Why and When Do Probabilities Multiply?

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	Why and When Do Probabilities Multiply?

APPENDIX

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Table 1 (Concise). Intersections with Israel (י ש ר א ל ).

State	In Hebrew	Number of letters	Shared with י ש ר א ל	Number of occurrences, N	Lowest ELS, LELS	Lowest inter-section, LIELS	N?\|LIELS\| (P)
1	Bulgaria	ב ו ל ג ר י ה	7	3	21	-456	-456	9,576
2	Ireland	א י ר ל נ ד	6	4	745	32	32	23,840
3	Iceland	א י ס ל נ ד	6	3	100	276	276	27,600
4	Britain	ב ר י ט נ י ה	7	3	6	9,240	14,440	86,640
5	Turkey	ט ו ר ק י ה	6	2	97	-364	-978	94,866
6	Albania	א ל ב נ י ה	6	3	2,711	-2	-38	103,018
7	Denmark	ד נ מ ר ק	5	1	1,851	-19	97	179,547
8	Switzerland	ש ו ו י ץ	5	2	15,823	-10	-14	221,552
9	Sweden	ש ו ו ד י ה	6	2	2,031	7	121	245,751
10	America	א מ ר י ק ה	6	3	1,140	-33	233	265,620
Number of Letters in the Torah	304,805
11	Germany	ג ר מ נ י ה	6	2	259	161	1,263	327,117
12	France	צ ר פ ת	4	1	22,867	2	17	388,739
13	Poland	פ ו ל י ן	5	2	11,829	-6	36	425,844
14	Finland	פ י נ ל נ ד	6	2	110	27	3,945	433,950
15	Russia	ר ו ס י ה	5	2	8,006	-8	-62	496,372
16	Italy	א י ט ל י ה	6	4	722	13	-777	560,994
17	Belgium	ב ל ג י ה	5	2	5,636	-21	-108	608,688
18	Romania	ר ו מ נ י ה	6	2	3,885	-11	208	808,080
19	Holland	ה ו ל נ ד	5	1	16,046	-5	157	2,519,222
20	Spain	ס פ ר ד	4	1	4,262	7	700	2,983,400
21	Austria	א ו ס ט ר י ה	7	3	2	249	none
22	Norway	נ ו ר ב ג י ה	7	2	14	-580	none
23	Hungary	ה ו נ ג ר י ה	7	2	34	378	none

Table 2. P-values obtained with the Lowest ELS-s.

NOTE: For the sake of clarity, Tables 2, 2A and 2B, contain the results obtained with both groups of the risky and safe states examined in this Part 2 only.

State	In Hebrew	Number of letters	Number of occurrences, N	Lowest ELS, LELS	N?\|LELS\| (P)
1	Austria	א ו ס ט ר י ה	7	2	249	498
2	Finland	פ י נ ל נ ד	6	110	27	2,970
3	Albania	א ל ב נ י ה	6	2,711	-2	5,422
4	Norway	נ ו ר ב ג י ה	7	14	-580	8,120
5	Italy	א י ט ל י ה	6	722	13	9,386
6	Bulgaria	ב ו ל ג ר י ה	7	21	-456	9,576
7	Hungary	ה ו נ ג ר י ה	7	34	378	12,852
8	Sweden	ש ו ו ד י ה	6	2,031	7	14,217
Jerusalem	י ר ו ש ל י ם	7	280	71	19,880
9	Ireland	א י ר ל נ ד	6	745	32	23,840
10	Iceland	א י ס ל נ ד	6	100	276	27,600
11	Spain	ס פ ר ד	4	4,262	7	29,834
12	Denmark	ד נ מ ר ק	5	1,851	-19	35,169
13	Turkey	ט ו ר ק י ה	6	97	-364	35,301
14	America	א מ ר י ק ה	6	1,140	-33	37,620
15	Germany	ג ר מ נ י ה	6	259	161	41,699
16	Romania	ר ו מ נ י ה	6	3,885	-11	42,735
17	France	צ ר פ ת	4	22,867	2	45,734
18	Britain	ב ר י ט נ י ה	7	6	9,240	55,640
19	Russia	ר ו ס י ה	5	8,006	-8	64,048
20	Poland	פ ו ל י ן	5	11,829	-6	70,974
21	Holland	ה ו ל נ ד	5	16,046	-5	80,230
22	Belgium	ב ל ג י ה	5	5,636	-21	118,356
23	Switzerland	ש ו ו י ץ	5	15,823	-10	158,230
Number of Letters in the Torah	304,805

