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Last Updated: May 28, 2019

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Keno Sums

Introduction

This page shows the probability of any given sum of the number of 20 keno balls. For those not familiar with keno, it is a game in which the player bets on the draw of 20 balls, without replacement, numbered 1 to 80. I created this page when I noticed bets on the sum of the draw in Blitz Keno.

There are combin(80,20)= 3,535,316,142,212,174,320 ways to choose 20 items out of 80. The following table shows the number of combinations for any given possible total from 210 to 1,410, the probability of that total, and the probability of that total or less.

Keno Sum of Ball Draw

Total Combinations Probability
(Exact)
Probability
(This Total or Less)
210 1 0.000000000000 0.000000000000
211 1 0.000000000000 0.000000000000
212 2 0.000000000000 0.000000000000
213 3 0.000000000000 0.000000000000
214 5 0.000000000000 0.000000000000
215 7 0.000000000000 0.000000000000
216 11 0.000000000000 0.000000000000
217 15 0.000000000000 0.000000000000
218 22 0.000000000000 0.000000000000
219 30 0.000000000000 0.000000000000
220 42 0.000000000000 0.000000000000
221 56 0.000000000000 0.000000000000
222 77 0.000000000000 0.000000000000
223 101 0.000000000000 0.000000000000
224 135 0.000000000000 0.000000000000
225 176 0.000000000000 0.000000000000
226 231 0.000000000000 0.000000000000
227 297 0.000000000000 0.000000000000
228 385 0.000000000000 0.000000000000
229 490 0.000000000000 0.000000000000
230 627 0.000000000000 0.000000000000
231 791 0.000000000000 0.000000000000
232 1000 0.000000000000 0.000000000000
233 1251 0.000000000000 0.000000000000
234 1568 0.000000000000 0.000000000000
235 1946 0.000000000000 0.000000000000
236 2417 0.000000000000 0.000000000000
237 2980 0.000000000000 0.000000000000
238 3673 0.000000000000 0.000000000000
239 4498 0.000000000000 0.000000000000
240 5507 0.000000000000 0.000000000000
241 6703 0.000000000000 0.000000000000
242 8154 0.000000000000 0.000000000000
243 9871 0.000000000000 0.000000000000
244 11937 0.000000000000 0.000000000000
245 14375 0.000000000000 0.000000000000
246 17293 0.000000000000 0.000000000000
247 20722 0.000000000000 0.000000000000
248 24803 0.000000000000 0.000000000000
249 29588 0.000000000000 0.000000000000
250 35251 0.000000000000 0.000000000000
251 41869 0.000000000000 0.000000000000
252 49668 0.000000000000 0.000000000000
253 58754 0.000000000000 0.000000000000
254 69414 0.000000000000 0.000000000000
255 81801 0.000000000000 0.000000000000
256 96271 0.000000000000 0.000000000000
257 113039 0.000000000000 0.000000000000
258 132559 0.000000000000 0.000000000000
259 155112 0.000000000000 0.000000000000
260 181274 0.000000000000 0.000000000000
261 211428 0.000000000000 0.000000000000
262 246288 0.000000000000 0.000000000000
263 286364 0.000000000000 0.000000000001
264 332557 0.000000000000 0.000000000001
265 385528 0.000000000000 0.000000000001
266 446405 0.000000000000 0.000000000001
267 516054 0.000000000000 0.000000000001
268 595872 0.000000000000 0.000000000001
269 686983 0.000000000000 0.000000000001
270 791131 0.000000000000 0.000000000002
271 909741 0.000000000000 0.000000000002
272 1044984 0.000000000000 0.000000000002
273 1198689 0.000000000000 0.000000000003
274 1373524 0.000000000000 0.000000000003
275 1571812 0.000000000000 0.000000000003
276 1796855 0.000000000001 0.000000000004
277 2051569 0.000000000001 0.000000000004
278 2340024 0.000000000001 0.000000000005
279 2665885 0.000000000001 0.000000000006
280 3034135 0.000000000001 0.000000000007
281 3449359 0.000000000001 0.000000000008
282 3917670 0.000000000001 0.000000000009
283 4444748 0.000000000001 0.000000000010
284 5038070 0.000000000001 0.000000000011
285 5704686 0.000000000002 0.000000000013
286 6453684 0.000000000002 0.000000000015
