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Wizard
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Wizard
Joined: Oct 14, 2009
  • Threads: 1332
  • Posts: 21942
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

It's not whether you win or lose; it's whether or not you had a good bet.
7craps
7craps
Joined: Jan 23, 2010
  • Threads: 18
  • Posts: 1977
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
Joined: Oct 14, 2009
  • Threads: 1332
  • Posts: 21942
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
It's not whether you win or lose; it's whether or not you had a good bet.
Wizard
Administrator
Wizard
Joined: Oct 14, 2009
  • Threads: 1332
  • Posts: 21942
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
It's not whether you win or lose; it's whether or not you had a good bet.
7craps
7craps
Joined: Jan 23, 2010
  • Threads: 18
  • Posts: 1977
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
Administrator
Wizard
Joined: Oct 14, 2009
  • Threads: 1332
  • Posts: 21942
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.
It's not whether you win or lose; it's whether or not you had a good bet.
7craps
7craps
Joined: Jan 23, 2010
  • Threads: 18
  • Posts: 1977
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
Joined: Nov 17, 2009
  • Threads: 236
  • Posts: 6763
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
Joined: Jun 22, 2011
  • Threads: 95
  • Posts: 4289
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 ysum of the numbers.
To determine how many times 20 balls add up to some number N, count the number of times x20 yN 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
Joined: Oct 14, 2009
  • Threads: 1332
  • Posts: 21942
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.
It's not whether you win or lose; it's whether or not you had a good bet.

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