Table 2 A. P-values calculated on the basis of LELS in the text containing ISRAEL (from Genesis 32:28 to the end of the Torah). The non-shaded states, together with JERUSALEM, occupy the upper compartment in Table 1. The number P in the last column indicates the “best case” for each state.

State	In Hebrew	Number of letters	Number of occurrences, N	Lowest ELS from Gen. 32:28 on	Minimum possible P
1	Norway	נ ו ר ב ג י ה	7	14	-580	8,120
2	Italy	א י ט ל י ה	6	722	13	9,386
3	Bulgaria	ב ו ל ג ר י ה	7	21	-456	9,576
4	Hungary	ה ו נ ג ר י ה	7	34	378	12,852
5	Sweden	ש ו ו ד י ה	6	2,031	7	14,217
6	Austria	א ו ס ט ר י ה	7	2	9,763	19,526
7	Finland	פ י נ ל נ ד	6	110	-196	21,560
8	Albania	א ל ב נ י ה	6	2,711	8	21,688
9	Ireland	א י ר ל נ ד	6	745	32	23,840
10	Iceland	א י ס ל נ ד	6	100	276	27,600
11	Spain	ס פ ר ד	4	4,262	7	29,834
12	Denmark	ד נ מ ר ק	5	1,851	-19	35,169
13	Turkey	ט ו ר ק י ה	6	97	-364	35,301
14	America	א מ ר י ק ה	6	1,140	-33	37,620
15	Romania	ר ו מ נ י ה	6	3,885	-11	42,735
16	France	צ ר פ ת	4	22,867	2	45,734
17	Germany	ג ר מ נ י ה	6	259	-185	47,915
Jerusalem	י ר ו ש ל י ם	7	280	198	55,440
18	Britain	ב ר י ט נ י ה	7	6	9,240	55,640
19	Russia	ר ו ס י ה	5	8,006	-8	64,048
20	Poland	פ ו ל י ן	5	11,829	-6	70,974
21	Belgium	ב ל ג י ה	5	5,636	-21	118,356
22	Switzerland	ש ו ו י ץ	5	15,823	-10	158,230
23	Holland	ה ו ל נ ד	5	16,046	12	192,552
Number of Letters in the Torah	304,805

Table 2 B. P-values calculated on the basis of LELS in the text containing SONS OF ISRAEL (from Genesis 32:33 to the end of the Torah). The states, together with JERUSALEM, occupy the lower compartment in Table 3. The number P in the last column indicates the “best case” for each state.

State	In Hebrew	Number of letters	Number of occurrences, N	Lowest ELS from Gen. 32:33 on	Minimum possible P
1	Norway	נ ו ר ב ג י ה	7	14	-580	8,120
2	Italy	א י ט ל י ה	6	722	13	9,386
3	Bulgaria	ב ו ל ג ר י ה	7	21	-456	9,576
4	Hungary	ה ו נ ג ר י ה	7	34	378	12,852
5	Sweden	ש ו ו ד י ה	6	2,031	7	14,217
6	Austria	א ו ס ט ר י ה	7	2	9,763	19,526
7	Finland	פ י נ ל נ ד	6	110	-196	21,560
8	Albania	א ל ב נ י ה	6	2,711	8	21,688
9	Ireland	א י ר ל נ ד	6	745	32	23,840
10	Iceland	א י ס ל נ ד	6	100	276	27,600
11	Spain	ס פ ר ד	4	4,262	7	29,834
12	Denmark	ד נ מ ר ק	5	1,851	-19	35,169
13	Turkey	ט ו ר ק י ה	6	97	-364	35,301
14	America	א מ ר י ק ה	6	1,140	-33	37,620
15	Romania	ר ו מ נ י ה	6	3,885	-11	42,735
16	France	צ ר פ ת	4	22,867	2	45,734
17	Germany	ג ר מ נ י ה	6	259	-185	47,915
Jerusalem	י ר ו ש ל י ם	7	280	198	55,440
18	Britain	ב ר י ט נ י ה	7	6	9,240	55,640
19	Russia	ר ו ס י ה	5	8,006	-8	64,048
20	Poland	פ ו ל י ן	5	11,829	-6	70,974
21	Belgium	ב ל ג י ה	5	5,636	-21	118,356
22	Switzerland	ש ו ו י ץ	5	15,823	-10	158,230
23	Holland	ה ו ל נ ד	5	16,046	12	192,552
Number of Letters in the Torah	304,805