287 7293767 0.000000000002 0.000000000017
288 8236001 0.000000000002 0.000000000019
289 9291060 0.000000000003 0.000000000022
290 10472375 0.000000000003 0.000000000025
291 11793028 0.000000000003 0.000000000028
292 13269250 0.000000000004 0.000000000032
293 14917024 0.000000000004 0.000000000036
294 16755957 0.000000000005 0.000000000041
295 18805464 0.000000000005 0.000000000046
296 21089176 0.000000000006 0.000000000052
297 23630664 0.000000000007 0.000000000059
298 26458297 0.000000000007 0.000000000066
299 29600587 0.000000000008 0.000000000075
300 33091578 0.000000000009 0.000000000084
301 36965620 0.000000000010 0.000000000095
302 41263462 0.000000000012 0.000000000106
303 46026431 0.000000000013 0.000000000119
304 51303148 0.000000000015 0.000000000134
305 57143201 0.000000000016 0.000000000150
306 63604540 0.000000000018 0.000000000168
307 70746444 0.000000000020 0.000000000188
308 78637842 0.000000000022 0.000000000210
309 87349501 0.000000000025 0.000000000235
310 96963169 0.000000000027 0.000000000262
311 107563103 0.000000000030 0.000000000293
312 119246146 0.000000000034 0.000000000326
313 132112328 0.000000000037 0.000000000364
314 146276124 0.000000000041 0.000000000405
315 161856093 0.000000000046 0.000000000451
316 178987225 0.000000000051 0.000000000502
317 197809729 0.000000000056 0.000000000558
318 218482569 0.000000000062 0.000000000619
319 241171101 0.000000000068 0.000000000688
320 266062168 0.000000000075 0.000000000763
321 293350487 0.000000000083 0.000000000846
322 323255164 0.000000000091 0.000000000937
323 356005000 0.000000000101 0.000000001038
324 391856581 0.000000000111 0.000000001149
325 431078138 0.000000000122 0.000000001271
326 473969587 0.000000000134 0.000000001405
327 520844871 0.000000000147 0.000000001552
328 572053851 0.000000000162 0.000000001714
329 627963266 0.000000000178 0.000000001892
330 688980643 0.000000000195 0.000000002086
331 755533468 0.000000000214 0.000000002300
332 828095629 0.000000000234 0.000000002534
333 907164667 0.000000000257 0.000000002791
334 993290571 0.000000000281 0.000000003072
335 1087051347 0.000000000307 0.000000003379
336 1189084411 0.000000000336 0.000000003716
337 1300059954 0.000000000368 0.000000004084
338 1420715489 0.000000000402 0.000000004485
339 1551826889 0.000000000439 0.000000004924
340 1694245990 0.000000000479 0.000000005404
341 1848869496 0.000000000523 0.000000005927
342 2016679881 0.000000000570 0.000000006497
343 2198711618 0.000000000622 0.000000007119
344 2396096082 0.000000000678 0.000000007797
345 2610024936 0.000000000738 0.000000008535
346 2841798846 0.000000000804 0.000000009339
347 3092788314 0.000000000875 0.000000010214
348 3364486581 0.000000000952 0.000000011165
349 3658467191 0.000000001035 0.000000012200
350 3976441876 0.000000001125 0.000000013325
351 4320214692 0.000000001222 0.000000014547
352 4691744778 0.000000001327 0.000000015874
353 5093097335 0.000000001441 0.000000017315
354 5526511602 0.000000001563 0.000000018878
355 5994347965 0.000000001696 0.000000020573
356 6499162145 0.000000001838 0.000000022412
357 7043648186 0.000000001992 0.000000024404
358 7630718671 0.000000002158 0.000000026563
359 8263443950 0.000000002337 0.000000028900
360 8945138936 0.000000002530 0.000000031430
361 9679297643 0.000000002738 0.000000034168
362 10469687646 0.000000002961 0.000000037130
363 11320279729 0.000000003202 0.000000040332
364 12235349889 0.000000003461 0.000000043793
365 13219404469 0.000000003739 0.000000047532
366 14277290236 0.000000004038 0.000000051570
367 15414114032 0.000000004360 0.000000055930
368 16635362300 0.000000004705 0.000000060636
369 17946814901 0.000000005076 0.000000065712
370 19354673930 0.000000005475 0.000000071187
371 20865472184 0.000000005902 0.000000077089
372 22486212019 0.000000006360 0.000000083449
373 24224267278 0.000000006852 0.000000090301
374 26087533643 0.000000007379 0.000000097681
