Sum of all 20 keno balls discussed in Math/Questions and Answers at Wizard of Vegas

Wizard
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Posts: 27036

Joined: Oct 14, 2009

May 20th, 2019 at 3:04:04 PM permalink

I am doing a software review of a company that offers prop bets on the sum of the balls in keno. As a reminder, in keno the game draws 20 numbers from a 80 balls numbered 1 to 80. The problem at hand is finding a probability of any given total without using a simulation.

I do know simulations are a perfectly valid way of analyzing casino games, but I also feel more satisfied finding an exact answer. At first I wrote a simple program to loop through 3,535,316,142,212,180,000 possible ways to choose 20 out of 80 balls. Needless to say, it would have taken centuries to cycle through them all. I've also toyed with shortcut code, but I fear I'll die before the program is done running.

I've also just counted combinations by hand, hoping to find a pattern (maybe some variant of Fibonaci), but so far have not found anything. So, at this point I open it up to the forum. You do not have to put replies in spoiler tags and may refer to any outside sources. This isn't the usual math puzzle thread.

The question for the poll is what are your thoughts on it?

In this spoiler tag are the results of my simulation. I excluded extreme totals where the probability was less than 1 in a million. It's a big table, so I won't clutter up the thread with it, unless you want to see it. The third column is the expected count per million games.

Total	Count	Exp. per Mil.
441	3,435	1
442	3,744	1
443	3,838	1
444	3,969	1
445	4,323	1
446	4,534	1
447	4,739	1
448	5,044	1
449	5,321	1
450	5,545	1
451	5,796	1
452	6,293	1
453	6,611	1
454	6,999	1
455	7,282	1
456	7,860	1
457	7,940	1
458	8,452	1
459	9,068	1
460	9,480	1
461	10,009	2
462	10,574	2
463	11,129	2
464	11,514	2
465	12,115	2
466	12,753	2
467	13,486	2
468	14,189	2
469	14,828	2
470	15,359	2
471	16,377	3
472	17,251	3
473	18,045	3
474	19,033	3
475	19,760	3
476	20,672	3
477	21,744	3
478	22,939	4
479	24,007	4
480	25,196	4
481	26,055	4
482	27,080	4
483	28,661	4
484	29,927	5
485	31,384	5
486	32,913	5
487	34,522	5
488	36,028	6
489	37,678	6
490	39,692	6
491	41,261	6
492	43,166	7
493	45,217	7
494	47,199	7
495	48,906	8
496	51,285	8
497	53,787	8
498	56,207	9
499	58,636	9
500	61,251	9
501	63,618	10
502	66,347	10
503	69,196	11
504	72,198	11
505	75,956	12
506	78,432	12
507	81,814	13
508	85,550	13
509	89,007	14
510	92,998	14
511	96,468	15
512	101,191	16
513	104,498	16
514	109,120	17
515	113,433	17
516	118,328	18
517	122,687	19
518	127,485	20
519	134,015	21
520	138,739	21
521	144,682	22
522	149,379	23
523	155,583	24
524	161,377	25
525	167,986	26
526	174,464	27
527	181,674	28
528	188,399	29
529	196,313	30
530	202,955	31
531	211,699	32
532	217,869	33
533	228,639	35
534	235,939	36
535	245,385	38
536	253,054	39
537	262,573	40
538	272,414	42
539	282,960	43
540	293,067	45
541	303,583	47
542	315,744	48
543	325,370	50
544	339,232	52
545	350,113	54
546	362,701	56
547	376,191	58
548	388,543	60
549	402,589	62
550	416,646	64
551	431,632	66
552	446,715	69
553	461,115	71
554	476,083	73
555	493,312	76
556	509,761	78
557	527,002	81
558	545,283	84
559	563,285	86
560	579,537	89
561	600,939	92
562	620,205	95
563	640,774	98
564	663,018	102
565	682,220	105
566	705,101	108
567	728,213	112
568	751,145	115
569	773,003	119
570	800,365	123
571	822,985	126
572	849,868	130
573	877,063	135
574	903,857	139
575	933,334	143
576	960,447	147
577	990,405	152
578	1,021,121	157
579	1,051,043	161
580	1,081,462	166
581	1,115,569	171
582	1,149,091	176
583	1,182,506	181
584	1,218,923	187
585	1,252,892	192
586	1,289,475	198
587	1,326,081	203
588	1,365,000	209
589	1,404,936	215
590	1,447,113	222
591	1,486,588	228
592	1,527,549	234
593	1,572,339	241
594	1,612,520	247
595	1,659,021	254
596	1,705,695	262
597	1,754,146	269
598	1,801,657	276
599	1,850,683	284
600	1,899,207	291
601	1,953,142	300
602	2,005,515	308
603	2,057,840	316
604	2,111,828	324
605	2,165,195	332
606	2,222,992	341
607	2,283,008	350
608	2,339,728	359
609	2,399,839	368
610	2,462,058	378
611	2,522,864	387
612	2,590,626	397
613	2,653,379	407
614	2,720,129	417
615	2,789,066	428
616	2,857,378	438
617	2,927,708	449
618	3,005,437	461
619	3,076,651	472
620	3,145,799	482
621	3,223,678	494
622	3,298,823	506
623	3,375,968	518
624	3,458,561	530
625	3,540,982	543
626	3,621,286	555
627	3,706,336	568
628	3,793,415	582
629	3,879,513	595
630	3,968,574	609
631	4,059,927	623
632	4,147,717	636
633	4,240,094	650
634	4,333,209	665
635	4,427,877	679
636	4,526,070	694
637	4,627,644	710
638	4,727,946	725
639	4,830,073	741
640	4,930,411	756
641	5,035,769	772
642	5,144,626	789
643	5,249,616	805
644	5,363,903	823
645	5,469,158	839
646	5,584,422	857
647	5,698,184	874
648	5,821,229	893
649	5,931,005	910
650	6,053,589	928
651	6,170,477	946
652	6,296,285	966
653	6,419,240	985
654	6,547,043	1,004
655	6,667,303	1,023
656	6,796,655	1,042
657	6,927,675	1,063
658	7,055,827	1,082
659	7,189,622	1,103
660	7,326,894	1,124
661	7,456,552	1,144
662	7,595,410	1,165
663	7,741,620	1,187
664	7,878,448	1,208
665	8,024,240	1,231
666	8,164,932	1,252
667	8,310,177	1,275
668	8,456,983	1,297
669	8,607,591	1,320
670	8,762,016	1,344
671	8,900,482	1,365
672	9,061,785	1,390
673	9,209,800	1,413
674	9,370,622	1,437
675	9,530,537	1,462
676	9,682,898	1,485
677	9,840,473	1,509
678	10,002,377	1,534
679	10,166,739	1,559
680	10,332,192	1,585
681	10,489,367	1,609
682	10,656,721	1,634
683	10,833,480	1,662
684	10,997,379	1,687
685	11,164,607	1,712
686	11,334,202	1,738
687	11,511,102	1,766
688	11,683,505	1,792
689	11,854,861	1,818
690	12,026,976	1,845
691	12,206,568	1,872
692	12,378,607	1,899
693	12,560,691	1,926
694	12,737,586	1,954
695	12,921,619	1,982
696	13,099,750	2,009
697	13,281,463	2,037
698	13,464,349	2,065
699	13,659,987	2,095
700	13,831,868	2,121
701	14,015,126	2,150
702	14,205,072	2,179
703	14,392,695	2,207
704	14,581,750	2,236
705	14,757,007	2,263
706	14,956,306	2,294
707	15,144,221	2,323
708	15,330,541	2,351
709	15,523,000	2,381
710	15,707,557	2,409
711	15,901,310	2,439
712	16,090,644	2,468
713	16,277,131	2,496
714	16,474,557	2,527
715	16,657,263	2,555
716	16,859,581	2,586
717	17,044,543	2,614
718	17,241,315	2,644
719	17,431,764	2,674
720	17,621,957	2,703
721	17,820,359	2,733
722	18,007,127	2,762
723	18,196,152	2,791
724	18,390,894	2,821
725	18,581,770	2,850
726	18,776,054	2,880
727	18,952,584	2,907
728	19,150,208	2,937
729	19,340,928	2,966
730	19,530,508	2,995
731	19,711,150	3,023
732	19,909,926	3,054
733	20,090,359	3,081
734	20,276,212	3,110
735	20,458,780	3,138
736	20,645,075	3,166
737	20,826,094	3,194
738	21,010,505	3,222
739	21,184,684	3,249
740	21,374,281	3,278
741	21,542,413	3,304
742	21,729,944	3,333
743	21,905,866	3,360
744	22,084,327	3,387
745	22,253,227	3,413
746	22,419,160	3,439
747	22,594,055	3,465
748	22,775,467	3,493
749	22,936,395	3,518
750	23,101,841	3,543
751	23,278,659	3,570
752	23,442,393	3,595
753	23,594,853	3,619
754	23,764,936	3,645
755	23,915,848	3,668
756	24,072,575	3,692
757	24,219,496	3,715
758	24,382,858	3,740
759	24,537,006	3,763
760	24,683,616	3,786
761	24,822,836	3,807
762	24,986,511	3,832
763	25,116,433	3,852
764	25,255,618	3,874
765	25,398,888	3,896
766	25,526,481	3,915
767	25,672,383	3,937
768	25,791,251	3,956
769	25,915,025	3,975
770	26,050,458	3,995
771	26,178,986	4,015
772	26,296,271	4,033
773	26,413,333	4,051
774	26,528,066	4,069
775	26,636,170	4,085
776	26,748,652	4,103
777	26,857,717	4,119
778	26,969,571	4,136
779	27,055,593	4,150
780	27,167,435	4,167
781	27,261,216	4,181
782	27,355,200	4,196
783	27,453,032	4,211
784	27,520,647	4,221
785	27,609,338	4,235
786	27,683,154	4,246
787	27,783,913	4,261
788	27,851,963	4,272
789	27,920,350	4,282
790	27,985,234	4,292
791	28,042,985	4,301
792	28,118,461	4,313
793	28,177,419	4,322
794	28,230,306	4,330
795	28,279,969	4,337
796	28,330,982	4,345
797	28,376,165	4,352
798	28,405,742	4,357
799	28,457,425	4,365
800	28,508,733	4,373
801	28,522,647	4,375
802	28,548,374	4,379
803	28,585,242	4,384
804	28,592,528	4,385
805	28,623,406	4,390
806	28,642,581	4,393
807	28,643,977	4,393
808	28,650,200	4,394
809	28,662,962	4,396
810	28,663,683	4,396
811	28,666,272	4,397
812	28,659,446	4,396
813	28,642,707	4,393
814	28,636,339	4,392
815	28,623,072	4,390
816	28,602,453	4,387
817	28,577,793	4,383
818	28,558,620	4,380
819	28,525,239	4,375
820	28,484,774	4,369
821	28,454,276	4,364
822	28,424,946	4,360
823	28,369,113	4,351
824	28,336,596	4,346
825	28,280,514	4,338
826	28,229,969	4,330
827	28,179,004	4,322
828	28,115,128	4,312
829	28,050,436	4,302
830	27,978,178	4,291
831	27,923,254	4,283
832	27,851,480	4,272
833	27,775,285	4,260
834	27,696,831	4,248
835	27,614,904	4,235
836	27,523,482	4,221
837	27,446,944	4,210
838	27,350,877	4,195
839	27,259,610	4,181
840	27,166,593	4,167
841	27,077,387	4,153
842	26,959,292	4,135
843	26,857,548	4,119
844	26,737,880	4,101
845	26,644,740	4,087
846	26,530,087	4,069
847	26,409,425	4,051
848	26,293,674	4,033
849	26,179,346	4,015
850	26,043,665	3,994
851	25,924,860	3,976
852	25,786,873	3,955
853	25,667,874	3,937
854	25,534,938	3,916
855	25,394,716	3,895
856	25,266,400	3,875
857	25,117,582	3,852
858	24,974,923	3,831
859	24,827,566	3,808
860	24,681,658	3,786
861	24,530,307	3,762
862	24,380,059	3,739
863	24,229,392	3,716
864	24,067,458	3,691
865	23,916,986	3,668
866	23,763,351	3,645
867	23,595,872	3,619
868	23,435,026	3,594
869	23,272,670	3,569
870	23,106,061	3,544
871	22,943,980	3,519
872	22,768,413	3,492
873	22,605,841	3,467
874	22,430,941	3,440
875	22,250,856	3,413
876	22,084,770	3,387
877	21,909,607	3,360
878	21,734,749	3,334
879	21,552,728	3,306
880	21,368,700	3,277
881	21,193,387	3,251
882	20,999,640	3,221
883	20,822,422	3,194
884	20,643,153	3,166
885	20,462,048	3,138
886	20,272,437	3,109
887	20,090,583	3,081
888	19,908,019	3,053
889	19,718,060	3,024
890	19,525,110	2,995
891	19,333,451	2,965
892	19,148,662	2,937
893	18,959,539	2,908
894	18,764,734	2,878
895	18,579,508	2,850
896	18,393,824	2,821
897	18,199,883	2,791
898	18,014,782	2,763
899	17,820,332	2,733
900	17,622,591	2,703
901	17,431,435	2,674
902	17,237,539	2,644
903	17,047,999	2,615
904	16,851,334	2,585
905	16,671,357	2,557
906	16,464,875	2,525
907	16,285,487	2,498
908	16,088,821	2,468
909	15,901,591	2,439
910	15,712,769	2,410
911	15,523,156	2,381
912	15,328,998	2,351
913	15,142,681	2,322
914	14,948,183	2,293
915	14,768,973	2,265
916	14,581,616	2,236
917	14,393,031	2,208
918	14,201,510	2,178
919	14,022,741	2,151
920	13,840,535	2,123
921	13,637,420	2,092
922	13,469,777	2,066
923	13,279,699	2,037
924	13,103,752	2,010
925	12,918,928	1,981
926	12,737,099	1,954
927	12,565,003	1,927
928	12,382,420	1,899
929	12,202,822	1,872
930	12,034,624	1,846
931	11,852,584	1,818
932	11,679,705	1,791
933	11,506,740	1,765
934	11,332,814	1,738
935	11,171,123	1,713
936	10,989,047	1,685
937	10,827,757	1,661
938	10,656,739	1,634
939	10,488,145	1,609
940	10,333,650	1,585
941	10,166,315	1,559
942	9,999,714	1,534
943	9,842,566	1,510
944	9,685,425	1,485
945	9,524,252	1,461
946	9,371,438	1,437
947	9,213,035	1,413
948	9,063,459	1,390
949	8,903,743	1,366
950	8,760,545	1,344
951	8,607,733	1,320
952	8,454,229	1,297
953	8,309,658	1,274
954	8,165,969	1,252
955	8,024,407	1,231
956	7,884,097	1,209
957	7,741,275	1,187
958	7,602,600	1,166
959	7,467,518	1,145
960	7,326,924	1,124
961	7,186,025	1,102
962	7,059,347	1,083
963	6,925,203	1,062
964	6,797,473	1,043
965	6,671,659	1,023
966	6,542,112	1,003
967	6,417,141	984
968	6,296,293	966
969	6,172,646	947
970	6,054,444	929
971	5,934,001	910
972	5,816,099	892
973	5,696,308	874
974	5,585,518	857
975	5,470,602	839
976	5,363,579	823
977	5,250,182	805
978	5,143,910	789
979	5,036,934	773
980	4,934,663	757
981	4,829,209	741
982	4,727,906	725
983	4,629,768	710
984	4,524,975	694
985	4,427,515	679
986	4,335,792	665
987	4,240,216	650
988	4,147,607	636
989	4,055,769	622
990	3,965,357	608
991	3,879,116	595
992	3,793,847	582
993	3,708,386	569
994	3,620,849	555
995	3,539,124	543
996	3,456,841	530
997	3,378,144	518
998	3,299,675	506
999	3,223,531	494
1,000	3,149,948	483
1,001	3,073,548	471
1,002	2,999,209	460
1,003	2,930,573	449
1,004	2,860,976	439
1,005	2,791,039	428
1,006	2,721,344	417
1,007	2,656,566	407
1,008	2,589,961	397
1,009	2,524,701	387
1,010	2,462,746	378
1,011	2,405,928	369
1,012	2,342,078	359
1,013	2,283,811	350
1,014	2,222,728	341
1,015	2,163,391	332
1,016	2,111,514	324
1,017	2,058,080	316
1,018	2,002,275	307
1,019	1,952,564	299
1,020	1,899,055	291
1,021	1,852,096	284
1,022	1,800,065	276
1,023	1,752,832	269
1,024	1,705,132	262
1,025	1,657,875	254
1,026	1,617,609	248
1,027	1,569,189	241
1,028	1,528,429	234
1,029	1,485,557	228
1,030	1,445,678	222
1,031	1,402,081	215
1,032	1,364,455	209
1,033	1,327,805	204
1,034	1,289,037	198
1,035	1,254,056	192
1,036	1,216,909	187
1,037	1,183,935	182
1,038	1,148,374	176
1,039	1,113,641	171
1,040	1,081,610	166
1,041	1,049,225	161
1,042	1,019,412	156
1,043	990,011	152
1,044	960,185	147
1,045	932,152	143
1,046	903,127	139
1,047	878,067	135
1,048	849,403	130
1,049	825,986	127
1,050	800,391	123
1,051	774,569	119
1,052	750,355	115
1,053	727,303	112
1,054	705,731	108
1,055	681,719	105
1,056	662,209	102
1,057	642,753	99
1,058	620,171	95
1,059	601,326	92
1,060	581,228	89
1,061	563,149	86
1,062	543,921	83
1,063	527,559	81
1,064	510,221	78
1,065	492,829	76
1,066	476,649	73
1,067	461,857	71
1,068	446,006	68
1,069	431,170	66
1,070	416,983	64
1,071	403,009	62
1,072	388,740	60
1,073	375,730	58
1,074	362,744	56
1,075	349,799	54
1,076	337,515	52
1,077	325,578	50
1,078	315,035	48
1,079	304,850	47
1,080	293,324	45
1,081	282,717	43
1,082	272,825	42
1,083	262,148	40
1,084	253,864	39
1,085	244,561	38
1,086	236,658	36
1,087	227,144	35
1,088	219,325	34
1,089	211,325	32
1,090	203,347	31
1,091	196,097	30
1,092	188,406	29
1,093	182,111	28
1,094	174,976	27
1,095	167,875	26
1,096	161,607	25
1,097	156,189	24
1,098	149,548	23
1,099	144,428	22
1,100	138,467	21
1,101	133,467	20
1,102	128,131	20
1,103	123,875	19
1,104	118,380	18
1,105	113,401	17
1,106	109,324	17
1,107	105,067	16
1,108	100,365	15
1,109	96,211	15
1,110	92,518	14
1,111	89,418	14
1,112	85,559	13
1,113	82,129	13
1,114	78,608	12
1,115	75,594	12
1,116	72,488	11
1,117	69,493	11
1,118	65,888	10
1,119	63,739	10
1,120	60,695	9
1,121	58,616	9
1,122	56,145	9
1,123	53,379	8
1,124	51,672	8
1,125	49,369	8
1,126	46,770	7
1,127	44,880	7
1,128	43,118	7
1,129	41,257	6
1,130	39,401	6
1,131	37,876	6
1,132	35,977	6
1,133	34,743	5
1,134	32,984	5
1,135	31,269	5
1,136	29,765	5
1,137	28,461	4
1,138	27,738	4
1,139	26,374	4
1,140	24,998	4
1,141	23,762	4
1,142	22,931	4
1,143	21,563	3
1,144	20,568	3
1,145	19,825	3
1,146	18,735	3
1,147	17,862	3
1,148	17,258	3
1,149	16,271	2
1,150	15,606	2
1,151	14,719	2
1,152	14,241	2
1,153	13,258	2
1,154	12,837	2
1,155	11,988	2
1,156	11,648	2
1,157	10,916	2
1,158	10,411	2
1,159	9,810	2
1,160	9,449	1
1,161	8,923	1
1,162	8,499	1
1,163	8,170	1
1,164	7,715	1
1,165	7,342	1
1,166	6,914	1
1,167	6,632	1
1,168	6,243	1
1,169	5,866	1
1,170	5,757	1
1,171	5,388	1
1,172	4,976	1
1,173	4,800	1
1,174	4,605	1
1,175	4,282	1
1,176	3,958	1
1,177	3,703	1
1,178	3,722	1
1,179	3,478	1
Total	6,520,000,000	1,000,000