Table 3 (Concise). Intersections with SONS OF ISRAEL (ב נ י י ש ר א ל )

State	In Hebrew	Number of letters	Shrd with ב נ י י ש ר א ל	Number of occurrences, N	Lowest ELS, LELS	Lowest inter-section, LIELS	N?\|LIELS\| (P)
1	Albania	א ל ב נ י ה	6	5	2,711	-2	8	21,688
2	Iceland	א י ס ל נ ד	6	4	100	276	-285	28,500
3	Romania	ר ו מ נ י ה	6	3	3,885	-11	-11	42,735
4	Bulgaria	ב ו ל ג ר י ה	7	4	21	-456	-3,768	77,826
5	Britain	ב ר י ט נ י ה	7	5	6	9,240	14,440	86,640
6	Turkey	ט ו ר ק י ה	6	2	97	-364	-978	94,866
7	Germany	ג ר מ נ י ה	6	3	259	161	-406	105,154
8	Finland	פ י נ ל נ ד	6	4	110	27	1,339	147,290
9	Norway	נ ו ר ב ג י ה	7	4	14	-580	10,561	147,854
10	Switzerland	ש ו ו י ץ	5	2	15,823	-10	-14	221,552
11	Sweden	ש ו ו ד י ה	6	2	2,031	7	121	245,751
12	Hungary	ה ו נ ג ר י ה	7	3	34	378	7,944	270,096
Number of Letters in the Torah	304,805
13	Holland	ה ו ל נ ד	5	2	16,046	-5	22	353,012
14	Poland	פ ו ל י ן	5	3	11,829	-6	36	425,844
15	Russia	ר ו ס י ה	5	2	8,006	-8	-62	496,372
16	Ireland	א י ר ל נ ד	6	5	745	32	-735	547,575
17	America	א מ ר י ק ה	6	3	1,140	-33	-512	583,680
18	Belgium	ב ל ג י ה	5	3	5,636	-21	-108	608,688
19	Italy	א י ט ל י ה	6	4	722	13	1,303	940,766
20	Denmark	ד נ מ ר ק	5	2	1,851	-19	-555	1,027,305
21	France	צ ר פ ת	4	1	22,867	2	129	2,949,843
22	Spain	ס פ ר ד	4	1	4,262	7	700	2,983,400
23	Austria	א ו ס ט ר י ה	7	3	2	249	none

The table below contains the χ ²-values corresponding to the first six degrees of freedom at various significance levels. A significance level is the probability that, in a case of random distribution (i.e. in the absence of a “driving force”), the χ ²-value obtained in the calculation carried out according to the formula given in the text, will exceed the figure in the respective column. The shaded row corresponds to the number of degrees of freedom and significance level with respect to the results obtained with the intersections of the safe states with digrams and discussed in the text (see Table 30). In the case of yod-shin (י ש ) the probability that the distribution obtained with the safe states is random is below 0.05 = 5% but still above 0.01 = 1%. (χ ² = 10.56 corresponds to about 3 % probability).

Significance Level
Degrees of Freedom	0.05	0.01	0.001
1	3.84	6.54	10.83
2	5.99	9.21	13.82
3	7.82	11.34	16.27
4	9.49	13.28	18.47
5	11.07	15.09	20.52
6	12.59	16.81	22.46

Why and When Do Probabilities Multiply?

