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Last Updated: May 28, 2019

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Lottery Sums

Introduction

This page shows the probability of any given sum of in a 6-49 lottery. For those not familiar with a 6-49 keno, it is a game in which the player bets on the draw of six balls, without replacement, numbered 1 to 49. It is a classic form of the lottery and one found in most states with a lottery. I created this page when I noticed bets on the sum of the draw in the game 49 Gold Rush.

There are combin(49,6)= 13,983,816 ways to choose 6 items out of 49. The following table shows the number of combinations for any given possible total from 21 to 279, the probability of that total, and the probability of that total or less.

6/49 Lottery Sum of Ball Draw

Total Combinations Probability
(Exact)
Probability
(This Total or Less)
2110.0000000715110.000000071511
2210.0000000715110.000000143022
2320.0000001430220.000000286045
2430.0000002145340.000000500579
2550.0000003575560.000000858135
2670.0000005005790.000001358714
27110.0000007866240.000002145337
28140.0000010011570.000003146494
29200.0000014302250.000004576719
30260.0000018592920.000006436011
31350.0000025028930.000008938905
32440.0000031464940.000012085399
33580.0000041476520.000016233051
34710.0000050772980.000021310349
35900.0000064360110.000027746361
361100.0000078662360.000035612597
371360.0000097255280.000045338125
381630.0000116563320.000056994457
391990.0000142307360.000071225193
402350.0000168051410.000088030334
412820.0000201661690.000108196504
423310.0000236702200.000131866724
433910.0000279608940.000159827618
444540.0000324661020.000192293720
455320.0000380439790.000230337699
466120.0000437648780.000274102577
477090.0000507014680.000324804045
488110.0000579956140.000382799659
499310.0000665769630.000449376622
5010570.0000755873790.000524964001
5112060.0000862425540.000611206555
5213600.0000972552840.000708461839
5315400.0001101273070.000818589146
5417290.0001236429310.000942232077
5519450.0001390893590.001081321436
5621720.0001553224100.001236643846
5724320.0001739153320.001410559178
5827020.0001932233660.001603782544
5930090.0002151773160.001818959860
6033310.0002382039350.002057163796
6136920.0002640194920.002321183288
6240700.0002910507400.002612234028
6344940.0003213715050.002933605534
6449350.0003529079620.003286513495
6554260.0003880199800.003674533475
6659400.0004247767560.004099310231
6765060.0004652521170.004564562348
6870970.0005075152590.005072077607
6977480.0005540690750.005626146683
7084230.0006023391610.006228485844
7191630.0006552574780.006883743322
7299330.0007103211310.007594064453
73107690.0007701045270.008364168979
74116370.0008321762810.009196345261
75125790.0008995398680.010095885129
76135520.0009691203030.011065005432
77146030.0010442786150.012109284047
78156900.0011220113310.013231295377
79168560.0012053934350.014436688812
80180590.0012914214550.015728110267
81193490.0013836709520.017111781219
82206730.0014783518320.018590133051
83220870.0015794687230.020169601774
84235400.0016833745520.021852976326
85250820.0017936448820.023646621208
86266630.0019067041500.025553325358
87283400.0020266284970.027579953855
88300510.0021489842260.029728938081
89318600.0022783480560.032007286137
90337060.0024103578020.034417643939
91356480.0025492326270.036966876566
92376250.0026906103460.039657486912
93397030.0028392106990.042496697611
94418090.0029898133670.045486510978
95440160.0031476386700.048634149648
96462530.0033076093110.051941758959
97485860.0034744450300.055416203989
98509440.0036430685300.059059272519
99534020.0038188431540.062878115673
100558750.0039956904470.066873806120
101584460.0041795458410.071053351961
102610310.0043644023920.075417754353
103637060.0045556949550.079973449307
104663880.0047474880960.084720937404
105691610.0049457887600.089666726164
106719280.0051436603570.094810386521
107747810.0053476819200.100158068441
108776240.0055509883710.105709056813
109805420.0057596581650.111468714977
110834400.0059668977340.117435612711
111864120.0061794291340.123615041846
112893480.0063893861300.130004427976
113923500.0066040628680.136608490844
114953110.0068158076450.143424298489
115983240.0070312710060.150455569495
1161012850.0072430157830.157698585279
1171042950.0074582646110.165156849890
1181072350.0076685076520.172825357542
1191102150.0078816111420.180706968684
1201131190.0080892797790.188796248463
1211160480.0082987361960.197094984659
1221188890.0085018996250.205596884284
1231217510.0087065647890.214303449073
1241245070.0089036497620.223207098835
1251272740.0091015213590.232308620194
1261299300.0092914552080.241600075401
1271325810.0094810315010.251081106902
1281351090.0096618119120.260742918814
1291376290.0098420202330.270584939047
1301400080.0100121454690.280597084515
1311423700.0101810550140.290778139529
1321445870.0103395954290.301117734959
1331467710.0104957759740.311613510933
1341488000.0106408722770.322254383210
1351507940.0107834656860.333037848896
1361526170.0109138306740.343951679570
1371543970.0110411206780.354992800249
1381560040.0111560392390.366148839487
1391575540.0112668816580.377415721145
1401589230.0113647805430.388780501689
1411602360.0114586747990.400239176488
1421613540.0115386243640.411777800852
1431624100.0116141402320.423391941084
1441632730.0116758544310.435067795514
1451640620.0117322767980.446800072312