"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)

7craps

7craps

Threads: 18
Posts: 1977

Joined: Jan 23, 2010

May 20th, 2019 at 3:21:59 PM permalink

Quote: Wizard
I am doing a software review of a company that offers prop bets on the sum of the balls in keno. As a reminder, in keno the game draws 20 numbers from a 80 balls numbered 1 to 80.

The problem at hand is finding a probability of any given total without using a simulation.

Been done as is very simple
uses generating function (I know that much)

"or in Pari Gp (free)
prod(i=1, 80, 1 + x*y^i);"

one needs only the y values for x^20 and I have not verified this method. I think this was found online.

https://wizardofvegas.com/forum/questions-and-answers/gambling/10526-what-is-keno-exact-scores-house-edge/3/#post652407

the results exactly for all the sums can be found here
that page has yet to be cleaned up. text file and Excel

https://sites.google.com/view/krapstuff/keno

winsome johnny (not Win some johnny)

Wizard
Administrator

Wizard

Threads: 1518
Posts: 27036

Joined: Oct 14, 2009

May 20th, 2019 at 3:38:56 PM permalink

Interesting. I wish Sally were still around to convert that to more plain simple English. Can anyone else help? Here are the combinations from Sally's source. Again, putting table in spoiler tags because it's so big. I can't post the whole table because there is a maximum size per post.

Total	Combinations
210	1
211	1
212	2
213	3
214	5
215	7
216	11
217	15
218	22
219	30
220	42
221	56
222	77
223	101
224	135
225	176
226	231
227	297
228	385
229	490
230	627
231	791
232	1,000
233	1,251
234	1,568
235	1,946
236	2,417
237	2,980
238	3,673
239	4,498
240	5,507
241	6,703
242	8,154
243	9,871
244	11,937
245	14,375
246	17,293
247	20,722
248	24,803
249	29,588
250	35,251
251	41,869
252	49,668
253	58,754
254	69,414
255	81,801
256	96,271
257	113,039
258	132,559
259	155,112
260	181,274
261	211,428
262	246,288
263	286,364
264	332,557
265	385,528
266	446,405
267	516,054
268	595,872
269	686,983
270	791,131
271	909,741
272	1,044,984
273	1,198,689
274	1,373,524
275	1,571,812
276	1,796,855
277	2,051,569
278	2,340,024
279	2,665,885
280	3,034,135
281	3,449,359
282	3,917,670
283	4,444,748
284	5,038,070
285	5,704,686
286	6,453,684
287	7,293,767
288	8,236,001
289	9,291,060
290	10,472,375
291	11,793,028
292	13,269,250
293	14,917,024
294	16,755,957
295	18,805,464
296	21,089,176
297	23,630,664
298	26,458,297
299	29,600,587
300	33,091,578
301	36,965,620
302	41,263,462
303	46,026,431
304	51,303,148
305	57,143,201
306	63,604,540
307	70,746,444
308	78,637,842
309	87,349,501
310	96,963,169
311	107,563,103
312	119,246,146
313	132,112,328
314	146,276,124
315	161,856,093
316	178,987,225
317	197,809,729
318	218,482,569
319	241,171,101
320	266,062,168
321	293,350,487
322	323,255,164
323	356,005,000
324	391,856,581
325	431,078,138
326	473,969,587
327	520,844,871
328	572,053,851
329	627,963,266
330	688,980,643
331	755,533,468
332	828,095,629
333	907,164,667
334	993,290,571
335	1,087,051,347
336	1,189,084,411
337	1,300,059,954
338	1,420,715,489
339	1,551,826,889
340	1,694,245,990
341	1,848,869,496
342	2,016,679,881
343	2,198,711,618
344	2,396,096,082
345	2,610,024,936
346	2,841,798,846
347	3,092,788,314
348	3,364,486,581
349	3,658,467,191
350	3,976,441,876
351	4,320,214,692
352	4,691,744,778
353	5,093,097,335
354	5,526,511,602
355	5,994,347,965
356	6,499,162,145
357	7,043,648,186
358	7,630,718,671
359	8,263,443,950
360	8,945,138,936
361	9,679,297,643
362	10,469,687,646
363	11,320,279,729
364	12,235,349,889
365	13,219,404,469
366	14,277,290,236
367	15,414,114,032
368	16,635,362,300
369	17,946,814,901
370	19,354,673,930
371	20,865,472,184
372	22,486,212,019
373	24,224,267,278
374	26,087,533,643
375	28,084,323,773
376	30,223,529,082
377	32,514,508,546
378	34,967,262,684
379	37,592,314,751
380	40,400,898,786
381	43,404,832,757
382	46,616,720,518
383	50,049,817,550
384	53,718,247,833
385	57,636,860,670
386	61,821,464,449
387	66,288,674,187
388	71,056,162,189
389	76,142,496,873
390	81,567,411,419
391	87,351,632,249
392	93,517,168,095
393	100,087,127,594
394	107,086,028,628
395	114,539,605,867
396	122,475,141,756
397	130,921,262,045
398	139,908,291,043
399	149,468,034,620
400	159,634,159,836
401	170,441,966,142
402	181,928,790,813
403	194,133,766,301
404	207,098,254,570
405	220,865,589,927
406	235,481,542,190
407	250,994,045,973
408	267,453,694,512
409	284,913,452,879
410	303,429,185,857
411	323,059,354,425
412	343,865,577,844
413	365,912,314,318
414	389,267,459,099
415	414,002,006,749
416	440,190,689,221
417	467,911,618,783
418	497,246,966,079
419	528,282,584,833
420	561,108,732,221
421	595,819,672,189
422	632,514,442,353
423	671,296,435,063
424	712,274,211,009
425	755,561,059,108
426	801,275,859,118
427	849,542,617,012
428	900,491,381,529
429	954,257,753,733
430	1,010,983,857,544
431	1,070,817,824,773
432	1,133,914,822,434
433	1,200,436,509,276
434	1,270,552,124,993
435	1,344,437,917,067
436	1,422,278,292,091
437	1,504,265,214,046
438	1,590,599,420,536
439	1,681,489,788,194
440	1,777,154,619,819
441	1,877,820,975,168
442	1,983,726,028,916
443	2,095,116,368,246
444	2,212,249,424,868
445	2,335,392,734,382
446	2,464,825,448,513
447	2,600,837,554,495
448	2,743,731,467,669
449	2,893,821,211,946
450	3,051,434,096,031
451	3,216,909,849,508
452	3,390,602,389,686
453	3,572,878,910,940
454	3,764,121,741,811
455	3,964,727,389,027
456	4,175,108,488,550
457	4,395,692,797,816
458	4,626,925,247,931
459	4,869,266,881,582
460	5,123,197,006,289
461	5,389,212,078,927
462	5,667,827,963,566
463	5,959,578,755,669
464	6,265,019,152,487
465	6,584,723,213,704
466	6,919,286,843,610
467	7,269,326,489,361
468	7,635,481,738,958
469	8,018,413,948,848
470	8,418,808,965,914
471	8,837,375,680,898
472	9,274,848,873,719
473	9,731,987,693,363
474	10,209,578,630,140
475	10,708,433,913,262
476	11,229,394,619,008
477	11,773,328,981,483
478	12,341,135,635,235
479	12,933,741,839,899
480	13,552,106,861,442
481	14,197,220,100,303
482	14,870,104,620,270
483	15,571,815,175,325
484	16,303,441,884,554
485	17,066,108,157,849
486	17,860,974,520,443
487	18,689,236,426,307
488	19,552,128,242,181
489	20,450,920,942,555
490	21,386,926,250,737
491	22,361,494,215,565
492	23,376,017,513,498
493	24,431,928,894,154
494	25,530,705,653,070
495	26,673,866,939,713
496	27,862,978,398,196
497	29,099,649,337,087
498	30,385,537,541,434
499	31,722,346,290,416
500	33,111,829,351,342
501	34,555,787,837,266
502	36,056,075,379,039
503	37,614,594,822,327
504	39,233,303,581,272
505	40,914,210,159,126
506	42,659,379,693,854
507	44,470,930,294,083
508	46,351,038,772,509
509	48,301,936,795,528
510	50,325,916,807,381
511	52,425,327,976,532
512	54,602,582,321,801
513	56,860,150,445,667
514	59,200,567,856,702
515	61,626,430,488,492
516	64,140,401,221,500
517	66,745,205,168,528
518	69,443,636,407,019
519	72,238,553,020,681
520	75,132,884,036,461
521	78,129,624,219,915
522	81,231,841,219,122
523	84,442,670,093,739
524	87,765,320,677,722
525	91,203,071,830,294
526	94,759,279,010,024
527	98,437,368,245,632
528	102,240,843,924,128
529	106,173,282,459,017
530	110,238,340,304,328
531	114,439,747,308,027
532	118,781,314,944,316
533	123,266,929,349,329
534	127,900,559,773,363
535	132,686,251,275,763
536	137,628,133,409,981
537	142,730,412,563,985
538	147,997,380,868,998
539	153,433,408,182,971
540	159,042,951,225,485
541	164,830,545,178,892
542	170,800,813,062,207
543	176,958,456,936,318
544	183,308,267,507,535
545	189,855,114,934,011
546	196,603,958,660,512
547	203,559,837,800,120
548	210,727,881,214,639
549	218,113,297,456,569
550	225,721,385,084,601
551	233,557,522,163,930
552	241,627,176,819,381
553	249,935,896,266,269
554	258,489,317,615,563
555	267,293,156,421,037
556	276,353,217,726,061
557	285,675,384,124,940
558	295,265,627,053,394
559	305,129,994,336,046
560	315,274,621,737,283
561	325,705,719,979,621
562	336,429,586,543,371
563	347,452,592,155,562
564	358,781,192,841,696
565	370,421,915,855,964
566	382,381,372,002,154
567	394,666,240,991,089
568	407,283,284,019,903
569	420,239,328,556,562
570	433,541,281,180,913
571	447,196,111,767,398
572	461,210,866,607,609
573	475,592,651,976,889
574	490,348,647,526,558
575	505,486,089,237,058
576	521,012,283,085,036
577	536,934,587,353,256
578	553,260,426,593,322
579	569,997,273,279,789
580	587,152,662,058,258
581	604,734,170,747,184
582	622,749,434,876,858
583	641,206,128,010,489
584	660,111,976,597,920
585	679,474,739,604,064
586	699,302,223,666,421
587	719,602,262,037,791
588	740,382,730,055,707
589	761,651,523,370,470
590	783,416,573,752,126
591	805,685,826,594,439
592	828,467,257,050,350
593	851,768,846,820,577
594	875,598,600,626,154
595	899,964,522,255,079
596	924,874,631,401,209
597	950,336,938,960,611
598	976,359,464,227,661
599	1,002,950,209,453,770
600	1,030,117,177,412,690
601	1,057,868,345,195,040
602	1,086,211,682,173,950
603	1,115,155,123,030,640
604	1,144,706,586,113,290
605	1,174,873,945,708,900
606	1,205,665,050,808,210
607	1,237,087,696,599,980
608	1,269,149,643,677,450
609	1,301,858,588,751,630
610	1,335,222,184,292,440
611	1,369,248,008,480,890
612	1,403,943,585,301,200
613	1,439,316,353,708,700
614	1,475,373,688,210,190
615	1,512,122,867,248,400
616	1,549,571,094,266,050
617	1,587,725,465,329,520
618	1,626,592,990,693,830
619	1,666,180,561,645,290
620	1,706,494,972,609,520
621	1,747,542,887,215,360
622	1,789,330,860,941,280
623	1,831,865,306,428,570
624	1,875,152,516,686,540
625	1,919,198,629,635,730
626	1,964,009,651,915,660
627	2,009,591,422,664,540
628	2,055,949,637,927,470
629	2,103,089,813,703,010
630	2,151,017,310,973,090
631	2,199,737,298,000,370
632	2,249,254,776,027,770
633	2,299,574,540,834,780
634	2,350,701,209,103,640
635	2,402,639,179,270,460
636	2,455,392,658,582,210
637	2,508,965,623,226,930
638	2,563,361,846,130,740
639	2,618,584,856,379,090
640	2,674,637,967,752,940
641	2,731,524,237,478,950
642	2,789,246,495,528,880
643	2,847,807,302,685,920
644	2,907,208,980,659,850
645	2,967,453,569,482,800
646	3,028,542,858,437,460
647	3,090,478,342,825,040
648	3,153,261,255,732,850
649	3,216,892,524,160,980
650	3,281,372,801,682,290
651	3,346,702,423,944,310
652	3,412,881,442,219,840
653	3,479,909,578,328,190
654	3,547,786,259,101,070
655	3,616,510,570,712,250
656	3,686,081,294,113,980
657	3,756,496,858,791,470
658	3,827,755,379,165,350
659	3,899,854,607,818,590
660	3,972,791,972,891,010
661	4,046,564,530,766,010
662	4,121,169,004,511,070
663	4,196,601,736,043,210
664	4,272,858,725,610,060
665	4,349,935,583,482,260
666	4,427,827,570,497,980
667	4,506,529,549,270,560
668	4,586,036,025,851,500
669	4,666,341,100,470,600
670	4,747,438,510,307,980
671	4,829,321,579,817,140
672	4,911,983,264,628,410
673	4,995,416,101,440,400
674	5,079,612,253,106,590
675	5,164,563,458,113,970
676	5,250,261,076,840,900
677	5,336,696,040,671,980
678	5,423,858,899,445,150
679	5,511,739,770,187,560
680	5,600,328,385,803,350
681	5,689,614,043,446,160
682	5,779,585,654,429,600
683	5,870,231,692,286,640
684	5,961,540,243,918,360
685	6,053,498,957,320,960
686	6,146,095,094,019,120
687	6,239,315,476,476,150
688	6,333,146,541,790,360
689	6,427,574,288,835,360
690	6,522,584,333,229,140
691	6,618,161,854,183,300
692	6,714,291,650,789,160
693	6,810,958,088,586,950
694	6,908,145,157,139,450
695	7,005,836,416,368,480
696	7,104,015,055,421,620
697	7,202,663,838,749,380
698	7,301,765,166,303,070
699	7,401,301,019,361,720
700	7,501,253,022,027,260
701	7,601,602,386,843,980
702	7,702,329,977,589,080
703	7,803,416,254,655,220
704	7,904,841,339,170,870
705	8,006,584,958,150,180
706	8,108,626,509,899,880
707	8,210,945,008,978,190
708	8,313,519,152,881,930
709	8,416,327,266,778,080
710	8,519,347,371,498,560
711	8,622,557,128,046,050
712	8,725,933,906,847,840
713	8,829,454,732,072,480
714	8,933,096,352,127,630
715	9,036,835,183,748,460
716	9,140,647,383,762,520
717	9,244,508,792,945,670
718	9,348,395,008,996,000
719	9,452,281,330,191,890
720	9,556,142,829,555,640
721	9,659,954,298,267,300
722	9,763,690,321,033,400
723	9,867,325,219,249,980
724	9,970,833,127,512,510
725	10,074,187,936,558,100
726	10,177,363,370,888,400
727	10,280,332,931,441,700
728	10,383,069,974,340,200
729	10,485,547,653,279,600
730	10,587,738,999,328,200
731	10,689,616,863,061,700
732	10,791,153,995,382,700
733	10,892,322,989,344,500
734	10,993,096,361,994,600
735	11,093,446,495,874,100
736	11,193,345,721,807,600
737	11,292,766,260,101,800
738	11,391,680,304,241,600
739	11,490,059,961,730,100
740	11,587,877,338,692,700
741	11,685,104,480,338,700
742	11,781,713,456,388,200
743	11,877,676,301,179,600
744	11,972,965,099,875,100
745	12,067,551,928,151,000
746	12,161,408,939,178,500
747	12,254,508,302,876,300
748	12,346,822,293,578,000
749	12,438,323,228,868,000
750	12,528,983,557,886,000
751	12,618,775,799,682,300
752	12,707,672,632,154,500
753	12,795,646,829,899,000
754	12,882,671,353,685,000
755	12,968,719,287,824,400
756	13,053,763,930,133,200
757	13,137,778,728,752,300
758	13,220,737,372,587,100
759	13,302,613,727,557,800
760	13,383,381,927,422,100
761	13,463,016,309,475,000
762	13,541,491,505,700,400
763	13,618,782,377,856,200
764	13,694,864,108,945,900
765	13,769,712,137,666,200
766	13,843,302,250,100,100
767	13,915,610,513,550,400
768	13,986,613,368,402,600
769	14,056,287,561,278,000
770	14,124,610,237,055,900
771	14,191,558,871,327,400
772	14,257,111,362,481,200
773	14,321,245,963,481,600
774	14,383,941,373,963,900
775	14,445,176,671,275,600
776	14,504,931,402,574,800
777	14,563,185,515,116,400
778	14,619,919,448,256,100
779	14,675,114,063,011,100
780	14,728,750,733,921,000
781	14,780,811,277,822,800
782	14,831,278,045,561,900
783	14,880,133,849,969,200
784	14,927,362,057,331,000
785	14,972,946,514,599,900
786	15,016,871,640,574,700
787	15,059,122,352,292,200
788	15,099,684,155,916,700
789	15,138,543,072,303,300
790	15,175,685,727,506,900
791	15,211,099,277,556,400
792	15,244,771,498,541,800
793	15,276,690,710,548,100
794	15,306,845,867,323,800
795	15,335,226,479,373,200
796	15,361,822,703,122,200
797	15,386,625,263,213,800
798	15,409,625,541,131,800
799	15,430,815,496,656,900
800	15,450,187,755,959,900

