Lottery pairings reduce millions of lotto combinations.

Less is More: Lottery pairings reduce millions of lotto combinations

Web site by Ion Saliu, founder of lottery software science, gambling mathematics.

Written by Nik Barker on 3/16/02.

Dear all,

You will see below the test I've just completed for the Least Pairings for lotto. This is an important test. It is almost as arduous as the test I did for Util-6 lotto 6 software. The aim of this analysis was to look at the relationship between the least pairings and the numbers drawn in 100 consecutive lottery drawings.

The headings in the table below are fairly self-explanatory, but I will clarify for lack of confusion… by the way, the way to read the table is from bottom to top…and the +/- relates whether the 'overall' figure was higher or lower than the previous draw…

“Draw No.” – that's obvious, right?!

“No. of least pairings that could have been discarded for each lotto ball” – this means that each time, I have run a frequency report and looked at the Pairs6 file, gone to each number that occurred in the relevant draw, and counted back from the END of the Pairs6 file to see how many least pairings for EACH lottery number could have been eliminated before finding another number that came up in that draw…example…in draw number 649 I looked at the Pairs6 file and found the B1 number, went to the end of the line and then counted back until I found the occurrence of either B2, B3, B4, B5, or B6 – whichever came first…then did the same for EACH lotto ball that came up in that draw.

“Overall Least pairings that could have been used” – this is the lowest 'common denominator'. I.E. B1 may have been able to have 10 of its worst lottery pairings eliminated without affecting it coming up with the other 5 lotto balls drawn, but B6 could only 'survive' 2 worst lotto pairings being eliminated without that affecting it coming up with the other 5 balls drawn. So the end result – to still be able to eliminate worst pairings and still leave the 6 winning numbers available – is the lowest value for any of the 6 balls' least pairings.

This table relates to the UK National Lottery and is for the most recent (at time of going to press – 14/03/02) 100 lotto draws. I should also say that each time I ran the frequency report I used the default of 147 draws to check against.

		 No. of Least Pairings that could have	Overall Least
		         been discarded for each ball		pairings that could
Draw No	B1	B2	B3	B4	B5	B6	have been used	+/-