375 28084323773 0.000000007944 0.000000105624
376 30223529082 0.000000008549 0.000000114173
377 32514508546 0.000000009197 0.000000123371
378 34967262684 0.000000009891 0.000000133261
379 37592314751 0.000000010633 0.000000143895
380 40400898786 0.000000011428 0.000000155323
381 43404832757 0.000000012277 0.000000167600
382 46616720518 0.000000013186 0.000000180786
383 50049817550 0.000000014157 0.000000194943
384 53718247833 0.000000015195 0.000000210138
385 57636860670 0.000000016303 0.000000226441
386 61821464449 0.000000017487 0.000000243928
387 66288674187 0.000000018750 0.000000262678
388 71056162189 0.000000020099 0.000000282777
389 76142496873 0.000000021538 0.000000304315
390 81567411419 0.000000023072 0.000000327387
391 87351632249 0.000000024708 0.000000352095
392 93517168095 0.000000026452 0.000000378548
393 100087127594 0.000000028311 0.000000406858
394 107086028628 0.000000030290 0.000000437149
395 114539605867 0.000000032399 0.000000469547
396 122475141756 0.000000034643 0.000000504191
397 130921262045 0.000000037032 0.000000541223
398 139908291043 0.000000039574 0.000000580798
399 149468034620 0.000000042279 0.000000623076
400 159634159836 0.000000045154 0.000000668230
401 170441966142 0.000000048211 0.000000716442
402 181928790813 0.000000051460 0.000000767902
403 194133766301 0.000000054913 0.000000822815
404 207098254570 0.000000058580 0.000000881395
405 220865589927 0.000000062474 0.000000943869
406 235481542190 0.000000066608 0.000001010477
407 250994045973 0.000000070996 0.000001081473
408 267453694512 0.000000075652 0.000001157125
409 284913452879 0.000000080591 0.000001237716
410 303429185857 0.000000085828 0.000001323544
411 323059354425 0.000000091381 0.000001414924
412 343865577844 0.000000097266 0.000001512190
413 365912314318 0.000000103502 0.000001615692
414 389267459099 0.000000110108 0.000001725801
415 414002006749 0.000000117105 0.000001842905
416 440190689221 0.000000124512 0.000001967418
417 467911618783 0.000000132354 0.000002099771
418 497246966079 0.000000140651 0.000002240422
419 528282584833 0.000000149430 0.000002389853
420 561108732221 0.000000158715 0.000002548568
421 595819672189 0.000000168534 0.000002717101
422 632514442353 0.000000178913 0.000002896015
423 671296435063 0.000000189883 0.000003085898
424 712274211009 0.000000201474 0.000003287372
425 755561059108 0.000000213718 0.000003501090
426 801275859118 0.000000226649 0.000003727739
427 849542617012 0.000000240302 0.000003968040
428 900491381529 0.000000254713 0.000004222754
429 954257753733 0.000000269921 0.000004492675
430 1010983857544 0.000000285967 0.000004778642
431 1070817824773 0.000000302892 0.000005081534
432 1133914822434 0.000000320739 0.000005402273
433 1200436509276 0.000000339556 0.000005741829
434 1270552124993 0.000000359389 0.000006101217
435 1344437917067 0.000000380288 0.000006481505
436 1422278292091 0.000000402306 0.000006883811
437 1504265214046 0.000000425497 0.000007309308
438 1590599420536 0.000000449917 0.000007759225
439 1681489788194 0.000000475626 0.000008234851
440 1777154619819 0.000000502686 0.000008737537
441 1877820975168 0.000000531161 0.000009268698
442 1983726028916 0.000000561117 0.000009829815
443 2095116368246 0.000000592625 0.000010422440
444 2212249424868 0.000000625757 0.000011048197
445 2335392734382 0.000000660590 0.000011708787
446 2464825448513 0.000000697201 0.000012405988
447 2600837554495 0.000000735673 0.000013141661
448 2743731467669 0.000000776092 0.000013917753
449 2893821211946 0.000000818547 0.000014736300
450 3051434096031 0.000000863129 0.000015599429
451 3216909849508 0.000000909936 0.000016509364
452 3390602389686 0.000000959066 0.000017468431
453 3572878910940 0.000001010625 0.000018479056
454 3764121741811 0.000001064720 0.000019543776
455 3964727389027 0.000001121463 0.000020665239
456 4175108488550 0.000001180972 0.000021846211