Source: https://sites.google.com/view/krapstuff/keno

"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)

Wizard
Administrator

Wizard

Threads: 1518
Posts: 27036

Joined: Oct 14, 2009

May 20th, 2019 at 3:40:29 PM permalink

Here is the rest of the table.

801	15,467,735,532,221,000
802	15,483,452,713,112,300
803	15,497,333,780,632,900
804	15,509,373,897,951,000
805	15,519,568,828,416,200
806	15,527,915,021,776,200
807	15,534,409,532,379,100
808	15,539,050,104,692,300
809	15,541,835,090,752,100
810	15,542,763,534,960,500
811	15,541,835,090,752,100
812	15,539,050,104,692,300
813	15,534,409,532,379,100
814	15,527,915,021,776,200
815	15,519,568,828,416,200
816	15,509,373,897,951,000
817	15,497,333,780,632,900
818	15,483,452,713,112,300
819	15,467,735,532,221,000
820	15,450,187,755,959,900
821	15,430,815,496,656,900
822	15,409,625,541,131,800
823	15,386,625,263,213,800
824	15,361,822,703,122,200
825	15,335,226,479,373,200
826	15,306,845,867,323,800
827	15,276,690,710,548,100
828	15,244,771,498,541,800
829	15,211,099,277,556,400
830	15,175,685,727,506,900
831	15,138,543,072,303,300
832	15,099,684,155,916,700
833	15,059,122,352,292,200
834	15,016,871,640,574,700
835	14,972,946,514,599,900
836	14,927,362,057,331,000
837	14,880,133,849,969,200
838	14,831,278,045,561,900
839	14,780,811,277,822,800
840	14,728,750,733,921,000
841	14,675,114,063,011,100
842	14,619,919,448,256,100
843	14,563,185,515,116,400
844	14,504,931,402,574,800
845	14,445,176,671,275,600
846	14,383,941,373,963,900
847	14,321,245,963,481,600
848	14,257,111,362,481,200
849	14,191,558,871,327,400
850	14,124,610,237,055,900
851	14,056,287,561,278,000
852	13,986,613,368,402,600
853	13,915,610,513,550,400
854	13,843,302,250,100,100
855	13,769,712,137,666,200
856	13,694,864,108,945,900
857	13,618,782,377,856,200
858	13,541,491,505,700,400
859	13,463,016,309,475,000
860	13,383,381,927,422,100
861	13,302,613,727,557,800
862	13,220,737,372,587,100
863	13,137,778,728,752,300
864	13,053,763,930,133,200
865	12,968,719,287,824,400
866	12,882,671,353,685,000
867	12,795,646,829,899,000
868	12,707,672,632,154,500
869	12,618,775,799,682,300
870	12,528,983,557,886,000
871	12,438,323,228,868,000
872	12,346,822,293,578,000
873	12,254,508,302,876,300
874	12,161,408,939,178,500
875	12,067,551,928,151,000
876	11,972,965,099,875,100
877	11,877,676,301,179,600
878	11,781,713,456,388,200
879	11,685,104,480,338,700
880	11,587,877,338,692,700
881	11,490,059,961,730,100
882	11,391,680,304,241,600
883	11,292,766,260,101,800
884	11,193,345,721,807,600
885	11,093,446,495,874,100
886	10,993,096,361,994,600
887	10,892,322,989,344,500
888	10,791,153,995,382,700
889	10,689,616,863,061,700
890	10,587,738,999,328,200
891	10,485,547,653,279,600
892	10,383,069,974,340,200
893	10,280,332,931,441,700
894	10,177,363,370,888,400
895	10,074,187,936,558,100
896	9,970,833,127,512,510
897	9,867,325,219,249,980
898	9,763,690,321,033,400
899	9,659,954,298,267,300
900	9,556,142,829,555,640
901	9,452,281,330,191,890
902	9,348,395,008,996,000
903	9,244,508,792,945,670
904	9,140,647,383,762,520
905	9,036,835,183,748,460
906	8,933,096,352,127,630
907	8,829,454,732,072,480
908	8,725,933,906,847,840
909	8,622,557,128,046,050
910	8,519,347,371,498,560
911	8,416,327,266,778,080
912	8,313,519,152,881,930
913	8,210,945,008,978,190
914	8,108,626,509,899,880
915	8,006,584,958,150,180
916	7,904,841,339,170,870
917	7,803,416,254,655,220
918	7,702,329,977,589,080
919	7,601,602,386,843,980
920	7,501,253,022,027,260
921	7,401,301,019,361,720
922	7,301,765,166,303,070
923	7,202,663,838,749,380
924	7,104,015,055,421,620
925	7,005,836,416,368,480
926	6,908,145,157,139,450
927	6,810,958,088,586,950
928	6,714,291,650,789,160
929	6,618,161,854,183,300
930	6,522,584,333,229,140
931	6,427,574,288,835,360
932	6,333,146,541,790,360
933	6,239,315,476,476,150
934	6,146,095,094,019,120
935	6,053,498,957,320,960
936	5,961,540,243,918,360
937	5,870,231,692,286,640
938	5,779,585,654,429,600
939	5,689,614,043,446,160
940	5,600,328,385,803,350
941	5,511,739,770,187,560
942	5,423,858,899,445,150
943	5,336,696,040,671,980
944	5,250,261,076,840,900
945	5,164,563,458,113,970
946	5,079,612,253,106,590
947	4,995,416,101,440,400
948	4,911,983,264,628,410
949	4,829,321,579,817,140
950	4,747,438,510,307,980
951	4,666,341,100,470,600
952	4,586,036,025,851,500
953	4,506,529,549,270,560
954	4,427,827,570,497,980
955	4,349,935,583,482,260
956	4,272,858,725,610,060
957	4,196,601,736,043,210
958	4,121,169,004,511,070
959	4,046,564,530,766,010
960	3,972,791,972,891,010
961	3,899,854,607,818,590
962	3,827,755,379,165,350
963	3,756,496,858,791,470
964	3,686,081,294,113,980
965	3,616,510,570,712,250
966	3,547,786,259,101,070
967	3,479,909,578,328,190
968	3,412,881,442,219,840
969	3,346,702,423,944,310
970	3,281,372,801,682,290
971	3,216,892,524,160,980
972	3,153,261,255,732,850
973	3,090,478,342,825,040
974	3,028,542,858,437,460
975	2,967,453,569,482,800
976	2,907,208,980,659,850
977	2,847,807,302,685,920
978	2,789,246,495,528,880
979	2,731,524,237,478,950
980	2,674,637,967,752,940
981	2,618,584,856,379,090
982	2,563,361,846,130,740
983	2,508,965,623,226,930
984	2,455,392,658,582,210
985	2,402,639,179,270,460
986	2,350,701,209,103,640
987	2,299,574,540,834,780
988	2,249,254,776,027,770
989	2,199,737,298,000,370
990	2,151,017,310,973,090
991	2,103,089,813,703,010
992	2,055,949,637,927,470
993	2,009,591,422,664,540
994	1,964,009,651,915,660
995	1,919,198,629,635,730
996	1,875,152,516,686,540
997	1,831,865,306,428,570
998	1,789,330,860,941,280
999	1,747,542,887,215,360
1000	1,706,494,972,609,520
1001	1,666,180,561,645,290
1002	1,626,592,990,693,830
1003	1,587,725,465,329,520
1004	1,549,571,094,266,050
1005	1,512,122,867,248,400
1006	1,475,373,688,210,190
1007	1,439,316,353,708,700
1008	1,403,943,585,301,200
1009	1,369,248,008,480,890
1010	1,335,222,184,292,440
1011	1,301,858,588,751,630
1012	1,269,149,643,677,450
1013	1,237,087,696,599,980
1014	1,205,665,050,808,210
1015	1,174,873,945,708,900
1016	1,144,706,586,113,290
1017	1,115,155,123,030,640
1018	1,086,211,682,173,950
1019	1,057,868,345,195,040
1020	1,030,117,177,412,690
1021	1,002,950,209,453,770
1022	976,359,464,227,661
1023	950,336,938,960,611
1024	924,874,631,401,209
1025	899,964,522,255,079
1026	875,598,600,626,154
1027	851,768,846,820,577
1028	828,467,257,050,350
1029	805,685,826,594,439
1030	783,416,573,752,126
1031	761,651,523,370,470
1032	740,382,730,055,707
1033	719,602,262,037,791
1034	699,302,223,666,421
1035	679,474,739,604,064
1036	660,111,976,597,920
1037	641,206,128,010,489
1038	622,749,434,876,858
1039	604,734,170,747,184
1040	587,152,662,058,258
1041	569,997,273,279,789
1042	553,260,426,593,322
1043	536,934,587,353,256
1044	521,012,283,085,036
1045	505,486,089,237,058
1046	490,348,647,526,558
1047	475,592,651,976,889
1048	461,210,866,607,609
1049	447,196,111,767,398
1050	433,541,281,180,913
1051	420,239,328,556,562
1052	407,283,284,019,903
1053	394,666,240,991,089
1054	382,381,372,002,154
1055	370,421,915,855,964
1056	358,781,192,841,696
1057	347,452,592,155,562
1058	336,429,586,543,371
1059	325,705,719,979,621
1060	315,274,621,737,283
1061	305,129,994,336,046
1062	295,265,627,053,394
1063	285,675,384,124,940
1064	276,353,217,726,061
1065	267,293,156,421,037
1066	258,489,317,615,563
1067	249,935,896,266,269
1068	241,627,176,819,381
1069	233,557,522,163,930
1070	225,721,385,084,601
1071	218,113,297,456,569
1072	210,727,881,214,639
1073	203,559,837,800,120
1074	196,603,958,660,512
1075	189,855,114,934,011
1076	183,308,267,507,535
1077	176,958,456,936,318
1078	170,800,813,062,207
1079	164,830,545,178,892
1080	159,042,951,225,485
1081	153,433,408,182,971
1082	147,997,380,868,998
1083	142,730,412,563,985
1084	137,628,133,409,981
1085	132,686,251,275,763
1086	127,900,559,773,363
1087	123,266,929,349,329
1088	118,781,314,944,316
1089	114,439,747,308,027
1090	110,238,340,304,328
1091	106,173,282,459,017
1092	102,240,843,924,128
1093	98,437,368,245,632
1094	94,759,279,010,024
1095	91,203,071,830,294
1096	87,765,320,677,722
1097	84,442,670,093,739
1098	81,231,841,219,122
1099	78,129,624,219,915
1100	75,132,884,036,461
1101	72,238,553,020,681
1102	69,443,636,407,019
1103	66,745,205,168,528
1104	64,140,401,221,500
1105	61,626,430,488,492
1106	59,200,567,856,702
1107	56,860,150,445,667
1108	54,602,582,321,801
1109	52,425,327,976,532
1110	50,325,916,807,381
1111	48,301,936,795,528
1112	46,351,038,772,509
1113	44,470,930,294,083
1114	42,659,379,693,854
1115	40,914,210,159,126
1116	39,233,303,581,272
1117	37,614,594,822,327
1118	36,056,075,379,039
1119	34,555,787,837,266
1120	33,111,829,351,342
1121	31,722,346,290,416
1122	30,385,537,541,434
1123	29,099,649,337,087
1124	27,862,978,398,196
1125	26,673,866,939,713
1126	25,530,705,653,070
1127	24,431,928,894,154
1128	23,376,017,513,498
1129	22,361,494,215,565
1130	21,386,926,250,737
1131	20,450,920,942,555
1132	19,552,128,242,181
1133	18,689,236,426,307
1134	17,860,974,520,443
1135	17,066,108,157,849
1136	16,303,441,884,554
1137	15,571,815,175,325
1138	14,870,104,620,270
1139	14,197,220,100,303
1140	13,552,106,861,442
1141	12,933,741,839,899
1142	12,341,135,635,235
1143	11,773,328,981,483
1144	11,229,394,619,008
1145	10,708,433,913,262
1146	10,209,578,630,140
1147	9,731,987,693,363
1148	9,274,848,873,719
1149	8,837,375,680,898
1150	8,418,808,965,914
1151	8,018,413,948,848
1152	7,635,481,738,958
1153	7,269,326,489,361
1154	6,919,286,843,610
1155	6,584,723,213,704
1156	6,265,019,152,487
1157	5,959,578,755,669
1158	5,667,827,963,566
1159	5,389,212,078,927
1160	5,123,197,006,289
1161	4,869,266,881,582
1162	4,626,925,247,931
1163	4,395,692,797,816
1164	4,175,108,488,550
1165	3,964,727,389,027
1166	3,764,121,741,811
1167	3,572,878,910,940
1168	3,390,602,389,686
1169	3,216,909,849,508
1170	3,051,434,096,031
1171	2,893,821,211,946
1172	2,743,731,467,669
1173	2,600,837,554,495
1174	2,464,825,448,513
1175	2,335,392,734,382
1176	2,212,249,424,868
1177	2,095,116,368,246
1178	1,983,726,028,916
1179	1,877,820,975,168
1180	1,777,154,619,819
1181	1,681,489,788,194
1182	1,590,599,420,536
1183	1,504,265,214,046
1184	1,422,278,292,091
1185	1,344,437,917,067
1186	1,270,552,124,993
1187	1,200,436,509,276
1188	1,133,914,822,434
1189	1,070,817,824,773
1190	1,010,983,857,544
1191	954,257,753,733
1192	900,491,381,529
1193	849,542,617,012
1194	801,275,859,118
1195	755,561,059,108
1196	712,274,211,009
1197	671,296,435,063
1198	632,514,442,353
1199	595,819,672,189
1200	561,108,732,221
1201	528,282,584,833
1202	497,246,966,079
1203	467,911,618,783
1204	440,190,689,221
1205	414,002,006,749
1206	389,267,459,099
1207	365,912,314,318
1208	343,865,577,844
1209	323,059,354,425
1210	303,429,185,857
1211	284,913,452,879
1212	267,453,694,512
1213	250,994,045,973
1214	235,481,542,190
1215	220,865,589,927
1216	207,098,254,570
1217	194,133,766,301
1218	181,928,790,813
1219	170,441,966,142
1220	159,634,159,836
1221	149,468,034,620
1222	139,908,291,043
1223	130,921,262,045
1224	122,475,141,756
1225	114,539,605,867
1226	107,086,028,628
1227	100,087,127,594
1228	93,517,168,095
1229	87,351,632,249
1230	81,567,411,419
1231	76,142,496,873
1232	71,056,162,189
1233	66,288,674,187
1234	61,821,464,449
1235	57,636,860,670
1236	53,718,247,833
1237	50,049,817,550
1238	46,616,720,518
1239	43,404,832,757
1240	40,400,898,786
1241	37,592,314,751
1242	34,967,262,684
1243	32,514,508,546
1244	30,223,529,082
1245	28,084,323,773
1246	26,087,533,643
1247	24,224,267,278
1248	22,486,212,019
1249	20,865,472,184
1250	19,354,673,930
1251	17,946,814,901
1252	16,635,362,300
1253	15,414,114,032
1254	14,277,290,236
1255	13,219,404,469
1256	12,235,349,889
1257	11,320,279,729
1258	10,469,687,646
1259	9,679,297,643
1260	8,945,138,936
1261	8,263,443,950
1262	7,630,718,671
1263	7,043,648,186
1264	6,499,162,145
1265	5,994,347,965
1266	5,526,511,602
1267	5,093,097,335
1268	4,691,744,778
1269	4,320,214,692
1270	3,976,441,876
1271	3,658,467,191
1272	3,364,486,581
1273	3,092,788,314
1274	2,841,798,846
1275	2,610,024,936
1276	2,396,096,082
1277	2,198,711,618
1278	2,016,679,881
1279	1,848,869,496
1280	1,694,245,990
1281	1,551,826,889
1282	1,420,715,489
1283	1,300,059,954
1284	1,189,084,411
1285	1,087,051,347
1286	993,290,571
1287	907,164,667
1288	828,095,629
1289	755,533,468
1290	688,980,643
1291	627,963,266
1292	572,053,851
1293	520,844,871
1294	473,969,587
1295	431,078,138
1296	391,856,581
1297	356,005,000
1298	323,255,164
1299	293,350,487
1300	266,062,168
1301	241,171,101
1302	218,482,569
1303	197,809,729
1304	178,987,225
1305	161,856,093
1306	146,276,124
1307	132,112,328
1308	119,246,146
1309	107,563,103
1310	96,963,169
1311	87,349,501
1312	78,637,842
1313	70,746,444
1314	63,604,540
1315	57,143,201
1316	51,303,148
1317	46,026,431
1318	41,263,462
1319	36,965,620
1320	33,091,578
1321	29,600,587
1322	26,458,297
1323	23,630,664
1324	21,089,176
1325	18,805,464
1326	16,755,957
1327	14,917,024
1328	13,269,250
1329	11,793,028
1330	10,472,375
1331	9,291,060
1332	8,236,001
1333	7,293,767
1334	6,453,684
1335	5,704,686
1336	5,038,070
1337	4,444,748
1338	3,917,670
1339	3,449,359
1340	3,034,135
1341	2,665,885
1342	2,340,024
1343	2,051,569
1344	1,796,855
1345	1,571,812
1346	1,373,524
1347	1,198,689
1348	1,044,984
1349	909,741
1350	791,131
1351	686,983
1352	595,872
1353	516,054
1354	446,405
1355	385,528
1356	332,557
1357	286,364
1358	246,288
1359	211,428
1360	181,274
1361	155,112
1362	132,559
1363	113,039
1364	96,271
1365	81,801
1366	69,414
1367	58,754
1368	49,668
1369	41,869
1370	35,251
1371	29,588
1372	24,803
1373	20,722
1374	17,293
1375	14,375
1376	11,937
1377	9,871
1378	8,154
1379	6,703
1380	5,507
1381	4,498
1382	3,673
1383	2,980
1384	2,417
1385	1,946
1386	1,568
1387	1,251
1388	1,000
1389	791
1390	627
1391	490
1392	385
1393	297
1394	231
1395	176
1396	135
1397	101
1398	77
1399	56
1400	42
1401	30
1402	22
1403	15
1404	11
1405	7
1406	5
1407	3
1408	2
1409	1
1410	1

