23X10^{-5}

- Using the randomization PRWL with 100,000,000 permutations we receive
a significance of:
**r**._{1}(WGP)=6.16X10^{-6}

Because of the enormous difference between the values of

- In place of the set
*K*_{i}(T) in step c. above, we will build a set*K'*_{i}(T) in which there are

n(*K*_{i},T) random numbers chosen by lottery from the segment (0,1] (this is the range of values of*c(w,w')*), produced by a random function in the computer.

- We then carry out steps d-f as above, using
*K'*_{i}(T) in place of*K*_{i}(T).

- In this way we receive the set Q'(T), for which we calculate the
value P'
_{1}(T). (Step g. above, which was needed to guard against the effects of any possible dependence between the values of*c(w,w')*, is unnecessary when random numbers are involved.)

- We repeat the series of lotteries and calculations in steps 1-3
numerous times, and examine the ranking of
**r**among the set of values of P'_{1}(T)_{1}(T).

- We performed 1,000,000 lotteries for Genesis and calculated
1,000,000 values of

P'_{1}(G). The ranking of**r**among this set of values was 420._{1}(G)

- We also performed 1,000,000 lotteries for
*WGP*, and calculated 1,000,000 values of P'_{1}(WGP). The ranking of**r**among this set of values was 353,949._{1}(WGP)

2. Section 3.4.

3. See

4. Section 7.1-2.

5. M. Bar-Hillel, D. Bar-Natan, B. McKay;

6. See Document 4 at this site.

7. M. Bar-Hillel, D. Bar-Natan, B. McKay;

8. See their comment about Table 4, and at the end of Chapter 4.

9. Copies of the programs were deposited with editorial board of

10. In section 3.2.

11. See Table 5, number 5.