457 4395692797816 0.000001243366 0.000023089577
458 4626925247931 0.000001308773 0.000024398350
459 4869266881582 0.000001377321 0.000025775671
460 5123197006289 0.000001449148 0.000027224819
461 5389212078927 0.000001524393 0.000028749212
462 5667827963566 0.000001603203 0.000030352415
463 5959578755669 0.000001685727 0.000032038142
464 6265019152487 0.000001772124 0.000033810266
465 6584723213704 0.000001862556 0.000035672822
466 6919286843610 0.000001957190 0.000037630013
467 7269326489361 0.000002056203 0.000039686215
468 7635481738958 0.000002159773 0.000041845989
469 8018413948848 0.000002268090 0.000044114078
470 8418808965914 0.000002381345 0.000046495424
471 8837375680898 0.000002499741 0.000048995165
472 9274848873719 0.000002623485 0.000051618650
473 9731987693363 0.000002752791 0.000054371441
474 10209578630140 0.000002887883 0.000057259324
475 10708433913262 0.000003028989 0.000060288313
476 11229394619008 0.000003176348 0.000063464661
477 11773328981483 0.000003330205 0.000066794867
478 12341135635235 0.000003490815 0.000070285682
479 12933741839899 0.000003658440 0.000073944122
480 13552106861442 0.000003833351 0.000077777473
481 14197220100303 0.000004015828 0.000081793300
482 14870104620270 0.000004206160 0.000085999460
483 15571815175325 0.000004404646 0.000090404106
484 16303441884554 0.000004611594 0.000095015700
485 17066108157849 0.000004827322 0.000099843021
486 17860974520443 0.000005052158 0.000104895179
487 18689236426307 0.000005286440 0.000110181619
488 19552128242181 0.000005530518 0.000115712136
489 20450920942555 0.000005784750 0.000121496887
490 21386926250737 0.000006049509 0.000127546396
491 22361494215565 0.000006325175 0.000133871571
492 23376017513498 0.000006612143 0.000140483714
493 24431928894154 0.000006910819 0.000147394533
494 25530705653070 0.000007221619 0.000154616152
495 26673866939713 0.000007544974 0.000162161125
496 27862978398196 0.000007881326 0.000170042451
497 29099649337087 0.000008231131 0.000178273582
498 30385537541434 0.000008594857 0.000186868439
499 31722346290416 0.000008972987 0.000195841426
500 33111829351342 0.000009366017 0.000205207443
501 34555787837266 0.000009774455 0.000214981898
502 36056075379039 0.000010198826 0.000225180724
503 37614594822327 0.000010639669 0.000235820393
504 39233303581272 0.000011097538 0.000246917931
505 40914210159126 0.000011572999 0.000258490930
506 42659379693854 0.000012066638 0.000270557568
507 44470930294083 0.000012579053 0.000283136621
508 46351038772509 0.000013110861 0.000296247482
509 48301936795528 0.000013662692 0.000309910175
510 50325916807381 0.000014235196 0.000324145370
511 52425327976532 0.000014829035 0.000338974405
512 54602582321801 0.000015444894 0.000354419299
513 56860150445667 0.000016083470 0.000370502769
514 59200567856702 0.000016745481 0.000387248250
515 61626430488492 0.000017431660 0.000404679910
516 64140401221500 0.000018142763 0.000422822673
517 66745205168528 0.000018879558 0.000441702230
518 69443636407019 0.000019642836 0.000461345067
519 72238553020681 0.000020433407 0.000481778473
520 75132884036461 0.000021252098 0.000503030571
521 78129624219915 0.000022099756 0.000525130327
522 81231841219122 0.000022977250 0.000548107577
523 84442670093739 0.000023885465 0.000571993042
524 87765320677722 0.000024825310 0.000596818352
525 91203071830294 0.000025797713 0.000622616065
526 94759279010024 0.000026803622 0.000649419688
527 98437368245632 0.000027844007 0.000677263695
528 102240843924128 0.000028919859 0.000706183554
529 106173282459017 0.000030032189 0.000736215743
530 110238340304328 0.000031182032 0.000767397775
531 114439747308027 0.000032370442 0.000799768217
532 118781314944316 0.000033598499 0.000833366716
533 123266929349329 0.000034867300 0.000868234016
534 127900559773363 0.000036177970 0.000904411986
535 132686251275763 0.000037531651 0.000941943636
536 137628133409981 0.000038929512 0.000980873149