An event is considered independent if the probability for its occurrence is not affected by any other event. A good example is throwing of dice. It can be easily realized that if we throw one fair die, the probability for falling on anyone of its sides is P = 1/6. If we throw a second die, the probability that it will fall on the same side is 1/6 again. When we throw both dice together, each die can fall on one of its six sides. Further, any result of the first die can combine with anyone of the six possible results of the second die. Therefore, any particular, preliminarily specified combination will occur with frequency

1/6 ? 1/6 = 1/36

This 1/36 probability is valid for any ordered pair of numbers. If we expect an occurrence of, say 4 and 5, and do not specify which die what side should fall on, the 1/6 probability for a first die fall on 4 combines with the 1/6 probability for a second dice fall on 5 and the overall probability is P_{(4 + 5)} = 1/36. But there are also 1/6 probabilities for the first die fall on 5 and 1/6 for the second die fall on 4. The result is the same: P_{(5 + 4)} = 1/36. So in the case of unspecified order of dice’s fall, the overall probability is the sum of the probabilities for each ordered pair:

P_{(4 + 5)} + P_{(5 + 4)} = 1/36 + 1/36 = 1/18

This is valid for any number of dice. Every additional die will decrease the overall probability for any ordered occurrence by 1/6. The important fact is that these calculations work in the same way with a set of dice or in case of one die thrown a number of times equal to the number of the dice in the set.

Now let us consider another typical means for studying probabilities: the fair coin. Suppose we throw the coin 4 times. What are the outcomes that can be recorded?

The probability for a head-up fall is 1/2. It is equal to that for a tail-up fall. Let us mark a head-up fall with 1 and the tail-up fall with 0. The order of digits in each outcome record is according to the serial number of the cast. For 4 casts, we obtain the following sets of results:

0000

0001 0010 0100 1000

0011 0101 0110 1001 1010 1100

0111 1011 1101 1110

1111

There are 16 outcomes altogether. Each individual outcome within a row contains the unchanged number of units and zeros. As in the example with the dice, every outcome has the same probability to occur: 1/16. But if we do not specify an ordered outcome, and are interested in the overall number of heads or tails, then again the rule described for dice applies: there are 4 outcomes with 1 head or 1 tail. The largest number of outcomes, 6, is in the case of the equal number of heads and tails.

Thus, the probability for 2 heads and 2 tails appears to be 6/16 or about 38 %. This result is lower than the “psychologically” expected 50%. The probability for 1 head or tail is 4/16 = 1/4 or exactly 25 %. Notice that single outcome forming the first row contains 0’s only while that in the last row consists entirely of 1’s. The second and the fourth rows are mirrored images: each outcome in the fourth row is obtained as 0’s are replaced by 1’s and written in reversed order. In the same way, the outcomes arranged in middle row arrayed in such way that the first one is a mirror image of the last one according to the procedure described for the rows before and after it. In the same way, the 2^nd outcome mirrors the next-to-the-last one, etc. These facts reflect a fundamental symmetry principle. So the probability for a “pure” outcome consisting of heads or tails only is the same:

1/2?1/2?1/2?1/2 = 1/16

As with the dice, instead of throwing a coin 4 times, we may put 4 coins in a box, shake it and check the outcome. However, we should mark the coins, say, by 4 colours, to be able to specify an ordered outcome.

Also as in the example of the dice, in the case with coins, analogically, every new coin decreases the overall probability for an ordered outcome by 1/2.

Now suppose we have a set of 5 coins and 3 dice. We put them in a box and shake. Then we open the box. What is the probability, P_overall, for an outcome of a six on all three dice and all 5 coins heads-up?

Answer:

P_overall = P(coins) ? P(dice) =

= [(1/2)?(1/2)?(1/2)?(1/2)?(1/2)] ? [(1/6)?(1/6)?(1/6)] =

= (1/2^5) ? (1/6^3) =

= 1/32 ? 1/216 = 1/6,912

The probability for such outcome is as low as almost 1 in 7 thousand!

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