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Colin F
Lotto Systems Tester Creator & Analyst
Lotto Systems Tester Creator & Analyst
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Quote Colin F Replybullet Topic: UNDERSTANDING LOTTO ODDS AND SIMPLIFICATION
    Posted: May 27 2009 at 3:28pm
UNDERSTANDING LOTTO ODDS AND SIMPLIFICATION
by Colin Fairbrother

Odds is a generic term which covers the different ways of expressing the chances of success for a given Lotto game and is best understood by adopting some flexibility towards the method of expression. Probability comes under the Odds umbrella and is the ratio of the chances of success for a given scenario with all the possibilities.

Using the classic and very popular Lotto game where 6 integers (whole numbers or balls) are randomly chosen from 49 then the total possiblities are 13,983,816. You will find this game referred to as a 6/49 Lotto game with the forward slash used as a seperator and not to indicate division. The odds in favor for the 6/49 Lotto game are usually expressed as the ratio of playing one combination or line against all the possibilties ie 1 in 13,983,816 and 1 divided by 13,983,816 gives a probability of success for the unique winning number of 0.00000007 when one line or number is played.

Some official Lotto sites will give Odds Against as in 13,983,816:1 or 13,983,816 to 1.

Since 2003 I have posted on the internet bits and pieces on Odds in Lotto and in this article I will pull as much as I can together to aid understanding. Probably, the most misunderstood and likely to draw swift dissension concept is that of reduction or simplification ie 50/100 = 25/50 = 1/2. Playing 1 line your chances of getting 1st prize are 1 in 13,983,816. No problem there - every one agrees. However, if you play 2 lines your chances of getting 1st prize are 2 in 13,983,816 or 1 in 6,491,908. This reduction is not accepted by some as being valid.

If we have 5 plays then 13,983,816 is not evenly divisible by 5 so it can't be simplified and we are stuck with 5 in 13,983,816. Playing 9 once again 13,983,816 can't be evenly divided by 9 but we can divide each side by 3 to get the simplified expression 3 in 4,661,272.

Below is a table for the 6/49 Lotto game showing 192 cases where the odds may be expressed as 1 in something because 13,983,816 is evenly divisible by the number of plays.

The concept becomes easier to understand when it is realized that this is just an average ratio dependent on adequate sampling where the difference is brought down to around the 5% mark compared with the theoretical probability calculation.

If we enumerate the 13,983,816 combinations for a 6/49 Lotto game in lexicographical or numerical order producing an index at the same time then we would have both 6,991,908 odd and even indexes. ie


                Index    Combination         Odd or Even Index

                   1    1  2  3  4  5  6           O

                   2    1  2  3  4  5  7           E

                   3    1  2  3  4  5  8           O

                   .    .  .  .  .  .  .           .

                   .    .  .  .  .  .  .           .

                   .    .  .  .  .  .  .           .

          13,983,816    44 45 46 47 48 49          E

Now, if you take all the 6/49 Lotto games around the world to date and put them into a table while still retaining the draw date order you will have some 19,577 lines with 24 repeats and a difference between the odds and even indexes from the full set as described previously of around 5% which is considered normal. Currently, it favours even indexes but over the next 10 to 20 years it could drift the other way.

Using our odds and even indexes the single winning number has either an odd or an even index - it can't have both. Allocating 1 ticket to the odd indexes and the other to the even we see the valid ratio is 1 in 6,991,908. But as previously explained we could have a run of 13 even or odd indexes for the draw numbers so this ratio must be considered as valid only within the context of adequate sampling. Playing just 1 line or ticket there is no guarantee your line will come up in 13,983,816 draws and in fact for that number of draws there will be repeats with only close to 63% unique. Playing 2 lines or tickets, maybe, no guarantee and how relevant is 140,000 years?