537 142730412563985 0.000040372744 0.001021245893
538 147997380868998 0.000041862559 0.001063108452
539 153433408182971 0.000043400194 0.001106508646
540 159042951225485 0.000044986911 0.001151495557
541 164830545178892 0.000046623990 0.001198119547
542 170800813062207 0.000048312741 0.001246432288
543 176958456936318 0.000050054493 0.001296486781
544 183308267507535 0.000051850601 0.001348337382
545 189855114934011 0.000053702443 0.001402039825
546 196603958660512 0.000055611422 0.001457651248
547 203559837800120 0.000057578963 0.001515230211
548 210727881214639 0.000059606517 0.001574836728
549 218113297456569 0.000061695557 0.001636532284
550 225721385084601 0.000063847581 0.001700379866
551 233557522163930 0.000066064112 0.001766443978
552 241627176819381 0.000068346696 0.001834790674
553 249935896266269 0.000070696901 0.001905487575
554 258489317615563 0.000073116323 0.001978603898
555 267293156421037 0.000075606578 0.002054210476
556 276353217726061 0.000078169308 0.002132379785
557 285675384124940 0.000080806178 0.002213185962
558 295265627053394 0.000083518875 0.002296704837
559 305129994336046 0.000086309111 0.002383013949
560 315274621737283 0.000089178622 0.002472192571
561 325705719979621 0.000092129164 0.002564321734
562 336429586543371 0.000095162518 0.002659484253
563 347452592155562 0.000098280487 0.002757764740
564 358781192841696 0.000101484897 0.002859249636
565 370421915855964 0.000104777593 0.002964027229
566 382381372002154 0.000108160446 0.003072187675
567 394666240991089 0.000111635346 0.003183823021
568 407283284019903 0.000115204205 0.003299027226
569 420239328556562 0.000118868953 0.003417896179
570 433541281180913 0.000122631545 0.003540527724
571 447196111767398 0.000126493952 0.003667021677
572 461210866607609 0.000130458168 0.003797479844
573 475592651976889 0.000134526202 0.003932006046
574 490348647526558 0.000138700084 0.004070706130
575 505486089237058 0.000142981863 0.004213687994
576 521012283085036 0.000147373605 0.004361061598
577 536934587353256 0.000151877390 0.004512938988
578 553260426593322 0.000156495319 0.004669434307
579 569997273279789 0.000161229505 0.004830663812
580 587152662058258 0.000166082081 0.004996745893
581 604734170747184 0.000171055189 0.005167801082
582 622749434876858 0.000176150989 0.005343952071
583 641206128010489 0.000181371652 0.005525323723
584 660111976597920 0.000186719363 0.005712043086
585 679474739604064 0.000192196316 0.005904239402
586 699302223666421 0.000197804721 0.006102044123
587 719602262037791 0.000203546793 0.006305590916
588 740382730055707 0.000209424759 0.006515015674
589 761651523370470 0.000215440852 0.006730456527
590 783416573752126 0.000221597318 0.006952053844
591 805685826594439 0.000227896401 0.007179950246
592 828467257050350 0.000234340360 0.007414290606
593 851768846820577 0.000240931451 0.007655222056
594 875598600626154 0.000247671938 0.007902893994
595 899964522255079 0.000254564086 0.008157458080
596 924874631401209 0.000261610163 0.008419068243
597 950336938960611 0.000268812435 0.008687880678
598 976359464227661 0.000276173170 0.008964053847
599 1002950209453776 0.000283694631 0.009247748478
600 1030117177412699 0.000291379084 0.009539127562
601 1057868345195045 0.000299228783 0.009838356345
602 1086211682173956 0.000307245983 0.010145602327
603 1115155123030642 0.000315432928 0.010461035255
604 1144706586113291 0.000323791859 0.010784827115
605 1174873945708909 0.000332325002 0.011117152117
606 1205665050808215 0.000341034579 0.011458186696
607 1237087696599984 0.000349922792 0.011808109488
608 1269149643677458 0.000358991839 0.012167101327
609 1301858588751637 0.000368243896 0.012535345223
610 1335222184292444 0.000377681127 0.012913026350
611 1369248008480895 0.000387305676 0.013300332026
612 1403943585301204 0.000397119672 0.013697451697
613 1439316353708705 0.000407125218 0.014104576915
