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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)
21 1 0.000000071511 0.000000071511
22 1 0.000000071511 0.000000143022
23 2 0.000000143022 0.000000286045
24 3 0.000000214534 0.000000500579
25 5 0.000000357556 0.000000858135
26 7 0.000000500579 0.000001358714
27 11 0.000000786624 0.000002145337
28 14 0.000001001157 0.000003146494
29 20 0.000001430225 0.000004576719
30 26 0.000001859292 0.000006436011
31 35 0.000002502893 0.000008938905
32 44 0.000003146494 0.000012085399
33 58 0.000004147652 0.000016233051
34 71 0.000005077298 0.000021310349
35 90 0.000006436011 0.000027746361
36 110 0.000007866236 0.000035612597
37 136 0.000009725528 0.000045338125
38 163 0.000011656332 0.000056994457
39 199 0.000014230736 0.000071225193
40 235 0.000016805141 0.000088030334
41 282 0.000020166169 0.000108196504
42 331 0.000023670220 0.000131866724
43 391 0.000027960894 0.000159827618
44 454 0.000032466102 0.000192293720
45 532 0.000038043979 0.000230337699
46 612 0.000043764878 0.000274102577
47 709 0.000050701468 0.000324804045
48 811 0.000057995614 0.000382799659
49 931 0.000066576963 0.000449376622
50 1057 0.000075587379 0.000524964001
51 1206 0.000086242554 0.000611206555
52 1360 0.000097255284 0.000708461839
53 1540 0.000110127307 0.000818589146
54 1729 0.000123642931 0.000942232077
55 1945 0.000139089359 0.001081321436
56 2172 0.000155322410 0.001236643846
57 2432 0.000173915332 0.001410559178
58 2702 0.000193223366 0.001603782544
59 3009 0.000215177316 0.001818959860
60 3331 0.000238203935 0.002057163796
61 3692 0.000264019492 0.002321183288
62 4070 0.000291050740 0.002612234028
63 4494 0.000321371505 0.002933605534
64 4935 0.000352907962 0.003286513495
65 5426 0.000388019980 0.003674533475
66 5940 0.000424776756 0.004099310231
67 6506 0.000465252117 0.004564562348
68 7097 0.000507515259 0.005072077607
69 7748 0.000554069075 0.005626146683
70 8423 0.000602339161 0.006228485844
71 9163 0.000655257478 0.006883743322
72 9933 0.000710321131 0.007594064453
73 10769 0.000770104527 0.008364168979
74 11637 0.000832176281 0.009196345261
75 12579 0.000899539868 0.010095885129
76 13552 0.000969120303 0.011065005432
77 14603 0.001044278615 0.012109284047
78 15690 0.001122011331 0.013231295377
79 16856 0.001205393435 0.014436688812
80 18059 0.001291421455 0.015728110267
81 19349 0.001383670952 0.017111781219
82 20673 0.001478351832 0.018590133051
83 22087 0.001579468723 0.020169601774
84 23540 0.001683374552 0.021852976326
85 25082 0.001793644882 0.023646621208
86 26663 0.001906704150 0.025553325358
87 28340 0.002026628497 0.027579953855
88 30051 0.002148984226 0.029728938081
89 31860 0.002278348056 0.032007286137
90 33706 0.002410357802 0.034417643939
91 35648 0.002549232627 0.036966876566
92 37625 0.002690610346 0.039657486912
93 39703 0.002839210699 0.042496697611
94 41809 0.002989813367 0.045486510978
95 44016 0.003147638670 0.048634149648
96 46253 0.003307609311 0.051941758959
97 48586 0.003474445030 0.055416203989
98 50944 0.003643068530 0.059059272519
99 53402 0.003818843154 0.062878115673
100 55875 0.003995690447 0.066873806120
101 58446 0.004179545841 0.071053351961
102 61031 0.004364402392 0.075417754353
103 63706 0.004555694955 0.079973449307
104 66388 0.004747488096 0.084720937404
105 69161 0.004945788760 0.089666726164
106 71928 0.005143660357 0.094810386521
107 74781 0.005347681920 0.100158068441
108 77624 0.005550988371 0.105709056813