Checking out the various 6/49 games you will find even though you are theoretically playing half the possibilities you can still have a run of 13 odd or even indexes for 19,577 draws (see table below), Therefore, an adequate sample would have to be at least around 100 draws. In other words valuing 1 ticket at $1 and 1st prize at $5,000,000 you spend $600,491,908 to get $250,000,000 plus some secondary prizes. In the USA you are taxed on winnings so you can half that amount. Your stake for such a foolhardy endeavour would have to be enormous as at times you could lose consecutively some $80,000,000 all efforts progressing towards a loss of some $350,000,000 - good for the Lotto operators bad for anyone contemplating the idea of deliberately setting out to get a 1st prize win with an unlimited budget.


RunCnt 	  Cnt
  13 	   1
  12 	   3
  11 	   5
  10 	   9
   9 	  20
   8 	  39
   7 	  62
   6 	 163
   5 	 278
   4 	 640
   3 	1237
   2 	2489
   1 	4838     


Odds for winning 1st Prize in 6/49 Lotto game as plays per draw are increased.

PlaysSimplified OddsProbability
1 1 in 13983816 0.00000007
2 1 in 6991908 0.00000014
3 1 in 4661272 0.00000021
4 1 in 3495954 0.00000028
6 1 in 2330636 0.00000042
7 1 in 1997688 0.00000050
8 1 in 1747977 0.00000057
11 1 in 1271256 0.00000078
12 1 in 1165318 0.00000085
14 1 in 998844 0.00000100
21 1 in 665896 0.00000150
22 1 in 635628 0.00000157
23 1 in 607992 0.00000164
24 1 in 582659 0.00000171
28 1 in 499422 0.00000200
33 1 in 423752 0.00000235
42 1 in 332948 0.00000300
44 1 in 317814 0.00000314
46 1 in 303996 0.00000328
47 1 in 297528 0.00000336
49 1 in 285384 0.00000350
56 1 in 249711 0.00000400
66 1 in 211876 0.00000471
69 1 in 202664 0.00000493
77 1 in 181608 0.00000550
84 1 in 166474 0.00000600
88 1 in 158907 0.00000629
92 1 in 151998 0.00000657
94 1 in 148764 0.00000672
98 1 in 142692 0.00000701
132 1 in 105938 0.00000943
138 1 in 101332 0.00000986
141 1 in 99176 0.00001008
147 1 in 95128 0.00001051
154 1 in 90804 0.00001101
161 1 in 86856 0.00001151
168 1 in 83237 0.00001201
184 1 in 75999 0.00001315
188 1 in 74382 0.00001344
196 1 in 71346 0.00001401
231 1 in 60536 0.00001651
253 1 in 55272 0.00001809
264 1 in 52969 0.00001887
276 1 in 50666 0.00001973
282 1 in 49588 0.00002016
294 1 in 47564 0.00002102
308 1 in 45402 0.00002202
322 1 in 43428 0.00002302
329 1 in 42504 0.00002352
376 1 in 37191 0.00002688
392 1 in 35673 0.00002803
462 1 in 30268 0.00003303
483 1 in 28952 0.00003453
506 1 in 27636 0.00003618
517 1 in 27048 0.00003697
539 1 in 25944 0.00003854
552 1 in 25333 0.00003947
564 1 in 24794 0.00004033
588 1 in 23782 0.00004204
616 1 in 22701 0.00004405
644 1 in 21714 0.00004605
658 1 in 21252 0.00004705
759 1 in 18424 0.00005427
924 1 in 15134 0.00006607
966 1 in 14476 0.00006907
987 1 in 14168 0.00007058
1012 1 in 13818 0.00007236