614 1475373688210195 0.000417324400 0.014521901315
615 1512122867248405 0.000427719278 0.014949620593
616 1549571094266055 0.000438311888 0.015387932481
617 1587725465329523 0.000449104239 0.015837036720
618 1626592990693830 0.000460098312 0.016297135032
619 1666180561645293 0.000471296058 0.016768431090
620 1706494972609528 0.000482699398 0.017251130488
621 1747542887215360 0.000494310216 0.017745440705
622 1789330860941288 0.000506130368 0.018251571073
623 1831865306428571 0.000518161667 0.018769732740
624 1875152516686548 0.000530405893 0.019300138633
625 1919198629635734 0.000542864783 0.019843003416
626 1964009651915660 0.000555540035 0.020398543451
627 2009591422664545 0.000568433300 0.020966976751
628 2055949637927477 0.000581546191 0.021548522943
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1337 4444748 0.000000000001 0.999999999991
1338 3917670 0.000000000001 0.999999999992
1339 3449359 0.000000000001 0.999999999993
1340 3034135 0.000000000001 0.999999999994
1341 2665885 0.000000000001 0.999999999995
1342 2340024 0.000000000001 0.999999999996
1343 2051569 0.000000000001 0.999999999996
1344 1796855 0.000000000001 0.999999999997
1345 1571812 0.000000000000 0.999999999997
1346 1373524 0.000000000000 0.999999999997
1347 1198689 0.000000000000 0.999999999998
1348 1044984 0.000000000000 0.999999999998
1349 909741 0.000000000000 0.999999999998
1350 791131 0.000000000000 0.999999999999
1351 686983 0.000000000000 0.999999999999
1352 595872 0.000000000000 0.999999999999
1353 516054 0.000000000000 0.999999999999
1354 446405 0.000000000000 0.999999999999
1355 385528 0.000000000000 0.999999999999
1356 332557 0.000000000000 0.999999999999
1357 286364 0.000000000000 1.000000000000
1358 246288 0.000000000000 1.000000000000
1359 211428 0.000000000000 1.000000000000
1360 181274 0.000000000000 1.000000000000
1361 155112 0.000000000000 1.000000000000
1362 132559 0.000000000000 1.000000000000
1363 113039 0.000000000000 1.000000000000
1364 96271 0.000000000000 1.000000000000
1365 81801 0.000000000000 1.000000000000
1366 69414 0.000000000000 1.000000000000
1367 58754 0.000000000000 1.000000000000
1368 49668 0.000000000000 1.000000000000
1369 41869 0.000000000000 1.000000000000
1370 35251 0.000000000000 1.000000000000
1371 29588 0.000000000000 1.000000000000
1372 24803 0.000000000000 1.000000000000
1373 20722 0.000000000000 1.000000000000
1374 17293 0.000000000000 1.000000000000
1375 14375 0.000000000000 1.000000000000
1376 11937 0.000000000000 1.000000000000
1377 9871 0.000000000000 1.000000000000
1378 8154 0.000000000000 1.000000000000
1379 6703 0.000000000000 1.000000000000
1380 5507 0.000000000000 1.000000000000
1381 4498 0.000000000000 1.000000000000
1382 3673 0.000000000000 1.000000000000
1383 2980 0.000000000000 1.000000000000
1384 2417 0.000000000000 1.000000000000
1385 1946 0.000000000000 1.000000000000
1386 1568 0.000000000000 1.000000000000
1387 1251 0.000000000000 1.000000000000
1388 1000 0.000000000000 1.000000000000
1389 791 0.000000000000 1.000000000000
1390 627 0.000000000000 1.000000000000
1391 490 0.000000000000 1.000000000000
1392 385 0.000000000000 1.000000000000
1393 297 0.000000000000 1.000000000000
1394 231 0.000000000000 1.000000000000
1395 176 0.000000000000 1.000000000000
1396 135 0.000000000000 1.000000000000
1397 101 0.000000000000 1.000000000000
1398 77 0.000000000000 1.000000000000
1399 56 0.000000000000 1.000000000000
1400 42 0.000000000000 1.000000000000
1401 30 0.000000000000 1.000000000000
1402 22 0.000000000000 1.000000000000
1403 15 0.000000000000 1.000000000000
1404 11 0.000000000000 1.000000000000
1405 7 0.000000000000 1.000000000000
1406 5 0.000000000000 1.000000000000
1407 3 0.000000000000 1.000000000000
1408 2 0.000000000000 1.000000000000
1409 1 0.000000000000 1.000000000000
1410 1 0.000000000000 1.000000000000
Total 3,535,316,142,212,174,320 1.000000000000  