"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)

7craps

7craps

Threads: 18
Posts: 1977

Joined: Jan 23, 2010

May 20th, 2019 at 4:18:20 PM permalink

Quote: Wizard
Interesting. I wish Sally were still around to convert that to more plain simple English. Can anyone else help?

Sally seems to be long gone (moved to Kentucky) but you can try to email her. She stopped responding to mine.

here is where the formula was found
https://math.stackexchange.com/questions/2167789/ways-of-getting-a-sum-of-s-with-m-unique-numbers-in-the-range-1-n

This subject (Keno sums) does not interest me at the time. Maybe I will look closer into the generating function and see how it can be easily used.
I do not have Mathematica (but can use pari/gp). arbitrary precision is nice when it works within reason.
enjoy

winsome johnny (not Win some johnny)

Wizard
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Wizard

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Posts: 27036

Joined: Oct 14, 2009

May 20th, 2019 at 4:21:47 PM permalink

I'm afraid I just don't get that formula. I do have Sally's address, but don't want to bother her over this.

"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)

7craps

7craps

Threads: 18
Posts: 1977

Joined: Jan 23, 2010

May 20th, 2019 at 4:28:27 PM permalink

Quote: Wizard
I'm afraid I just don't get that formula.

prod(i=1, 80, 1 + x*y^i);
is a product function (like the sum())
i starts at 1 and goes to 80 and 1 + x*y^i is just the expression
that link I gave seems to give a better description than I could.
I am 100% certain you can code this in c++

back to baseball and NBA tonite

I do remember verifying her data at the time (I just did not do it myself)

winsome johnny (not Win some johnny)

Ayecarumba

Ayecarumba

Threads: 236
Posts: 6763

Joined: Nov 17, 2009

May 20th, 2019 at 5:00:27 PM permalink

Musings of a completely untrained mind...

Since there's only one way each to make the extreme totals 210 (balls 1-20) and 1410 (balls 61-80), I was expecting to see a perfectly smooth distribution of the instances of combinations to make the other totals with a peak at ~810. However, there are some totals with many more ways to make them compared to the one-off totals next to them. Is it because they are Primes? For example, there are 578 more ways to make a total of 456 than 455, but only 80 more ways to make 457 than 456. Can the existence of rarer totals be useful?

My gut tells me that a quick solution will involve assigning a weight to each ball reflecting the number of totals, not necessarily the totals themselves, that they can be part of. For example, ball #1 will be more valuable than ball #2 since it can be used to make the smallest incremental even and odd totals, but #2 can only make odd totals the next odd total, or even totals the next even.

Of course, that could be the Del Taco $4 value box I had for lunch talking.

Simplicity is the ultimate sophistication - Leonardo da Vinci

ThatDonGuy

ThatDonGuy

Threads: 122
Posts: 6678

Joined: Jun 22, 2011

May 20th, 2019 at 5:00:51 PM permalink

Quote: Wizard
I'm afraid I just don't get that formula. I do have Sally's address, but don't want to bother her over this.

I think I sort of do.
The formula is the product of:
1 + x y
1 + x y²
1 + x y³
1 + x y⁴
...
1 + x y⁷⁹
1 + x y⁸⁰

If you draw 20 particular balls, multiply the product of the left sides of the terms of the balls you did not draw and the right sides of the balls you did draw together; you will get x²⁰ and y^{sum of the numbers}.
To determine how many times 20 balls add up to some number N, count the number of times x²⁰ y^N is such a product.

How to get such a count without having to go through all 2⁸⁰ products, I don't know, and presumably, that's why things like Mathematica exist.

Wizard
Administrator

Wizard

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May 20th, 2019 at 5:02:54 PM permalink

Quote: 7craps
prod(i=1, 80, 1 + x*y^i);
is a product function (like the sum())
i starts at 1 and goes to 80 and 1 + x*y^i is just the expression
that link I gave seems to give a better description than I could.
I am 100% certain you can code this in c++

I'm afraid I just don't understand that post. Hopefully this thread will get more interest from the other math geniuses of the forum.

"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)

7craps

7craps

Threads: 18
Posts: 1977

Joined: Jan 23, 2010

May 20th, 2019 at 5:25:31 PM permalink

Quote: ThatDonGuy
How to get such a count without having to go through all 2⁸⁰ products, I don't know, and presumably, that's why things like Mathematica exist.

what you say is so true.
I looked for the original pari/gp text file that is created from that formula and I do not have it and do not want to make it. I think it was very large and a program as notepad++ only could open it and make things happen.
added: I made that file and it is only 1.9MB in a text file and took less than 1 second to make.

Here is what I do have for x^20 (draw 20)
y^sum is the sum and the coefficient is the number of ways that sum can occur.
this can be compared to by one who really wants to do this

This concept is about partitions

here is 2 numbers and 3 numbers expanded and it should make sense.

gp > a=prod(i=1, 2, 1 + x*y^i);
gp > b=prod(i=1, 3, 1 + x*y^i);
gp > a
%3 = y^3*x^2 + (y^2 + y)*x + 1
gp > b
%4 = y^6*x^3 + (y^5 + y^4 + y^3)*x^2 + (y^3 + y^2 + y)*x + 1

a=prod(i=1, 2, 1 + x*y^i); means we only have 2 numbers (1 and 2)
y^3*x^2 + (y^2 + y)*x + 1 means drawing 2 numbers (x^2) we have only 1 sum and that is y^3 = 3 and only 1 way to get it (1+2)
(y^2 + y)*x means we draw 1 number (x) and the sums are 2 and 1 and 1 way to get each