649		10	  5	  3	12	  9	  2		  2		+
648		  9	  8	  0	  1	  1	  3		  0
647		  0	  0	  3	  4	  6	  1		  0		-
646		  8	  2	14	24	  2	  2		  2		-
645		  8	11	13	  9	10	  8		  8		+
644		  4	11	10	  6	17	29		  4		+
643		  0	  0	  1	  9	  0	  0		  0
642		19	  9	22	  5	12	  0		  0		-
641		13	10	  9	  9  	  3	  4		  3		-
640		24	16	13	  9	18	28		  9		+
639		  2	  6	  3	  2	  4	18		  2
638		  2	  3	20	11	14	10		  2		+
637		  2	  0	13	  7	11	  8		  0
636		10	  5	  6	  0	14	12		  0
635		  9	20	10	  0	  5	  7		  0		-
634		12	  3	23	  3	  5	10		  3		-
633		13	  5	23	14	  4	  7		  4		+
632		  4	  3	  2	  7	11 	  1		  1
631		  2	  1	24	13	  5	  8		  1
630		  1	  8	  7	16	  9	  6		  1		+
629		  2	  4	  9	  0	  0	  1		  0		-
628		13	  1	  7	15	23	  1		  1		+
627		  0	12	  4	18	  2	  1	   	  0		-
626		  2	15	  3	  5	  9	  1		  1		
625		  6	  1	  5	  1	  9	  3		  1		+
624		  8	10	  0	  3	  0	  4		  0		-
623		  4	  7	  3	12	24	17		  3		+
622		  0	  4	  4	  4	  1	  2		  0
621		  1	12	  3	  1	  0	  6		  0		-
620		  8	  6	  5	  1	  1	  1		  1		+
619		  7	  4	11	  2	  4	  0		  0		-
618		10	  4	  7	  2	  1	10		  1		+
617		13	  0	29	  4	12	10		  0		-
616		10	10	23	23	22	32		10		+
615		  3	  4	  9	12	  4	23		  3		+
614		  5	  0	  3	  7	  8	  0		  0
613		  4	  0	  4	  0	  0	  2		  0		-
612		16	  6	  5	17	  5	12		  5		+
611		  5	  4	  6	  5	  0	  4		  0		-
610		15	14	  1	  9	  9	14		  1		+
609		  6	  2	  4	  0	  0	  5		  0
608		  0	  3	  7	  0	11	  3		  0		-
607		  6	11	  5	  8	  4	  2		  2		+
606		17	  5	  3	  4	  0	  0		  0		-
605		  3	  2	  8	  7	  5	  4		  2		-
604		  6	11	10	  5	13	12		  6		+
603		16	10	10	  6	10	  5		  5		+
602		  3	  6	17	  3	  4	  8		  3		+
601		  2	  4	11	15	17	  8		  2		-
600		  5	12	11	  3	  8	11		  3		+
599		18	  4	14	  0	19	  3		  0
598		  1	  0	15	  1	  1	  1		  0		-
597		13	27	11	  6	12	  3		  3		+
596		  5	  0 	  0	  2	  3	  0		  0		-
595		16	  4	12	  9	  8	  8		  4		+
594		  0	  0	  4	  2	  5	  0		  0
593		  0  	  1	  1	  1	  2	  1		  0		-
592	  	  6	25	27	  3	  3	  3		  3		+
591		  0	  3	  1	  3	  4	  3		  0
590		29	25	  1	  0	21	  6   		  0
589		  0	  8	  3	23	10	15		  0
588	  	  4	  0	18	16	11	  3		  0		-
587		  6	  3	16	  3	17	  1		  1		-
586		22	11	15	  7	  6	15		  6		+
585		  8	  5	  7	  4	  5	11		  4		+
584		  2	  4	  1	  4	29	  6		  1		
583	  	  4	10	23	21	  1	10		  1		-
582		  3	12	  5	  6	12	10		  3		+
581		15	  0	28	  2	  4	10		  0		-
580		  8	  9	  5	  6	  1	  3		  1		-
579		27	20	  5	  6	21	22		  5		+
578		  2	  6	  1	  2	  2	18		  1		-
577		25	  4	14	  2	11	23		  2
576		  3	  5	11	  3	  2	  6		  2		+
575		  1	  6	  2	  1	  0	  3		  0		-
574		  5	14	  8	  5	  3	  3		  3		+
573		  0	  3	  0	  3	  0	  2		  0		-
572		  4	11	11	20	17	  3		  3		+
571		  2	16	  4	  1	  1	  7		  1		+
570		  0	  3	  1	  0	  9	  5		  0		-
569		  2	  7	  3	18	  6	17		  2		+
568		28	28	  5	  2	  0	  3		  0		-
567	  	  2	  7	12	14	  9	22		  2		+
566		  2	  0	  0	  0	  0	  5		  0		-
565		  2	  9	10	10	  4	  9		  2		+
564		  4	  0	  7	  7	  3	  1		  0		-
563		  4	  4	  4	  3	  3	  3		  3		+
562	  	  2	12	22	  6	  3	  8		  2		+
561		10	  1	  7	10	  6	  0		  0		-
560		27	  7	  5	22	  7	  3		  3		+
559		  9	  2	17	  8	  1	  8		  1		-
558		22	13	  9	21	  5	  7		  5		-
557		16	11	18	14	  6	24		  6		+
556		  8	  3	16	  2	10	  0		  0		-
555		22	  8	24	  4	  7	16		  4		-
554		  6	12	20	  7	  7	  8		  6		+
553		  0	  6	  6	  1	  0	  1		  0		-
552		  3	15	  9	16	20	26		  3		+
551		  5	  2	16	  1	  5	  8		  1		-
550		  5	  6	14	  7	  5	  4		  4

>=10		29	27	42	25	29	29

0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 = 39
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 = 17
2,2,2,2,2,2,2,2,2,2,2,2,2 = 13
3,3,3,3,3,3,3,3,3,3,3,3,3,3 = 14
4,4,4,4,4,4 = 6
5,5,5,5 = 4
6,6,6,6 = 4
8 = 1
9 = 1
10 = 1

Total = 100

Conclusions

Initially I was a little perplexed. I was expecting much higher figures for the 'overall' category. After all, if we can say with some surety that 'on average' we could eliminate, say, 10 worst lotto pairings for each lotto number and still 'enable' the 6 winning numbers, then this would be a good thing…because to eliminate 10 worst pairings each draw for each number (and be right the majority of the time) would be a powerful tool in eliminating 'unlikely' lotto combinations.

However, the results are not like that at all. But that is not to say that they are not useful. Indeed, I believe they are most powerful. If we disregard the single occurrences for '8', '9', and '10' – we can say with 97%(!!!) assurance that each draw, one of the lotto numbers drawn will appear with one of its 6 worst pairings!

For a random event, this assuredness of 97% is phenomenal. For those of you that pick a pool of lotto numbers to play with each draw, I bet you that no one ever before picked their numbers that way – using some of LEAST PAIRINGS!

Also you will notice at the bottom of the table I have totted up the scores for all the occurrences for each lottery numbered ball where the individual least pairings that could have been eliminated was greater than or equal to 10. For some reason, the 3rd ball in numerical order has a figure significantly higher than any of the other balls – 42% of the time the third lotto ball in any draw is not drawn with AT LEAST 10 of its worst lottery pairings. Perhaps others can comment as to why this might be, or check it with their own lottery and post their results.

I'm not even sure right now of the whole implications of this test. Maybe others can comment. Maybe this fits into the jigsaw of others' thinking. I feel it is significant somehow, but I don't know quite how yet. I am researching further, but for now, I just wanted to let you know of my results.

I hope it helps.

Best wishes all.

Nik Kulai Barker

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Lottery pairings reduce millions of lotto combinations.

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