109 80542 0.005759658165 0.111468714977
110 83440 0.005966897734 0.117435612711
111 86412 0.006179429134 0.123615041846
112 89348 0.006389386130 0.130004427976
113 92350 0.006604062868 0.136608490844
114 95311 0.006815807645 0.143424298489
115 98324 0.007031271006 0.150455569495
116 101285 0.007243015783 0.157698585279
117 104295 0.007458264611 0.165156849890
118 107235 0.007668507652 0.172825357542
119 110215 0.007881611142 0.180706968684
120 113119 0.008089279779 0.188796248463
121 116048 0.008298736196 0.197094984659
122 118889 0.008501899625 0.205596884284
123 121751 0.008706564789 0.214303449073
124 124507 0.008903649762 0.223207098835
125 127274 0.009101521359 0.232308620194
126 129930 0.009291455208 0.241600075401
127 132581 0.009481031501 0.251081106902
128 135109 0.009661811912 0.260742918814
129 137629 0.009842020233 0.270584939047
130 140008 0.010012145469 0.280597084515
131 142370 0.010181055014 0.290778139529
132 144587 0.010339595429 0.301117734959
133 146771 0.010495775974 0.311613510933
134 148800 0.010640872277 0.322254383210
135 150794 0.010783465686 0.333037848896
136 152617 0.010913830674 0.343951679570
137 154397 0.011041120678 0.354992800249
138 156004 0.011156039239 0.366148839487
139 157554 0.011266881658 0.377415721145
140 158923 0.011364780543 0.388780501689
141 160236 0.011458674799 0.400239176488
142 161354 0.011538624364 0.411777800852
143 162410 0.011614140232 0.423391941084
144 163273 0.011675854431 0.435067795514
145 164062 0.011732276798 0.446800072312
146 164654 0.011774611451 0.458574683763
147 165176 0.011811940317 0.470386624080
148 165490 0.011834394846 0.482221018926
149 165732 0.011851700566 0.494072719492
150 165772 0.011854561015 0.505927280508
151 165732 0.011851700566 0.517778981074
152 165490 0.011834394846 0.529613375920
153 165176 0.011811940317 0.541425316237
154 164654 0.011774611451 0.553199927688
155 164062 0.011732276798 0.564932204486
156 163273 0.011675854431 0.576608058916
157 162410 0.011614140232 0.588222199148
158 161354 0.011538624364 0.599760823512
159 160236 0.011458674799 0.611219498311
160 158923 0.011364780543 0.622584278855
161 157554 0.011266881658 0.633851160513
162 156004 0.011156039239 0.645007199751
163 154397 0.011041120678 0.656048320430
164 152617 0.010913830674 0.666962151104
165 150794 0.010783465686 0.677745616790
166 148800 0.010640872277 0.688386489067
167 146771 0.010495775974 0.698882265041
168 144587 0.010339595429 0.709221860471
169 142370 0.010181055014 0.719402915485
170 140008 0.010012145469 0.729415060953
171 137629 0.009842020233 0.739257081186
172 135109 0.009661811912 0.748918893098
173 132581 0.009481031501 0.758399924599
174 129930 0.009291455208 0.767691379806
175 127274 0.009101521359 0.776792901165
176 124507 0.008903649762 0.785696550927
177 121751 0.008706564789 0.794403115716
178 118889 0.008501899625 0.802905015341
179 116048 0.008298736196 0.811203751537
180 113119 0.008089279779 0.819293031316
181 110215 0.007881611142 0.827174642458
182 107235 0.007668507652 0.834843150110
183 104295 0.007458264611 0.842301414721
184 101285 0.007243015783 0.849544430505
185 98324 0.007031271006 0.856575701511
186 95311 0.006815807645 0.863391509156
187 92350 0.006604062868 0.869995572024
188 89348 0.006389386130 0.876384958154
189 86412 0.006179429134 0.882564387289
190 83440 0.005966897734 0.888531285023
191 80542 0.005759658165 0.894290943187
192 77624 0.005550988371 0.899841931559