same concept with numbers {1,2,3}

x^20
1*y^1410
1*y^1409
2*y^1408
3*y^1407
5*y^1406
7*y^1405
11*y^1404
15*y^1403
22*y^1402
30*y^1401
42*y^1400
56*y^1399
77*y^1398
101*y^1397
135*y^1396
176*y^1395
231*y^1394
297*y^1393
385*y^1392
490*y^1391
627*y^1390
791*y^1389
1000*y^1388
1251*y^1387
1568*y^1386
1946*y^1385
2417*y^1384
2980*y^1383
3673*y^1382
4498*y^1381
5507*y^1380
6703*y^1379
8154*y^1378
9871*y^1377
11937*y^1376
14375*y^1375
17293*y^1374
20722*y^1373
24803*y^1372
29588*y^1371
35251*y^1370
41869*y^1369
49668*y^1368
58754*y^1367
69414*y^1366
81801*y^1365
96271*y^1364
113039*y^1363
132559*y^1362
155112*y^1361
181274*y^1360
211428*y^1359
246288*y^1358
286364*y^1357
332557*y^1356
385528*y^1355
446405*y^1354
516054*y^1353
595872*y^1352
686983*y^1351
791131*y^1350
909741*y^1349
1044984*y^1348
1198689*y^1347
1373524*y^1346
1571812*y^1345
1796855*y^1344
2051569*y^1343
2340024*y^1342
2665885*y^1341
3034135*y^1340
3449359*y^1339
3917670*y^1338
4444748*y^1337
5038070*y^1336
5704686*y^1335
6453684*y^1334
7293767*y^1333
8236001*y^1332
9291060*y^1331
10472375*y^1330
11793028*y^1329
13269250*y^1328
14917024*y^1327
16755957*y^1326
18805464*y^1325
21089176*y^1324
23630664*y^1323
26458297*y^1322
29600587*y^1321
33091578*y^1320
36965620*y^1319
41263462*y^1318
46026431*y^1317
51303148*y^1316
57143201*y^1315
63604540*y^1314
70746444*y^1313
78637842*y^1312
87349501*y^1311
96963169*y^1310
107563103*y^1309
119246146*y^1308
132112328*y^1307
146276124*y^1306
161856093*y^1305
178987225*y^1304
197809729*y^1303
218482569*y^1302
241171101*y^1301
266062168*y^1300
293350487*y^1299
323255164*y^1298
356005000*y^1297
391856581*y^1296
431078138*y^1295
473969587*y^1294
520844871*y^1293
572053851*y^1292
627963266*y^1291
688980643*y^1290
755533468*y^1289
828095629*y^1288
907164667*y^1287
993290571*y^1286
1087051347*y^1285
1189084411*y^1284
1300059954*y^1283
1420715489*y^1282
1551826889*y^1281
1694245990*y^1280
1848869496*y^1279
2016679881*y^1278
2198711618*y^1277
2396096082*y^1276
2610024936*y^1275
2841798846*y^1274
3092788314*y^1273
3364486581*y^1272
3658467191*y^1271
3976441876*y^1270
4320214692*y^1269
4691744778*y^1268
5093097335*y^1267
5526511602*y^1266
5994347965*y^1265
6499162145*y^1264
7043648186*y^1263
7630718671*y^1262
8263443950*y^1261
8945138936*y^1260
9679297643*y^1259
10469687646*y^1258
11320279729*y^1257
12235349889*y^1256
13219404469*y^1255
14277290236*y^1254
15414114032*y^1253
16635362300*y^1252
17946814901*y^1251
19354673930*y^1250
20865472184*y^1249
22486212019*y^1248
24224267278*y^1247
26087533643*y^1246
28084323773*y^1245
30223529082*y^1244
32514508546*y^1243
34967262684*y^1242
37592314751*y^1241
40400898786*y^1240
43404832757*y^1239
46616720518*y^1238
50049817550*y^1237
53718247833*y^1236
57636860670*y^1235
61821464449*y^1234
66288674187*y^1233
71056162189*y^1232
76142496873*y^1231
81567411419*y^1230
87351632249*y^1229
93517168095*y^1228
100087127594*y^1227
107086028628*y^1226
114539605867*y^1225
122475141756*y^1224
130921262045*y^1223
139908291043*y^1222
149468034620*y^1221
159634159836*y^1220
170441966142*y^1219
181928790813*y^1218
194133766301*y^1217
207098254570*y^1216
220865589927*y^1215
235481542190*y^1214
250994045973*y^1213
267453694512*y^1212
284913452879*y^1211
303429185857*y^1210
323059354425*y^1209
343865577844*y^1208
365912314318*y^1207
389267459099*y^1206
414002006749*y^1205
440190689221*y^1204
467911618783*y^1203
497246966079*y^1202
528282584833*y^1201
561108732221*y^1200
595819672189*y^1199
632514442353*y^1198
671296435063*y^1197
712274211009*y^1196
755561059108*y^1195
801275859118*y^1194
849542617012*y^1193
900491381529*y^1192
954257753733*y^1191
1010983857544*y^1190
1070817824773*y^1189
1133914822434*y^1188
1200436509276*y^1187
1270552124993*y^1186
1344437917067*y^1185
1422278292091*y^1184
1504265214046*y^1183
1590599420536*y^1182
1681489788194*y^1181
1777154619819*y^1180
1877820975168*y^1179
1983726028916*y^1178
2095116368246*y^1177
2212249424868*y^1176
2335392734382*y^1175
2464825448513*y^1174
2600837554495*y^1173
2743731467669*y^1172
2893821211946*y^1171
3051434096031*y^1170
3216909849508*y^1169
3390602389686*y^1168
3572878910940*y^1167
3764121741811*y^1166
3964727389027*y^1165
4175108488550*y^1164
4395692797816*y^1163
4626925247931*y^1162
4869266881582*y^1161
5123197006289*y^1160
5389212078927*y^1159
5667827963566*y^1158
5959578755669*y^1157
6265019152487*y^1156
6584723213704*y^1155
6919286843610*y^1154
7269326489361*y^1153
7635481738958*y^1152
8018413948848*y^1151
8418808965914*y^1150
8837375680898*y^1149
9274848873719*y^1148
9731987693363*y^1147
10209578630140*y^1146
10708433913262*y^1145
11229394619008*y^1144
11773328981483*y^1143
12341135635235*y^1142
12933741839899*y^1141
13552106861442*y^1140
14197220100303*y^1139
14870104620270*y^1138
15571815175325*y^1137
16303441884554*y^1136
17066108157849*y^1135
17860974520443*y^1134
18689236426307*y^1133
19552128242181*y^1132
20450920942555*y^1131
21386926250737*y^1130
22361494215565*y^1129
23376017513498*y^1128
24431928894154*y^1127
25530705653070*y^1126
26673866939713*y^1125
27862978398196*y^1124
29099649337087*y^1123
30385537541434*y^1122
31722346290416*y^1121
33111829351342*y^1120
34555787837266*y^1119
36056075379039*y^1118
37614594822327*y^1117
39233303581272*y^1116
40914210159126*y^1115
42659379693854*y^1114
44470930294083*y^1113
46351038772509*y^1112
48301936795528*y^1111
50325916807381*y^1110
52425327976532*y^1109
54602582321801*y^1108
56860150445667*y^1107
59200567856702*y^1106
61626430488492*y^1105
64140401221500*y^1104
66745205168528*y^1103
69443636407019*y^1102
72238553020681*y^1101
75132884036461*y^1100
78129624219915*y^1099
81231841219122*y^1098
84442670093739*y^1097
87765320677722*y^1096
91203071830294*y^1095
94759279010024*y^1094
98437368245632*y^1093
102240843924128*y^1092
106173282459017*y^1091
110238340304328*y^1090
114439747308027*y^1089
118781314944316*y^1088
123266929349329*y^1087
127900559773363*y^1086
132686251275763*y^1085
137628133409981*y^1084
142730412563985*y^1083
147997380868998*y^1082
153433408182971*y^1081
159042951225485*y^1080
164830545178892*y^1079
170800813062207*y^1078
176958456936318*y^1077
183308267507535*y^1076
189855114934011*y^1075
196603958660512*y^1074
203559837800120*y^1073
210727881214639*y^1072
218113297456569*y^1071
225721385084601*y^1070
233557522163930*y^1069
241627176819381*y^1068
249935896266269*y^1067
258489317615563*y^1066
267293156421037*y^1065
276353217726061*y^1064
285675384124940*y^1063
295265627053394*y^1062
305129994336046*y^1061
315274621737283*y^1060
325705719979621*y^1059
336429586543371*y^1058
347452592155562*y^1057
358781192841696*y^1056
370421915855964*y^1055
382381372002154*y^1054
394666240991089*y^1053
407283284019903*y^1052
420239328556562*y^1051
433541281180913*y^1050
447196111767398*y^1049
461210866607609*y^1048
475592651976889*y^1047
490348647526558*y^1046
505486089237058*y^1045
521012283085036*y^1044
536934587353256*y^1043
553260426593322*y^1042
569997273279789*y^1041
587152662058258*y^1040
604734170747184*y^1039
622749434876858*y^1038
641206128010489*y^1037
660111976597920*y^1036
679474739604064*y^1035
699302223666421*y^1034
719602262037791*y^1033
740382730055707*y^1032
761651523370470*y^1031
783416573752126*y^1030
805685826594439*y^1029
828467257050350*y^1028
851768846820577*y^1027
875598600626154*y^1026
899964522255079*y^1025
924874631401209*y^1024
950336938960611*y^1023
976359464227661*y^1022
1002950209453776*y^1021
1030117177412699*y^1020
1057868345195045*y^1019
1086211682173956*y^1018
1115155123030642*y^1017
1144706586113291*y^1016
1174873945708909*y^1015
1205665050808215*y^1014
1237087696599984*y^1013
1269149643677458*y^1012
1301858588751637*y^1011
1335222184292444*y^1010
1369248008480895*y^1009
1403943585301204*y^1008
1439316353708705*y^1007
1475373688210195*y^1006
1512122867248405*y^1005
1549571094266055*y^1004
1587725465329523*y^1003
1626592990693830*y^1002
1666180561645293*y^1001
1706494972609528*y^1000
1747542887215360*y^999
1789330860941288*y^998
1831865306428571*y^997
1875152516686548*y^996
1919198629635734*y^995
1964009651915660*y^994
2009591422664545*y^993
2055949637927477*y^992
2103089813703016*y^991
2151017310973098*y^990
2199737298000372*y^989
2249254776027773*y^988
2299574540834782*y^987
2350701209103646*y^986
2402639179270463*y^985
2455392658582214*y^984
2508965623226933*y^983
2563361846130743*y^982
2618584856379097*y^981
2674637967752948*y^980
2731524237478957*y^979
2789246495528881*y^978
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572053851*y^328
520844871*y^327
473969587*y^326
431078138*y^325
391856581*y^324
356005000*y^323
323255164*y^322
293350487*y^321
266062168*y^320
241171101*y^319
218482569*y^318
197809729*y^317
178987225*y^316
161856093*y^315
146276124*y^314
132112328*y^313
119246146*y^312
107563103*y^311
96963169*y^310
87349501*y^309
78637842*y^308
70746444*y^307
63604540*y^306
57143201*y^305
51303148*y^304
46026431*y^303
41263462*y^302
36965620*y^301
33091578*y^300
29600587*y^299
26458297*y^298
23630664*y^297
21089176*y^296
18805464*y^295
16755957*y^294
14917024*y^293
13269250*y^292
11793028*y^291
10472375*y^290
9291060*y^289
8236001*y^288
7293767*y^287
6453684*y^286
5704686*y^285
5038070*y^284
4444748*y^283
3917670*y^282
3449359*y^281
3034135*y^280
2665885*y^279
2340024*y^278
2051569*y^277
1796855*y^276
1571812*y^275
1373524*y^274
1198689*y^273
1044984*y^272
909741*y^271
791131*y^270
686983*y^269
595872*y^268
516054*y^267
446405*y^266
385528*y^265
332557*y^264
286364*y^263
246288*y^262
211428*y^261
181274*y^260
155112*y^259
132559*y^258
113039*y^257
96271*y^256
81801*y^255
69414*y^254
58754*y^253
49668*y^252
41869*y^251
35251*y^250
29588*y^249
24803*y^248
20722*y^247
17293*y^246
14375*y^245
11937*y^244
9871*y^243
8154*y^242
6703*y^241
5507*y^240
4498*y^239
3673*y^238
2980*y^237
2417*y^236
1946*y^235
1568*y^234
1251*y^233
1000*y^232
791*y^231
627*y^230
490*y^229
385*y^228
297*y^227
231*y^226
176*y^225
135*y^224
101*y^223
77*y^222
56*y^221
42*y^220
30*y^219
22*y^218
15*y^217
11*y^216
7*y^215
5*y^214
3*y^213
2*y^212
1*y^211
1*y^210

the distribution is symmetrical. 810 being the center value (well known to Keno folks)

Last edited by: 7craps on May 20, 2019

winsome johnny (not Win some johnny)

Wizard
Administrator

Wizard

Threads: 1518
Posts: 27036

Joined: Oct 14, 2009

May 20th, 2019 at 8:08:11 PM permalink

There is blood all over the wall from me trying to make sense of the above post. Please keep in mind I don't have Mathematica and I'm looking for a way to express the combinations in ordinary mathematical language.

"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)

7craps

7craps

Threads: 18
Posts: 1977

Joined: Jan 23, 2010

May 20th, 2019 at 8:37:37 PM permalink

Quote: Wizard
Please keep in mind I don't have Mathematica

LOL
Mathematica it is not required.
a program that can expand a product function accurately is all that is needed. That is why I use pari/gp as it is free (windows 64 version) and the learning curve is not steep at all if you already know another language.

Quote: Wizard
and I'm looking for a way to express the combinations in ordinary mathematical language.

Don showed why the formula works and it is actually very simple once you understand the concept of a generating function.

I guess I just got lucky as the light switch went on for me. On a scale of 1 to 10 , my math skills are about a 2
your math skills may be way different from mine, but this method of using a generating function is very basic, as generating functions go.

IF you can understand how a GF works with 2d6 and dice sums, you can get this product function.

(x+x^2+x^3+x^4+x^5+x^6)^2. most know how to expand this the long way. we learned this way back in school.
now we have computers that do it for us in 0 ms

you will get it in no time.

winsome johnny (not Win some johnny)

ThatDonGuy

ThatDonGuy

Threads: 122
Posts: 6678

Joined: Jun 22, 2011

May 20th, 2019 at 8:58:42 PM permalink

Quote: ThatDonGuy
How to get such a count without having to go through all 2⁸⁰ products, I don't know, and presumably, that's why things like Mathematica exist.

I did find this, however, which explains the problem and a way to find the solution

ZPP

ZPP

Threads: 2
Posts: 31

Joined: Feb 7, 2010

May 21st, 2019 at 12:12:49 AM permalink

Using a computer, this can be calculated using a recurrence:
Let A[n][k][m] be the number of ways to choose k numbers from {1..n} that sum to m.
Suppose we are trying to calculate A[n][k][m] and have already calculated all values for A[n-1][k][m].
We add together the number of ways that do not include n and the number of ways that do include n. This is exhaustive and disjoint.
1. The number of ways that do not include n is A[n-1][k][m], the number of ways to choose k numbers from {1..n-1} that sum to m.
2. If m>=n, the number of ways that do include n is A[n-1][k-1][m-n], the number of ways to choose k-1 numbers from {1..n-1} that sum to m-n. (If m<n, the number of ways to sum to m that do include n is 0.)

This is very fast: using bounds for n, k, and m of 80, 20, and 80*20 (which is overkill, could be 1410), the number of elements to calculate is 80*20*80*20=2,560,000, and each element is at most two array lookups and an addition. It takes less than 10 milliseconds on my laptop.

Here is a C program that does this calculation:

#include <stdio.h>
#include <stdint.h>

#define MAXK 20
#define MAXN 80
#define MAXM (MAXK*MAXN)

// Since 80 choose 20 < 2^64, we can get exact counts with 64-bit ints
// (but note that 80 choose 22 >= 2^64)
typedef uint64_t num_t;
// Just index by size_t
typedef size_t index_t;

// a[n][k][m] is the number of ways to choose k numbers from {1..n} that sum to m
num_t a[MAXN+1][MAXK+1][MAXM+1];
// (Note that because we only reference a[n-1] when calculating a[n],
// we only need two (current and previous) 2-dimensional arrays of size
// MAXK+1 by MAXM+1, not a 3-dimensional array, but nowadays the
// memory needed is nothing.)

int main(void)
{
    for (index_t n = 0; n <= MAXN; n++) {
        // For any set (including the empty set for n==0), there is 1
        // way to choose k==0 numbers and get a sum of 0 and no other
        // sum is possible by choosing 0 numbers.
        a[n][0][0] = 1;
        // Now calculate for choosing more than k>0 numbers.
        // We iterate k from 1 to min(n,MAXK).
        // Note that therefore we skip this loop if n==0.
        index_t maxk = (n < MAXK ? n : MAXK);
        for (index_t k = 1; k <= maxk; k++) {
            // maxm could be tighter, but this is already fast enough.
            index_t maxm = n * k;
            for (index_t m = 0; m <= maxm; m++) {
                // Calculate a[n][k][m] by adding the number of ways
                // that do not include n in the sum and the number of
                // ways that do include n in the sum, both of which
                // are already calculated in the array.
                //
                // The number of ways that do not include n is the
                // number of ways to choose k numbers from {1..n-1}
                // that sum to m.
                num_t c = a[n-1][k][m];
                // The number of ways that do include n is the number of
                // ways to choose k-1 numbers from {1..n-1} that sum
                // to m-n, or 0 if m<n.
                if (n <= m)
                    c += a[n-1][k-1][m-n];
                a[n][k][m] = c;
            }
        }
    }
    // Print results, which are a table listing M and the number of
    // ways of choosing MAXK numbers from {1..MAXN} that sum to M
    num_t total = 0;
    num_t *p = a[MAXN][MAXK];
    for (index_t m = 0; m <= MAXM; m++) {
        total += p[m];
        if (p[m] != 0)
            printf("%4zu %19llu\n", m, p[m]);
    }
    // Total number of ways can be checked to equal MAXN choose MAXK
    printf("Sum: %19llu\n", total);
    return 0;
}

Here are the results I get:

 210                   1
 211                   1
 212                   2
 213                   3
 214                   5
 215                   7
 216                  11
 217                  15
 218                  22
 219                  30
 220                  42
 221                  56
 222                  77
 223                 101
 224                 135
 225                 176
 226                 231
 227                 297
 228                 385
 229                 490
 230                 627
 231                 791
 232                1000
 233                1251
 234                1568
 235                1946
 236                2417
 237                2980
 238                3673
 239                4498
 240                5507
 241                6703
 242                8154
 243                9871
 244               11937
 245               14375
 246               17293
 247               20722
 248               24803
 249               29588
 250               35251
 251               41869
 252               49668
 253               58754
 254               69414
 255               81801
 256               96271
 257              113039
 258              132559
 259              155112
 260              181274
 261              211428
 262              246288
 263              286364
 264              332557
 265              385528
 266              446405
 267              516054
 268              595872
 269              686983
 270              791131
 271              909741
 272             1044984
 273             1198689
 274             1373524
 275             1571812
 276             1796855
 277             2051569
 278             2340024
 279             2665885
 280             3034135
 281             3449359
 282             3917670
 283             4444748
 284             5038070
 285             5704686
 286             6453684
 287             7293767
 288             8236001
 289             9291060
 290            10472375
 291            11793028
 292            13269250
 293            14917024
 294            16755957
 295            18805464
 296            21089176
 297            23630664
 298            26458297
 299            29600587
 300            33091578
 301            36965620
 302            41263462
 303            46026431
 304            51303148
 305            57143201
 306            63604540
 307            70746444
 308            78637842
 309            87349501
 310            96963169
 311           107563103
 312           119246146
 313           132112328
 314           146276124
 315           161856093
 316           178987225
 317           197809729
 318           218482569
 319           241171101
 320           266062168
 321           293350487
 322           323255164
 323           356005000
 324           391856581
 325           431078138
 326           473969587
 327           520844871
 328           572053851
 329           627963266
 330           688980643
 331           755533468
 332           828095629
 333           907164667
 334           993290571
 335          1087051347
 336          1189084411
 337          1300059954
 338          1420715489
 339          1551826889
 340          1694245990
 341          1848869496
 342          2016679881
 343          2198711618
 344          2396096082
 345          2610024936
 346          2841798846
 347          3092788314
 348          3364486581
 349          3658467191
 350          3976441876
 351          4320214692
 352          4691744778
 353          5093097335
 354          5526511602
 355          5994347965
 356          6499162145
 357          7043648186
 358          7630718671
 359          8263443950
 360          8945138936
 361          9679297643
 362         10469687646
 363         11320279729
 364         12235349889
 365         13219404469
 366         14277290236
 367         15414114032
 368         16635362300
 369         17946814901
 370         19354673930
 371         20865472184
 372         22486212019
 373         24224267278
 374         26087533643
 375         28084323773
 376         30223529082
 377         32514508546
 378         34967262684
 379         37592314751
 380         40400898786
 381         43404832757
 382         46616720518
 383         50049817550
 384         53718247833
 385         57636860670
 386         61821464449
 387         66288674187
 388         71056162189
 389         76142496873
 390         81567411419
 391         87351632249
 392         93517168095
 393        100087127594
 394        107086028628
 395        114539605867
 396        122475141756
 397        130921262045
 398        139908291043
 399        149468034620
 400        159634159836
 401        170441966142
 402        181928790813
 403        194133766301
 404        207098254570
 405        220865589927
 406        235481542190
 407        250994045973
 408        267453694512
 409        284913452879
 410        303429185857
 411        323059354425
 412        343865577844
 413        365912314318
 414        389267459099
 415        414002006749
 416        440190689221
 417        467911618783
 418        497246966079
 419        528282584833
 420        561108732221
 421        595819672189
 422        632514442353
 423        671296435063
 424        712274211009
 425        755561059108
 426        801275859118
 427        849542617012
 428        900491381529
 429        954257753733
 430       1010983857544
 431       1070817824773
 432       1133914822434
 433       1200436509276
 434       1270552124993
 435       1344437917067
 436       1422278292091
 437       1504265214046
 438       1590599420536
 439       1681489788194
 440       1777154619819
 441       1877820975168
 442       1983726028916
 443       2095116368246
 444       2212249424868
 445       2335392734382
 446       2464825448513
 447       2600837554495
 448       2743731467669
 449       2893821211946
 450       3051434096031
 451       3216909849508
 452       3390602389686
 453       3572878910940
 454       3764121741811
 455       3964727389027
 456       4175108488550
 457       4395692797816
 458       4626925247931
 459       4869266881582
 460       5123197006289
 461       5389212078927
 462       5667827963566
 463       5959578755669
 464       6265019152487
 465       6584723213704
 466       6919286843610
 467       7269326489361
 468       7635481738958
 469       8018413948848
 470       8418808965914
 471       8837375680898
 472       9274848873719
 473       9731987693363
 474      10209578630140
 475      10708433913262
 476      11229394619008
 477      11773328981483
 478      12341135635235
 479      12933741839899
 480      13552106861442
 481      14197220100303
 482      14870104620270
 483      15571815175325
 484      16303441884554
 485      17066108157849
 486      17860974520443
 487      18689236426307
 488      19552128242181
 489      20450920942555
 490      21386926250737
 491      22361494215565
 492      23376017513498
 493      24431928894154
 494      25530705653070
 495      26673866939713
 496      27862978398196
 497      29099649337087
 498      30385537541434
 499      31722346290416
 500      33111829351342
 501      34555787837266
 502      36056075379039
 503      37614594822327
 504      39233303581272
 505      40914210159126
 506      42659379693854
 507      44470930294083
 508      46351038772509
 509      48301936795528
 510      50325916807381
 511      52425327976532
 512      54602582321801
 513      56860150445667
 514      59200567856702
 515      61626430488492
 516      64140401221500
 517      66745205168528
 518      69443636407019
 519      72238553020681
 520      75132884036461
 521      78129624219915
 522      81231841219122
 523      84442670093739
 524      87765320677722
 525      91203071830294
 526      94759279010024
 527      98437368245632
 528     102240843924128
 529     106173282459017
 530     110238340304328
 531     114439747308027
 532     118781314944316
 533     123266929349329
 534     127900559773363
 535     132686251275763
 536     137628133409981
 537     142730412563985
 538     147997380868998
 539     153433408182971
 540     159042951225485
 541     164830545178892
 542     170800813062207
 543     176958456936318
 544     183308267507535
 545     189855114934011
 546     196603958660512
 547     203559837800120
 548     210727881214639
 549     218113297456569
 550     225721385084601
 551     233557522163930
 552     241627176819381
 553     249935896266269
 554     258489317615563
 555     267293156421037
 556     276353217726061
 557     285675384124940
 558     295265627053394
 559     305129994336046
 560     315274621737283
 561     325705719979621
 562     336429586543371
 563     347452592155562
 564     358781192841696
 565     370421915855964
 566     382381372002154
 567     394666240991089
 568     407283284019903
 569     420239328556562
 570     433541281180913
 571     447196111767398
 572     461210866607609
 573     475592651976889
 574     490348647526558
 575     505486089237058
 576     521012283085036
 577     536934587353256
 578     553260426593322
 579     569997273279789
 580     587152662058258
 581     604734170747184
 582     622749434876858
 583     641206128010489
 584     660111976597920
 585     679474739604064
 586     699302223666421
 587     719602262037791
 588     740382730055707
 589     761651523370470
 590     783416573752126
 591     805685826594439
 592     828467257050350
 593     851768846820577
 594     875598600626154
 595     899964522255079
 596     924874631401209
 597     950336938960611
 598     976359464227661
 599    1002950209453776
 600    1030117177412699
 601    1057868345195045
 602    1086211682173956
 603    1115155123030642
 604    1144706586113291
 605    1174873945708909
 606    1205665050808215
 607    1237087696599984
 608    1269149643677458
 609    1301858588751637
 610    1335222184292444
 611    1369248008480895
 612    1403943585301204
 613    1439316353708705
 614    1475373688210195
 615    1512122867248405
 616    1549571094266055
 617    1587725465329523
 618    1626592990693830
 619    1666180561645293
 620    1706494972609528
 621    1747542887215360
 622    1789330860941288
 623    1831865306428571
 624    1875152516686548
 625    1919198629635734
 626    1964009651915660
 627    2009591422664545
 628    2055949637927477
 629    2103089813703016
 630    2151017310973098
 631    2199737298000372
 632    2249254776027773
 633    2299574540834782
 634    2350701209103646
 635    2402639179270463
 636    2455392658582214
 637    2508965623226933
 638    2563361846130743
 639    2618584856379097
 640    2674637967752948
 641    2731524237478957
 642    2789246495528881
 643    2847807302685929
 644    2907208980659856
 645    2967453569482805
 646    3028542858437463
 647    3090478342825046
 648    3153261255732853
 649    3216892524160984
 650    3281372801682291
 651    3346702423944313
 652    3412881442219846
 653    3479909578328192
 654    3547786259101073
 655    3616510570712255
 656    3686081294113980
 657    3756496858791478
 658    3827755379165355
 659    3899854607818599
 660    3972791972891010
 661    4046564530766019
 662    4121169004511073
 663    4196601736043217
 664    4272858725610066
 665    4349935583482269
 666    4427827570497986
 667    4506529549270567
 668    4586036025851501
 669    4666341100470600
 670    4747438510307985
 671    4829321579817147
 672    4911983264628415
 673    4995416101440403
 674    5079612253106599
 675    5164563458113973
 676    5250261076840903
 677    5336696040671980
 678    5423858899445150
 679    5511739770187560
 680    5600328385803359
 681    5689614043446161
 682    5779585654429600
 683    5870231692286645
 684    5961540243918360
 685    6053498957320960
 686    6146095094019128
 687    6239315476476153
 688    6333146541790366
 689    6427574288835366
 690    6522584333229145
 691    6618161854183309
 692    6714291650789160
 693    6810958088586958
 694    6908145157139459
 695    7005836416368480
 696    7104015055421622
 697    7202663838749386
 698    7301765166303075
 699    7401301019361722
 700    7501253022027266
 701    7601602386843987
 702    7702329977589080
 703    7803416254655222
 704    7904841339170873
 705    8006584958150181
 706    8108626509899889
 707    8210945008978195
 708    8313519152881939
 709    8416327266778085
 710    8519347371498567
 711    8622557128046053
 712    8725933906847840
 713    8829454732072484
 714    8933096352127635
 715    9036835183748460
 716    9140647383762528
 717    9244508792945678
 718    9348395008996007
 719    9452281330191897
 720    9556142829555643
 721    9659954298267308
 722    9763690321033408
 723    9867325219249981
 724    9970833127512512
 725   10074187936558122
 726   10177363370888455
 727   10280332931441733
 728   10383069974340297
 729   10485547653279647
 730   10587738999328217
 731   10689616863061737
 732   10791153995382729
 733   10892322989344543
 734   10993096361994634
 735   11093446495874170
 736   11193345721807668
 737   11292766260101878
 738   11391680304241666
 739   11490059961730183
 740   11587877338692722
 741   11685104480338701
 742   11781713456388257
 743   11877676301179640
 744   11972965099875122
 745   12067551928151070
 746   12161408939178544
 747   12254508302876326
 748   12346822293578054
 749   12438323228868060
 750   12528983557886059
 751   12618775799682379
 752   12707672632154594
 753   12795646829899064
 754   12882671353685064
 755   12968719287824486
 756   13053763930133222
 757   13137778728752338
 758   13220737372587114
 759   13302613727557896
 760   13383381927422176
 761   13463016309475063
 762   13541491505700427
 763   13618782377856223
 764   13694864108945970
 765   13769712137666260
 766   13843302250100143
 767   13915610513550433
 768   13986613368402674
 769   14056287561278035
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 772   14257111362481292
 773   14321245963481667
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 781   14780811277822859
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 785   14972946514599961
 786   15016871640574771
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 803   15497333780632956
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 813   15534409532379142
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 815   15519568828416239
 816   15509373897951036
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 824   15361822703122288
 825   15335226479373283
 826   15306845867323822
 827   15276690710548182
 828   15244771498541818
 829   15211099277556439
 830   15175685727506919
 831   15138543072303315
 832   15099684155916740
 833   15059122352292240
 834   15016871640574771
 835   14972946514599961
 836   14927362057331041
 837   14880133849969280
 838   14831278045561927
 839   14780811277822859
 840   14728750733921081
 841   14675114063011110
 842   14619919448256126
 843   14563185515116493
 844   14504931402574878
 845   14445176671275633
 846   14383941373963924
 847   14321245963481667
 848   14257111362481292
 849   14191558871327486
 850   14124610237055954
 851   14056287561278035
 852   13986613368402674
 853   13915610513550433
 854   13843302250100143
 855   13769712137666260
 856   13694864108945970
 857   13618782377856223
 858   13541491505700427
 859   13463016309475063
 860   13383381927422176
 861   13302613727557896
 862   13220737372587114
 863   13137778728752338
 864   13053763930133222
 865   12968719287824486
 866   12882671353685064
 867   12795646829899064
 868   12707672632154594
 869   12618775799682379
 870   12528983557886059
 871   12438323228868060
 872   12346822293578054
 873   12254508302876326
 874   12161408939178544
 875   12067551928151070
 876   11972965099875122
 877   11877676301179640
 878   11781713456388257
 879   11685104480338701
 880   11587877338692722
 881   11490059961730183
 882   11391680304241666
 883   11292766260101878
 884   11193345721807668
 885   11093446495874170
 886   10993096361994634
 887   10892322989344543
 888   10791153995382729
 889   10689616863061737
 890   10587738999328217
 891   10485547653279647
 892   10383069974340297
 893   10280332931441733
 894   10177363370888455
 895   10074187936558122
 896    9970833127512512
 897    9867325219249981
 898    9763690321033408
 899    9659954298267308
 900    9556142829555643
 901    9452281330191897
 902    9348395008996007
 903    9244508792945678
 904    9140647383762528
 905    9036835183748460
 906    8933096352127635
 907    8829454732072484
 908    8725933906847840
 909    8622557128046053
 910    8519347371498567
 911    8416327266778085
 912    8313519152881939
 913    8210945008978195
 914    8108626509899889
 915    8006584958150181
 916    7904841339170873
 917    7803416254655222
 918    7702329977589080
 919    7601602386843987
 920    7501253022027266
 921    7401301019361722
 922    7301765166303075
 923    7202663838749386
 924    7104015055421622
 925    7005836416368480
 926    6908145157139459
 927    6810958088586958
 928    6714291650789160
 929    6618161854183309
 930    6522584333229145
 931    6427574288835366
 932    6333146541790366
 933    6239315476476153
 934    6146095094019128
 935    6053498957320960
 936    5961540243918360
 937    5870231692286645
 938    5779585654429600
 939    5689614043446161
 940    5600328385803359
 941    5511739770187560
 942    5423858899445150
 943    5336696040671980
 944    5250261076840903
 945    5164563458113973
 946    5079612253106599
 947    4995416101440403
 948    4911983264628415
 949    4829321579817147
 950    4747438510307985
 951    4666341100470600
 952    4586036025851501
 953    4506529549270567
 954    4427827570497986
 955    4349935583482269
 956    4272858725610066
 957    4196601736043217
 958    4121169004511073
 959    4046564530766019
 960    3972791972891010
 961    3899854607818599
 962    3827755379165355
 963    3756496858791478
 964    3686081294113980
 965    3616510570712255
 966    3547786259101073
 967    3479909578328192
 968    3412881442219846
 969    3346702423944313
 970    3281372801682291
 971    3216892524160984
 972    3153261255732853
 973    3090478342825046
 974    3028542858437463
 975    2967453569482805
 976    2907208980659856
 977    2847807302685929
 978    2789246495528881
 979    2731524237478957
 980    2674637967752948
 981    2618584856379097
 982    2563361846130743
 983    2508965623226933
 984    2455392658582214
 985    2402639179270463
 986    2350701209103646
 987    2299574540834782
 988    2249254776027773
 989    2199737298000372
 990    2151017310973098
 991    2103089813703016
 992    2055949637927477
 993    2009591422664545
 994    1964009651915660
 995    1919198629635734
 996    1875152516686548
 997    1831865306428571
 998    1789330860941288
 999    1747542887215360
1000    1706494972609528
1001    1666180561645293
1002    1626592990693830
1003    1587725465329523
1004    1549571094266055
1005    1512122867248405
1006    1475373688210195
1007    1439316353708705
1008    1403943585301204
1009    1369248008480895
1010    1335222184292444
1011    1301858588751637
1012    1269149643677458
1013    1237087696599984
1014    1205665050808215
1015    1174873945708909
1016    1144706586113291
1017    1115155123030642
1018    1086211682173956
1019    1057868345195045
1020    1030117177412699
1021    1002950209453776
1022     976359464227661
1023     950336938960611
1024     924874631401209
1025     899964522255079
1026     875598600626154
1027     851768846820577
1028     828467257050350
1029     805685826594439
1030     783416573752126
1031     761651523370470
1032     740382730055707
1033     719602262037791
1034     699302223666421
1035     679474739604064
1036     660111976597920
1037     641206128010489
1038     622749434876858
1039     604734170747184
1040     587152662058258
1041     569997273279789
1042     553260426593322
1043     536934587353256
1044     521012283085036
1045     505486089237058
1046     490348647526558
1047     475592651976889
1048     461210866607609
1049     447196111767398
1050     433541281180913
1051     420239328556562
1052     407283284019903
1053     394666240991089
1054     382381372002154
1055     370421915855964
1056     358781192841696
1057     347452592155562
1058     336429586543371
1059     325705719979621
1060     315274621737283
1061     305129994336046
1062     295265627053394
1063     285675384124940
1064     276353217726061
1065     267293156421037
1066     258489317615563
1067     249935896266269
1068     241627176819381
1069     233557522163930
1070     225721385084601
1071     218113297456569
1072     210727881214639
1073     203559837800120
1074     196603958660512
1075     189855114934011
1076     183308267507535
1077     176958456936318
1078     170800813062207
1079     164830545178892
1080     159042951225485
1081     153433408182971
1082     147997380868998
1083     142730412563985
1084     137628133409981
1085     132686251275763
1086     127900559773363
1087     123266929349329
1088     118781314944316
1089     114439747308027
1090     110238340304328
1091     106173282459017
1092     102240843924128
1093      98437368245632
1094      94759279010024
1095      91203071830294
1096      87765320677722
1097      84442670093739
1098      81231841219122
1099      78129624219915
1100      75132884036461
1101      72238553020681
1102      69443636407019
1103      66745205168528
1104      64140401221500
1105      61626430488492
1106      59200567856702
1107      56860150445667
1108      54602582321801
1109      52425327976532
1110      50325916807381
1111      48301936795528
1112      46351038772509
1113      44470930294083
1114      42659379693854
1115      40914210159126
1116      39233303581272
1117      37614594822327
1118      36056075379039
1119      34555787837266
1120      33111829351342
1121      31722346290416
1122      30385537541434
1123      29099649337087
1124      27862978398196
1125      26673866939713
1126      25530705653070
1127      24431928894154
1128      23376017513498
1129      22361494215565
1130      21386926250737
1131      20450920942555
1132      19552128242181
1133      18689236426307
1134      17860974520443
1135      17066108157849
1136      16303441884554
1137      15571815175325
1138      14870104620270
1139      14197220100303
1140      13552106861442
1141      12933741839899
1142      12341135635235
1143      11773328981483
1144      11229394619008
1145      10708433913262
1146      10209578630140
1147       9731987693363
1148       9274848873719
1149       8837375680898
1150       8418808965914
1151       8018413948848
1152       7635481738958
1153       7269326489361
1154       6919286843610
1155       6584723213704
1156       6265019152487
1157       5959578755669
1158       5667827963566
1159       5389212078927
1160       5123197006289
1161       4869266881582
1162       4626925247931
1163       4395692797816
1164       4175108488550
1165       3964727389027
1166       3764121741811
1167       3572878910940
1168       3390602389686
1169       3216909849508
1170       3051434096031
1171       2893821211946
1172       2743731467669
1173       2600837554495
1174       2464825448513
1175       2335392734382
1176       2212249424868
1177       2095116368246
1178       1983726028916
1179       1877820975168
1180       1777154619819
1181       1681489788194
1182       1590599420536
1183       1504265214046
1184       1422278292091
1185       1344437917067
1186       1270552124993
1187       1200436509276
1188       1133914822434
1189       1070817824773
1190       1010983857544
1191        954257753733
1192        900491381529
1193        849542617012
1194        801275859118
1195        755561059108
1196        712274211009
1197        671296435063
1198        632514442353
1199        595819672189
1200        561108732221
1201        528282584833
1202        497246966079
1203        467911618783
1204        440190689221
1205        414002006749
1206        389267459099
1207        365912314318
1208        343865577844
1209        323059354425
1210        303429185857
1211        284913452879
1212        267453694512
1213        250994045973
1214        235481542190
1215        220865589927
1216        207098254570
1217        194133766301
1218        181928790813
1219        170441966142
1220        159634159836
1221        149468034620
1222        139908291043
1223        130921262045
1224        122475141756
1225        114539605867
1226        107086028628
1227        100087127594
1228         93517168095
1229         87351632249
1230         81567411419
1231         76142496873
1232         71056162189
1233         66288674187
1234         61821464449
1235         57636860670
1236         53718247833
1237         50049817550
1238         46616720518
1239         43404832757
1240         40400898786
1241         37592314751
1242         34967262684
1243         32514508546
1244         30223529082
1245         28084323773
1246         26087533643
1247         24224267278
1248         22486212019
1249         20865472184
1250         19354673930
1251         17946814901
1252         16635362300
1253         15414114032
1254         14277290236
1255         13219404469
1256         12235349889
1257         11320279729
1258         10469687646
1259          9679297643
1260          8945138936
1261          8263443950
1262          7630718671
1263          7043648186
1264          6499162145
1265          5994347965
1266          5526511602
1267          5093097335
1268          4691744778
1269          4320214692
1270          3976441876
1271          3658467191
1272          3364486581
1273          3092788314
1274          2841798846
1275          2610024936
1276          2396096082
1277          2198711618
1278          2016679881
1279          1848869496
1280          1694245990
1281          1551826889
1282          1420715489
1283          1300059954
1284          1189084411
1285          1087051347
1286           993290571
1287           907164667
1288           828095629
1289           755533468
1290           688980643
1291           627963266
1292           572053851
1293           520844871
1294           473969587
1295           431078138
1296           391856581
1297           356005000
1298           323255164
1299           293350487
1300           266062168
1301           241171101
1302           218482569
1303           197809729
1304           178987225
1305           161856093
1306           146276124
1307           132112328
1308           119246146
1309           107563103
1310            96963169
1311            87349501
1312            78637842
1313            70746444
1314            63604540
1315            57143201
1316            51303148
1317            46026431
1318            41263462
1319            36965620
1320            33091578
1321            29600587
1322            26458297
1323            23630664
1324            21089176
1325            18805464
1326            16755957
1327            14917024
1328            13269250
1329            11793028
1330            10472375
1331             9291060
1332             8236001
1333             7293767
1334             6453684
1335             5704686
1336             5038070
1337             4444748
1338             3917670
1339             3449359
1340             3034135
1341             2665885
1342             2340024
1343             2051569
1344             1796855
1345             1571812
1346             1373524
1347             1198689
1348             1044984
1349              909741
1350              791131
1351              686983
1352              595872
1353              516054
1354              446405
1355              385528
1356              332557
1357              286364
1358              246288
1359              211428
1360              181274
1361              155112
1362              132559
1363              113039
1364               96271
1365               81801
1366               69414
1367               58754
1368               49668
1369               41869
1370               35251
1371               29588
1372               24803
1373               20722
1374               17293
1375               14375
1376               11937
1377                9871
1378                8154
1379                6703
1380                5507
1381                4498
1382                3673
1383                2980
1384                2417
1385                1946
1386                1568
1387                1251
1388                1000
1389                 791
1390                 627
1391                 490
1392                 385
1393                 297
1394                 231
1395                 176
1396                 135
1397                 101
1398                  77
1399                  56
1400                  42
1401                  30
1402                  22
1403                  15
1404                  11
1405                   7
1406                   5
1407                   3
1408                   2
1409                   1
1410                   1
Sum: 3535316142212174320