The average ball is 40.5, so the average sum is 20×40.5=810. As the table shows, the probability of getting exactly 810, which the mean, median, and mode, is 0.44%, or 1 in 227.46.

Programming Tips

My first attempt to get the combinations for any given total was by brute force, looping through all 3,535,316,142,212,174,320 ways to choose 20 balls out of 80 and taking a count of the sum of each. That program would have taken years to complete.

My second attempt was to look for a pattern. Perhaps a variant of the Fibonacci Sequence or line from a modified Pascal's Triangle. That was a noble attempt, but led to nothing.

My third attempt was at finding the generating function, but that would have been just as tedious as the first method. This can be done very easily in Mathematica, by the way, but I'm too cheap to buy it.

Finally, after some good programming tips from the math and computer geniuses at my forum, I made some programming shortcuts, which cut down the calculation time to under one second. Following is roughly the logic behind those shortcuts. This logic was inspired by code written by ZPP in this post on my forum.

As an example of this shortcut logic, the distribution of possible totals would be the same if the first three balls drawn were 1, 5, and 6 as if they were 2, 3, and 6. There is no need to loop through the other 18 balls twice. Likewise, the program never loops if the total of the remaining balls given the same number left and least possible ball has already been solved. This saves a HUGE amount of computer time.

I know that last statement is kind of heavy, so let me show you the critical line in my program:

a[k][level][m+t]+=a[m+1][level+1][t];

Here are the variable definitions:

Here is the full loop, after first filling in the 20th ball totals by hand.

	for (level=19; 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];
				}
			}
		}
	}

I hope that wasn't too confusing.

Internal Links

Lottery Sums — This same table, but based on a 6-49 lottery.

External Links

  • Assume the balls will be drawn in order, from smallest to largest.
  • Create an 81x21x1411 array. The first brackets will hold the smallest ball that can be drawn. The second the number of the next ball to be drawn (1 is the first, 20 is the last), and the third the total of all undrawn balls. The array will hold the count of the ways that total can be drawn, given the lowest possible ball and order of the next ball. The array could actually be 80x20x1410, but I prefer to not have to subtract one all the time to make the results fit.
  • Initialize all elements of that array to zero.
  • For the 20th ball, the number of combinations will be 1 for any total equal or greater than the smallest possible ball.
  • Starting with the 19th ball, and going down to the first, loop through all possible balls you can still get. For the array element for that least possible ball, ball order, and possible total of all remaining balls add the array element for one number greater than the least possible ball, one level up, and the total remaining from that level up (not counting the number just drawn.
    • a = name of array
    • k = least ball possible to be drawn (remember they are drawn in order)
    • level = ball order (where 20 is the last drawn and 1 is the first)
    • m = last number just drawn for the given level
    • t = total of all remaining balls AFTER the ball (m) just drawn, but not including m.

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