193 74781 0.005347681920 0.905189613479
194 71928 0.005143660357 0.910333273836
195 69161 0.004945788760 0.915279062596
196 66388 0.004747488096 0.920026550693
197 63706 0.004555694955 0.924582245647
198 61031 0.004364402392 0.928946648039
199 58446 0.004179545841 0.933126193880
200 55875 0.003995690447 0.937121884327
201 53402 0.003818843154 0.940940727481
202 50944 0.003643068530 0.944583796011
203 48586 0.003474445030 0.948058241041
204 46253 0.003307609311 0.951365850352
205 44016 0.003147638670 0.954513489022
206 41809 0.002989813367 0.957503302389
207 39703 0.002839210699 0.960342513088
208 37625 0.002690610346 0.963033123434
209 35648 0.002549232627 0.965582356061
210 33706 0.002410357802 0.967992713863
211 31860 0.002278348056 0.970271061919
212 30051 0.002148984226 0.972420046145
213 28340 0.002026628497 0.974446674642
214 26663 0.001906704150 0.976353378792
215 25082 0.001793644882 0.978147023674
216 23540 0.001683374552 0.979830398226
217 22087 0.001579468723 0.981409866949
218 20673 0.001478351832 0.982888218781
219 19349 0.001383670952 0.984271889733
220 18059 0.001291421455 0.985563311188
221 16856 0.001205393435 0.986768704623
222 15690 0.001122011331 0.987890715953
223 14603 0.001044278615 0.988934994568
224 13552 0.000969120303 0.989904114871
225 12579 0.000899539868 0.990803654739
226 11637 0.000832176281 0.991635831021
227 10769 0.000770104527 0.992405935547
228 9933 0.000710321131 0.993116256678
229 9163 0.000655257478 0.993771514156
230 8423 0.000602339161 0.994373853317
231 7748 0.000554069075 0.994927922393
232 7097 0.000507515259 0.995435437652
233 6506 0.000465252117 0.995900689769
234 5940 0.000424776756 0.996325466525
235 5426 0.000388019980 0.996713486505
236 4935 0.000352907962 0.997066394466
237 4494 0.000321371505 0.997387765972
238 4070 0.000291050740 0.997678816712
239 3692 0.000264019492 0.997942836204
240 3331 0.000238203935 0.998181040140
241 3009 0.000215177316 0.998396217456
242 2702 0.000193223366 0.998589440822
243 2432 0.000173915332 0.998763356154
244 2172 0.000155322410 0.998918678564
245 1945 0.000139089359 0.999057767923
246 1729 0.000123642931 0.999181410854
247 1540 0.000110127307 0.999291538161
248 1360 0.000097255284 0.999388793445
249 1206 0.000086242554 0.999475035999
250 1057 0.000075587379 0.999550623378
251 931 0.000066576963 0.999617200341
252 811 0.000057995614 0.999675195955
253 709 0.000050701468 0.999725897423
254 612 0.000043764878 0.999769662301
255 532 0.000038043979 0.999807706280
256 454 0.000032466102 0.999840172382
257 391 0.000027960894 0.999868133276
258 331 0.000023670220 0.999891803496
259 282 0.000020166169 0.999911969666
260 235 0.000016805141 0.999928774807
261 199 0.000014230736 0.999943005543
262 163 0.000011656332 0.999954661875
263 136 0.000009725528 0.999964387403
264 110 0.000007866236 0.999972253639
265 90 0.000006436011 0.999978689651
266 71 0.000005077298 0.999983766949
267 58 0.000004147652 0.999987914601
268 44 0.000003146494 0.999991061095
269 35 0.000002502893 0.999993563989
270 26 0.000001859292 0.999995423281
271 20 0.000001430225 0.999996853506
272 14 0.000001001157 0.999997854663
273 11 0.000000786624 0.999998641286
274 7 0.000000500579 0.999999141865
275 5 0.000000357556 0.999999499421
276 3 0.000000214534 0.999999713955
277 2 0.000000143022 0.999999856978
278 1 0.000000071511 0.999999928489
279 1 0.000000071511 1.000000000000

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).

Written by: Michael Shackleford