Comments:

There is a direct correspondence between this method and the method others have posted of expanding a very large polynomial and looking at certain coefficients. But I wrote this code before realizing how direct the correspondence was, and also no one else has really explained how an efficient calculation of the coefficients works, so here we are.

With regard to the part of the comment by ThatDonGuy specifically about looking at all 2⁸⁰ products, this is not necessary because the largest term, expanding the full polynomial that includes sums of more the 20 balls, is x⁸⁰y¹⁴¹⁰. If you collect like terms, there are "only" 81*1411=114291 terms in the full product. If you collect like terms at each stage (each multiplication by (1+xyⁿ)), the number of terms remains manageable at each stage.

However, I think ThatDonGuy has a great explanation of why the polynomial expansion works and I have nothing to add to it. (I just have an explanation of how to efficiently do this calculation.)

weezrDASvegas

weezrDASvegas

Threads: 2
Posts: 69

Joined: Feb 2, 2018

May 21st, 2019 at 2:25:23 AM permalink

Quote: Wizard
I am doing a software review of a company that offers prop bets on the sum of the balls in keno. As a reminder, in keno the game draws 20 numbers from a 80 balls numbered 1 to 80. The problem at hand is finding a probability of any given total without using a simulation.

I do know simulations are a perfectly valid way of analyzing casino games, but I also feel more satisfied finding an exact answer. At first I wrote a simple program to loop through 3,535,316,142,212,180,000 possible ways to choose 20 out of 80 balls. Needless to say, it would have taken centuries to cycle through them all. I've also toyed with shortcut code, but I fear I'll die before the program is done running.

I've also just counted combinations by hand, hoping to find a pattern (maybe some variant of Fibonaci), but so far have not found anything. So, at this point I open it up to the forum. You do not have to put replies in spoiler tags and may refer to any outside sources. This isn't the usual math puzzle thread.

The question for the poll is what are your thoughts on it?

In this spoiler tag are the results of my simulation. I excluded extreme totals where the probability was less than 1 in a million. It's a big table, so I won't clutter up the thread with it, unless you want to see it. The third column is the expected count per million games.

Aint a formula for that. gotta do it in soft code - iterate thru all combos in the game.

There is a website that shows the sums for many lottery games but no keno. probably keno has too many combos compared 2 lottos, even mega mills or powerball.

Total powerball combos (5 of 69 & 1 26)=292,201,338
Total keno combos (20 from 80)=3.53531614221217E+18 (19 digit number... very, very large for the computers).

But looks like the right algo exists and applicable to any lotto game.

https://saliu.com/forum/lottery-sums.html

Powerball 5/69 & 1/26 Lottery SUMS Chart

Sum-total Combinations Percentage

16 1 0.00%
17 2 0.00%
18 4 0.00%
19 7 0.00%
20 12 0.00%
21 19 0.00%
22 29 0.00%
23 42 0.00%
24 60 0.00%
25 83 0.00%
26 113 0.00%
27 150 0.00%
28 197 0.00%
29 254 0.00%
30 324 0.00%
31 408 0.00%
32 509 0.00%
33 628 0.00%
34 769 0.00%
35 933 0.00%
36 1125 0.00%
37 1346 0.00%
38 1601 0.00%
39 1892 0.00%
40 2225 0.00%
41 2602 0.00%
42 3028 0.00%
43 3507 0.00%
44 4045 0.00%
45 4645 0.00%
46 5314 0.00%
47 6055 0.00%
48 6876 0.00%
49 7781 0.00%
50 8777 0.00%
51 9869 0.00%
52 11065 0.00%
53 12370 0.00%
54 13792 0.00%
55 15337 0.01%
56 17014 0.01%
57 18828 0.01%
58 20789 0.01%
59 22903 0.01%
60 25180 0.01%
61 27627 0.01%
62 30253 0.01%
63 33066 0.01%
64 36077 0.01%
65 39293 0.01%
66 42725 0.01%
67 46381 0.02%
68 50273 0.02%
69 54409 0.02%
70 58801 0.02%
71 63458 0.02%
72 68392 0.02%
73 73613 0.03%
74 79133 0.03%
75 84962 0.03%
76 91114 0.03%
77 97598 0.03%
78 104428 0.04%
79 111615 0.04%
80 119173 0.04%
81 127112 0.04%
82 135446 0.05%
83 144186 0.05%
84 153346 0.05%
85 162936 0.06%
86 172970 0.06%
87 183457 0.06%
88 194412 0.07%
89 205843 0.07%
90 217763 0.07%
91 230181 0.08%
92 243109 0.08%
93 256554 0.09%
94 270528 0.09%
95 285037 0.10%
96 300092 0.10%
97 315697 0.11%
98 331862 0.11%
99 348590 0.12%
100 365890 0.13%
101 383763 0.13%
102 402215 0.14%
103 421247 0.14%
104 440864 0.15%
105 461063 0.16%
106 481848 0.16%
107 503215 0.17%
108 525166 0.18%
109 547695 0.19%
110 570802 0.20%
111 594480 0.20%
112 618727 0.21%
113 643534 0.22%
114 668897 0.23%
115 694806 0.24%
116 721256 0.25%
117 748233 0.26%
118 775731 0.27%
119 803736 0.28%
120 832239 0.28%
121 861224 0.29%
122 890680 0.30%
123 920590 0.32%
124 950942 0.33%
125 981716 0.34%
126 1012898 0.35%
127 1044467 0.36%
128 1076408 0.37%
129 1108697 0.38%
130 1141317 0.39%
131 1174244 0.40%
132 1207458 0.41%
133 1240933 0.42%
134 1274648 0.44%
135 1308575 0.45%
136 1342692 0.46%
137 1376968 0.47%
138 1411379 0.48%
139 1445894 0.49%
140 1480487 0.51%
141 1515124 0.52%
142 1549777 0.53%
143 1584412 0.54%
144 1618999 0.55%
145 1653501 0.57%
146 1687887 0.58%
147 1722119 0.59%
148 1756165 0.60%
149 1789986 0.61%
150 1823548 0.62%
151 1856813 0.64%
152 1889746 0.65%
153 1922308 0.66%
154 1954464 0.67%
155 1986176 0.68%
156 2017409 0.69%
157 2048124 0.70%
158 2078287 0.71%
159 2107861 0.72%
160 2136812 0.73%
161 2165104 0.74%
162 2192703 0.75%
163 2219576 0.76%
164 2245691 0.77%
165 2271015 0.78%
166 2295518 0.79%
167 2319170 0.79%
168 2341943 0.80%
169 2363809 0.81%
170 2384742 0.82%
171 2404718 0.82%
172 2423713 0.83%
173 2441705 0.84%
174 2458673 0.84%
175 2474598 0.85%
176 2489462 0.85%
177 2503247 0.86%
178 2515938 0.86%
179 2527521 0.86%
180 2537983 0.87%
181 2547312 0.87%
182 2555497 0.87%
183 2562530 0.88%
184 2568403 0.88%
185 2573109 0.88%
186 2576643 0.88%
187 2579001 0.88%