1461646540.0117746114510.458574683763
1471651760.0118119403170.470386624080
1481654900.0118343948460.482221018926
1491657320.0118517005660.494072719492
1501657720.0118545610150.505927280508
1511657320.0118517005660.517778981074
1521654900.0118343948460.529613375920
1531651760.0118119403170.541425316237
1541646540.0117746114510.553199927688
1551640620.0117322767980.564932204486
1561632730.0116758544310.576608058916
1571624100.0116141402320.588222199148
1581613540.0115386243640.599760823512
1591602360.0114586747990.611219498311
1601589230.0113647805430.622584278855
1611575540.0112668816580.633851160513
1621560040.0111560392390.645007199751
1631543970.0110411206780.656048320430
1641526170.0109138306740.666962151104
1651507940.0107834656860.677745616790
1661488000.0106408722770.688386489067
1671467710.0104957759740.698882265041
1681445870.0103395954290.709221860471
1691423700.0101810550140.719402915485
1701400080.0100121454690.729415060953
1711376290.0098420202330.739257081186
1721351090.0096618119120.748918893098
1731325810.0094810315010.758399924599
1741299300.0092914552080.767691379806
1751272740.0091015213590.776792901165
1761245070.0089036497620.785696550927
1771217510.0087065647890.794403115716
1781188890.0085018996250.802905015341
1791160480.0082987361960.811203751537
1801131190.0080892797790.819293031316
1811102150.0078816111420.827174642458
1821072350.0076685076520.834843150110
1831042950.0074582646110.842301414721
1841012850.0072430157830.849544430505
185983240.0070312710060.856575701511
186953110.0068158076450.863391509156
187923500.0066040628680.869995572024
188893480.0063893861300.876384958154
189864120.0061794291340.882564387289
190834400.0059668977340.888531285023
191805420.0057596581650.894290943187
192776240.0055509883710.899841931559
193747810.0053476819200.905189613479
194719280.0051436603570.910333273836
195691610.0049457887600.915279062596
196663880.0047474880960.920026550693
197637060.0045556949550.924582245647
198610310.0043644023920.928946648039
199584460.0041795458410.933126193880
200558750.0039956904470.937121884327
201534020.0038188431540.940940727481
202509440.0036430685300.944583796011
203485860.0034744450300.948058241041
204462530.0033076093110.951365850352
205440160.0031476386700.954513489022
206418090.0029898133670.957503302389
207397030.0028392106990.960342513088
208376250.0026906103460.963033123434
209356480.0025492326270.965582356061
210337060.0024103578020.967992713863
211318600.0022783480560.970271061919
212300510.0021489842260.972420046145
213283400.0020266284970.974446674642
214266630.0019067041500.976353378792
215250820.0017936448820.978147023674
216235400.0016833745520.979830398226
217220870.0015794687230.981409866949
218206730.0014783518320.982888218781
219193490.0013836709520.984271889733
220180590.0012914214550.985563311188
221168560.0012053934350.986768704623
222156900.0011220113310.987890715953
223146030.0010442786150.988934994568
224135520.0009691203030.989904114871
225125790.0008995398680.990803654739
226116370.0008321762810.991635831021
227107690.0007701045270.992405935547
22899330.0007103211310.993116256678
22991630.0006552574780.993771514156
23084230.0006023391610.994373853317
23177480.0005540690750.994927922393
23270970.0005075152590.995435437652
23365060.0004652521170.995900689769
23459400.0004247767560.996325466525
23554260.0003880199800.996713486505
23649350.0003529079620.997066394466
23744940.0003213715050.997387765972
23840700.0002910507400.997678816712
23936920.0002640194920.997942836204
24033310.0002382039350.998181040140
24130090.0002151773160.998396217456
24227020.0001932233660.998589440822
24324320.0001739153320.998763356154
24421720.0001553224100.998918678564
24519450.0001390893590.999057767923
24617290.0001236429310.999181410854
24715400.0001101273070.999291538161
24813600.0000972552840.999388793445
24912060.0000862425540.999475035999
25010570.0000755873790.999550623378
2519310.0000665769630.999617200341
2528110.0000579956140.999675195955
2537090.0000507014680.999725897423
2546120.0000437648780.999769662301
2555320.0000380439790.999807706280
2564540.0000324661020.999840172382
2573910.0000279608940.999868133276
2583310.0000236702200.999891803496
2592820.0000201661690.999911969666
2602350.0000168051410.999928774807
2611990.0000142307360.999943005543
2621630.0000116563320.999954661875
2631360.0000097255280.999964387403
2641100.0000078662360.999972253639
265900.0000064360110.999978689651
266710.0000050772980.999983766949
267580.0000041476520.999987914601
268440.0000031464940.999991061095
269350.0000025028930.999993563989
270260.0000018592920.999995423281
271200.0000014302250.999996853506
272140.0000010011570.999997854663
273110.0000007866240.999998641286
27470.0000005005790.999999141865
27550.0000003575560.999999499421
27630.0000002145340.999999713955
27720.0000001430220.999999856978
27810.0000000715110.999999928489
27910.0000000715111.000000000000

The average ball is 25, so the average sum is 25×6=150. As the table shows, the probability of getting exactly 150, which the mean, median, and mode, is 1.19%, or 1 in 84.4.

Internal Links

Keno Sums — The exact same topic, but based on a draw of 20 balls out of 80. Contains a section on programming tips.

External Links

Here are some links on the math behind find the number of combinations of each sum in keno (20 balls drawn out of 80).


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