188 2580181 0.88%
189 2580181 0.88%

190 2579001 0.88%
191 2576643 0.88%
192 2573109 0.88%
193 2568403 0.88%
194 2562530 0.88%
195 2555497 0.87%
196 2547312 0.87%
197 2537983 0.87%
198 2527521 0.86%
199 2515938 0.86%
200 2503247 0.86%
201 2489462 0.85%
202 2474598 0.85%
203 2458673 0.84%
204 2441705 0.84%
205 2423713 0.83%
206 2404718 0.82%
207 2384742 0.82%
208 2363809 0.81%
209 2341943 0.80%
210 2319170 0.79%
211 2295518 0.79%
212 2271015 0.78%
213 2245691 0.77%
214 2219576 0.76%
215 2192703 0.75%
216 2165104 0.74%
217 2136812 0.73%
218 2107861 0.72%
219 2078287 0.71%
220 2048124 0.70%
221 2017409 0.69%
222 1986176 0.68%
223 1954464 0.67%
224 1922308 0.66%
225 1889746 0.65%
226 1856813 0.64%
227 1823548 0.62%
228 1789986 0.61%
229 1756165 0.60%
230 1722119 0.59%
231 1687887 0.58%
232 1653501 0.57%
233 1618999 0.55%
234 1584412 0.54%
235 1549777 0.53%
236 1515124 0.52%
237 1480487 0.51%
238 1445894 0.49%
239 1411379 0.48%
240 1376968 0.47%
241 1342692 0.46%
242 1308575 0.45%
243 1274648 0.44%
244 1240933 0.42%
245 1207458 0.41%
246 1174244 0.40%
247 1141317 0.39%
248 1108697 0.38%
249 1076408 0.37%
250 1044467 0.36%
251 1012898 0.35%
252 981716 0.34%
253 950942 0.33%
254 920590 0.32%
255 890680 0.30%
256 861224 0.29%
257 832239 0.28%
258 803736 0.28%
259 775731 0.27%
260 748233 0.26%
261 721256 0.25%
262 694806 0.24%
263 668897 0.23%
264 643534 0.22%
265 618727 0.21%
266 594480 0.20%
267 570802 0.20%
268 547695 0.19%
269 525166 0.18%
270 503215 0.17%
271 481848 0.16%
272 461063 0.16%
273 440864 0.15%
274 421247 0.14%
275 402215 0.14%
276 383763 0.13%
277 365890 0.13%
278 348590 0.12%
279 331862 0.11%
280 315697 0.11%
281 300092 0.10%
282 285037 0.10%
283 270528 0.09%
284 256554 0.09%
285 243109 0.08%
286 230181 0.08%
287 217763 0.07%
288 205843 0.07%
289 194412 0.07%
290 183457 0.06%
291 172970 0.06%
292 162936 0.06%
293 153346 0.05%
294 144186 0.05%
295 135446 0.05%
296 127112 0.04%
297 119173 0.04%
298 111615 0.04%
299 104428 0.04%
300 97598 0.03%
301 91114 0.03%
302 84962 0.03%
303 79133 0.03%
304 73613 0.03%
305 68392 0.02%
306 63458 0.02%
307 58801 0.02%
308 54409 0.02%
309 50273 0.02%
310 46381 0.02%
311 42725 0.01%
312 39293 0.01%
313 36077 0.01%
314 33066 0.01%
315 30253 0.01%
316 27627 0.01%
317 25180 0.01%
318 22903 0.01%
319 20789 0.01%
320 18828 0.01%
321 17014 0.01%
322 15337 0.01%
323 13792 0.00%
324 12370 0.00%
325 11065 0.00%
326 9869 0.00%
327 8777 0.00%
328 7781 0.00%
329 6876 0.00%
330 6055 0.00%
331 5314 0.00%
332 4645 0.00%
333 4045 0.00%
334 3507 0.00%
335 3028 0.00%
336 2602 0.00%
337 2225 0.00%
338 1892 0.00%
339 1601 0.00%
340 1346 0.00%
341 1125 0.00%
342 933 0.00%
343 769 0.00%
344 628 0.00%
345 509 0.00%
346 408 0.00%
347 324 0.00%
348 254 0.00%
349 197 0.00%
350 150 0.00%
351 113 0.00%
352 83 0.00%
353 60 0.00%
354 42 0.00%
355 29 0.00%
356 19 0.00%
357 12 0.00%
358 7 0.00%
359 4 0.00%
360 2 0.00%
361 1 0.00%

Total: 346 292201338 100%

Quote: Wizard
Note: Post formatting corrected by management.

Last edited by: unnamed administrator on May 21, 2019

Wizard
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Wizard

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May 21st, 2019 at 6:40:07 AM permalink

I get the theory about finding the generating function, but to actually do it would be the equal amount of work as just cycling through all combin(80,20) balls by brute force.

ZPP, thanks for the code, I think I will try that method next. I started to go down that road, but didn't finishing, thinking it would only cut down centuries of computer time to years. Let's see what happens....

"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)

7craps

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May 21st, 2019 at 7:28:40 AM permalink

Quote: Wizard
I get the theory about finding the generating function, but to actually do it would be the equal amount of work as just cycling through all combin(80,20) balls by brute force.

the generating function is also easily changed for the values on the balls. say instead of 1 to 80 they were 0 to 79. simple adjustment

it takes 0ms on my 10 year old Dell 2 core laptop for all the calculations (according to pari/gp calculator) and it is actually doing all the draws from 0 to 80 inclusive and prints to a text file.

from there it is easy (for me) to get what is needed into Excel or work on the results further in pari.
generating functions go way back and it makes sense why.(they had no computers to do it then)
even poker gurus now (like BA) uses GFs to describe getting particular hands and so on.

of course there are 'expert computer guys and gals' that can do excellent code for almost any problem.

some get (and can use) generating functions and add them to their toolbox, others never get them for whatever reason(s).

added: reminds me of learning inclusion-exclusion and wondering why we have to do it this way. add and subtract over-counts seemed way confusing and not required as they were always another way to solve the problem. Most now, that even do not understand PIE, still use it in a computer to solve some problems because it IS so easy and follows basic counting methods. example code for coupon collecting problem: n=6;\\number of coupons
t=10;\\number of draws
x=sum(k=0,n,(-1)^k*binomial(n,k)*(1-(k/n))^t);

this was so confusing at 1st: (-1)^k*binomial(n,k)*(1-(k/n))^t

Last edited by: 7craps on May 21, 2019

winsome johnny (not Win some johnny)

charliepatrick

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Thanked by

May 21st, 2019 at 7:34:51 AM permalink

I'm about to go out (it's afternoon shoping here) but here's an idea I might look at later.
(i) Split the numbers into blocks 1-10, 11-20 etc. (consider 1-10 as 0*10+1 thru 0*10+10, 2-20 at 1*10+1 thru 1*10+10...)
(ii) Similar to working out poker hands, you can have a pattern of (a) 10 in one block, 10 in another (b) 10 9 1 (c) 10 8 2 (d) 10 8 1 1 etc.
(iii) Work out the permutations for various numbers of balls within a block for the digits. (e..g 1 number in block can be 1:1 2:1 ... 10:1).
(iv) Work out the permutations for various blocks being used, this will only affect the tens value of the total.
(v) Add together the digits part from (iii) and the tens part from (iv).
e.g. Looking at {10,10}, the sum of the digits part is fixed at 55+55=110. The sum of the tens part can be 10*0+10*10, 10*0+10*20, 10*10+10*10 etc. This gets 1 combination for 210 etc.
e.g. Looking at (10,9,1), the sum of the digits part can be {55}+{45,47,48,49...54)+(1,2,3,4,...,9,10}. Then the tens can be 10*0+9*10+1*20 etc.

gordonm888
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May 21st, 2019 at 10:21:34 AM permalink

Quote: charliepatrick
I'm about to go out (it's afternoon shoping here) but here's an idea I might look at later.
(i) Split the numbers into blocks 1-10, 11-20 etc. (consider 1-10 as 0*10+1 thru 0*10+10, 2-20 at 1*10+1 thru 1*10+10...)
(ii) Similar to working out poker hands, you can have a pattern of (a) 10 in one block, 10 in another (b) 10 9 1 (c) 10 8 2 (d) 10 8 1 1 etc.
(iii) Work out the permutations for various numbers of balls within a block for the digits. (e..g 1 number in block can be 1:1 2:1 ... 10:1).
(iv) Work out the permutations for various blocks being used, this will only affect the tens value of the total.
(v) Add together the digits part from (iii) and the tens part from (iv).
e.g. Looking at {10,10}, the sum of the digits part is fixed at 55+55=110. The sum of the tens part can be 10*0+10*10, 10*0+10*20, 10*10+10*10 etc. This gets 1 combination for 210 etc.
e.g. Looking at (10,9,1), the sum of the digits part can be {55}+{45,47,48,49...54)+(1,2,3,4,...,9,10}. Then the tens can be 10*0+9*10+1*20 etc.

I like this approach, ingenious. I wonder whether it might be slightly more efficient in base 9.

edit: Or 4-tier it in base 3.

Last edited by: gordonm888 on May 21, 2019

So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.

Wizard
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Wizard

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May 21st, 2019 at 2:29:50 PM permalink

I didn't get very far with the code above. My C++ compiler didn't like this statement and I don't understand what it is supposed to:

typedef uint64_t num_t;

However, I can't ask anyone to teach me a new computer language. I was half finished doing it this way anyway, so will try to complete it with my own code.

Thanks ZPP for your help!

"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)

ThatDonGuy

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May 21st, 2019 at 3:43:53 PM permalink

Quote: ThatDonGuy
I think I sort of do.
The formula is the product of:
1 + x y
1 + x y²
1 + x y³
1 + x y⁴
...
1 + x y⁷⁹
1 + x y⁸⁰

If you draw 20 particular balls, multiply the product of the left sides of the terms of the balls you did not draw and the right sides of the balls you did draw together; you will get x²⁰ and y^{sum of the numbers}.
To determine how many times 20 balls add up to some number N, count the number of times x²⁰ y^N is such a product.

How to get such a count without having to go through all 2⁸⁰ products, I don't know, and presumably, that's why things like Mathematica exist.

HOUSE LIGHTBULB MOMENT - this may be similar to ZPP's solution

Instead of treating those as 80 different sums, do the products from top to bottom - i.e.
Start with (1 + xy) (1 + x y²)
Multiply that by (1 + x y³)
Multiply that result by (1 + x y⁴)
and so on through (1 + x y⁸⁰)

This can be done with an array, where each row is one of the y exponents (from 0 to 1410) and each column is one of the x exponents (from 0 to 20)
Cell (a, b) is the current number of times x^a y^b appears in the solution so far
Start with all cells 0, except for 1s in (0,0) and (1,1); this is 1 + xy
Each step multiplies the existing value by x yⁿ n for some n from 1 to 80, then adding it to the existing result. This is done by making a copy of the array, shifting it one column to the right and n rows down, and adding those values to the existing values.
Once all 80 steps are done, cell (20, N) will contain the number of combinations of 20 balls that add up to N.
This took 1/2 second to calculate in C#.

rsactuary

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May 21st, 2019 at 6:36:32 PM permalink

In your WOO post, you don't define rain, cloud, lightning, fire and sprout.

I find it odd that small vs big has a different payout, given the symmetrical distribution. Any sense as to why they might do that?

Wizard
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Wizard

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May 21st, 2019 at 7:19:41 PM permalink

Quote: rsactuary
In your WOO post, you don't define rain, cloud, lightning, fire and sprout.

I find it odd that small vs big has a different payout, given the symmetrical distribution. Any sense as to why they might do that?

Good catches, thanks. The big had a typo in the win. I just added a description of the ranges covered by the bets you asked about.

For everyone else, here is my new page on the game with the prop bets on the total off all balls: Blitz Bingo.

"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)

7craps

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May 21st, 2019 at 7:36:37 PM permalink

Quote: Wizard
For everyone else, here is my new page on the game with the prop bets on the total off all balls: Blitz Bingo.

3,535,316,142,212,170,000
should be
3,535,316,142,212,174,320
Combin(80,20) on Wolfram Alpha

winsome johnny (not Win some johnny)

DogHand

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May 21st, 2019 at 8:37:45 PM permalink

Wizard,

Rule 5 has a typo on the payout for Big: it says 1.91, but on your EV table you have 1.95.

Dog Hand

Wizard
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May 21st, 2019 at 9:15:09 PM permalink

I'm happy to report my program worked and it runs in under a second. I'll include the code here, not that anyone will care.

#include <iostream>
#include <stdlib.h>
using namespace std;

void fast(void);

__int64 a[81][21][1601];

int main(void)
{
int i;
fast();
cerr << "Enter any number\t";
cin >> i;
return 1;
}

void fast(void)
{
int k,m,n,t;

for (k=0; k<=80; k++) // lowest available ball
{
for (m=0; m<=20; m++)
{
for (n=0; n<=1600; n++)
a[k][m][n]=0;
}
}
for (k=20; k<=80; k++)
{
for (m=k; m<=80; m++) // 20th ball
a[k][20][m]=1;
}
for (k=19; k<=79; k++) // lowest available ball
{
for (m=k; m<=79; m++) // 19th ball
{
for (t=0; t<=80; t++)
{
a[k][19][m+t]+=a[m+1][20][t];
}
}
}
for (k=18; k<=78; k++) // lowest available ball
{
for (m=k; m<=78; m++) // 18th ball
{
for (t=0; t<=160; t++)
{
a[k][18][m+t]+=a[m+1][19][t];
}
}
}
int level;
for (level=17; level>=1; level--)
{
for (k=level; k<=60+level; k++) // lowest available ball
{
for (m=k; m<=60+level; m++) // level-th ball
{
for (t=0; t<=1410; t++)
{
a[k][level][m+t]+=a[m+1][level+1][t];
}
}
}
}

printf("Raw totals\n");
for (n=0; n<=1410; n++)
{
if (a[1][1][n]>0)
printf("%i\t%I64i\n",n,a[1][1][n]);
}

printf("formatted totals\n");
for (n=0; n<=1410; n++)
{
if (a[1][1][n]>0)
printf("<tr><td>%i</td><td>%I64i</td></tr>\n",n,a[1][1][n]);
}
return;
}

I'd like to thank all who participated! This a great example of why I love this forum. I hope others have learned something as well.

Quote: 7craps
Quote: Wizard
For everyone else, here is my new page on the game with the prop bets on the total off all balls: Blitz Bingo.
3,535,316,142,212,170,000
should be
3,535,316,142,212,174,320
Combin(80,20) on Wolfram Alpha

Damn Microsoft! Excel allows for only so many significant digits (usually 15). I wish I had a dollar for every time I've had similar corrections.

Quote: DogHand
Wizard,

Rule 5 has a typo on the payout for Big: it says 1.91, but on your EV table you have 1.95.

Dog Hand

Thanks, got it.

"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)

Ace2

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Threads: 32
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May 22nd, 2019 at 12:17:04 AM permalink

Quote: Wizard
I get the theory about finding the generating function, but to actually do it would be the equal amount of work as just cycling through all combin(80,20) balls by brute force.
.

Exactly. This is why I�ve never spent much time studying generating functions.

I prefer slick, formulaic solutions and would be much more impressed by an estimate accurate within .1% to an exact solution that was so voluminous that it required programming.

You can probably just use the expectation (810) and standard deviation (90) to get decent estimates for values within 2 or 3 SDs of expectation.

Last edited by: Ace2 on May 22, 2019

It�s all about making that GTA

ZPP

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May 22nd, 2019 at 1:17:58 AM permalink

Quote: Wizard
I didn't get very far with the code above. My C++ compiler didn't like this statement and I don't understand what it is supposed to:

typedef uint64_t num_t;

However, I can't ask anyone to teach me a new computer language. I was half finished doing it this way anyway, so will try to complete it with my own code.

Thanks ZPP for your help!

You're welcome.
The typedef defines num_t as uint64_t. The point is that (80 choose 20) is near the limit of 64-bit int, so if I want to switch to double, for larger values, I only have to change that one typedef (and the printfs). Also, if I wanted to use an old version of MSVC that doesn't support uint64_t, I would only have to change the one typedef (and the printfs) to usigned __int64... :)

Quote: Wizard
Damn Microsoft! Excel allows for only so many significant digits (usually 15).

I'm pretty sure Excel is just using "double" for everything internally.

Wizard
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Wizard

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May 22nd, 2019 at 6:25:48 AM permalink

Somehow I think there is an elegant pattern to this problem somewhere. Some variant of Pascal's Triangle. However, if it were easy to find, I'm sure somebody would have by now.

"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)

Wizard
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Wizard

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May 22nd, 2019 at 7:05:53 PM permalink

Here is my new page on this topic, Keno Sums. I welcome all comments and corrections on it.

Again, thank you to all who contributed to this topic!

"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)

rsactuary

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May 22nd, 2019 at 7:39:58 PM permalink

Thanks for your work on this... I didn't check through the math, but it seems weird that the return for exactly 810 is about half of every other bet.

ETA: You seem to have the number of combinations correct.. must be accurate... weird.

Wizard
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Wizard

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May 22nd, 2019 at 9:44:10 PM permalink

Quote: rsactuary
Thanks for your work on this... I didn't check through the math, but it seems weird that the return for exactly 810 is about half of every other bet.

I figure that's because the payout is larger. More risk to the casino to offer it.

"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)

Klopp

Klopp

Threads: 7
Posts: 16

Joined: Mar 10, 2020

March 11th, 2020 at 12:07:52 PM permalink

An insightful approximate solution can be given. The probability distribution of the sum of the 20 keno balls can be very closely approximated by a normal distribution with expected value 810 and standard deviation 97.2.
In chapter 9 of the book Surprises in Probability written by Henk Tijms a more general situation is considered: a lottery in which r different numbers are randomly drawn from the numbers 1 up to s. Then the sum of the drawn numbers has an approximate normal distribution with expected value r(s+1)/2 and as standard deviation the square root of r(s+1)(s-r)/12 provided that s is not too small . An insightful and useful result.

Last edited by: Klopp on Mar 11, 2020

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Sum of all 20 keno balls

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