1034 1 in 13524 0.00007394
1078 1 in 12972 0.00007708
1081 1 in 12936 0.0000773
1127 1 in 12408 0.00008059
1128 1 in 12397 0.00008066
1176 1 in 11891 0.00008409
1288 1 in 10857 0.00009211
1316 1 in 10626 0.00009411
1518 1 in 9212 0.00010855
1551 1 in 9016 0.00011091
1617 1 in 8648 0.00011563
1771 1 in 7896 0.00012664
1848 1 in 7567 0.00013215
1932 1 in 7238 0.00013815
1974 1 in 7084 0.00014116
2024 1 in 6909 0.00014473
2068 1 in 6762 0.00014788
2156 1 in 6486 0.00015417
2162 1 in 6468 0.00015461
2254 1 in 6204 0.00016118
2303 1 in 6072 0.00016469
2632 1 in 5313 0.00018821
3036 1 in 4606 0.00021711
3102 1 in 4508 0.00022182
3234 1 in 4324 0.00023126
3243 1 in 4312 0.00023191
3381 1 in 4136 0.00024177
3542 1 in 3948 0.00025329
3619 1 in 3864 0.00025879
3864 1 in 3619 0.00027631
3948 1 in 3542 0.00028232
4136 1 in 3381 0.00029577
4312 1 in 3243 0.00030835
4324 1 in 3234 0.00030921
4508 1 in 3102 0.00032237
4606 1 in 3036 0.00032938
5313 1 in 2632 0.00037993
6072 1 in 2303 0.00043421
6204 1 in 2254 0.00044365
6468 1 in 2162 0.00046253
6486 1 in 2156 0.00046382
6762 1 in 2068 0.00048355
6909 1 in 2024 0.00049407
7084 1 in 1974 0.00050658
7238 1 in 1932 0.00051759
7567 1 in 1848 0.00054112
7896 1 in 1771 0.00056465
8648 1 in 1617 0.00061842
9016 1 in 1551 0.00064474
9212 1 in 1518 0.00065876
10626 1 in 1316 0.00075987
10857 1 in 1288 0.00077639
11891 1 in 1176 0.00085034
12397 1 in 1128 0.00088652
12408 1 in 1127 0.00088731
12936 1 in 1081 0.00092506
12972 1 in 1078 0.00092764
13524 1 in 1034 0.00096711
13818 1 in 1012 0.00098814
14168 1 in 987 0.00101317
14476 1 in 966 0.00103519
15134 1 in 924 0.00108225
18424 1 in 759 0.00131752
21252 1 in 658 0.00151975
21714 1 in 644 0.00155279
22701 1 in 616 0.00162337
23782 1 in 588 0.00170068
24794 1 in 564 0.00177304
25333 1 in 552 0.00181159
25944 1 in 539 0.00185528
27048 1 in 517 0.00193423
27636 1 in 506 0.00197628
28952 1 in 483 0.00207039
30268 1 in 462 0.0021645
35673 1 in 392 0.00255102
37191 1 in 376 0.00265957
42504 1 in 329 0.00303951
43428 1 in 322 0.00310559
45402 1 in 308 0.00324675
47564 1 in 294 0.00340136
49588 1 in 282 0.00354609
50666 1 in 276 0.00362318
52969 1 in 264 0.00378787
55272 1 in 253 0.00395256
60536 1 in 231 0.00432900
71346 1 in 196 0.00510204
74382 1 in 188 0.00531914
75999 1 in 184 0.00543478
83237 1 in 168 0.00595238
86856 1 in 161 0.00621118
90804 1 in 154 0.0064935
95128 1 in 147 0.00680272
99176 1 in 141 0.00709219
101332 1 in 138 0.00724637
105938 1 in 132 0.00757575
142692 1 in 98 0.01020408
148764 1 in 94 0.01063829
151998 1 in 92 0.01086956
158907 1 in 88 0.01136363
166474 1 in 84 0.01190476
181608 1 in 77 0.01298701
202664 1 in 69 0.01449275
211876 1 in 66 0.01515151
249711 1 in 56 0.01785714
285384 1 in 49 0.02040816
297528 1 in 47 0.02127659
303996 1 in 46 0.02173913
317814 1 in 44 0.02272727
332948 1 in 42 0.02380952
423752 1 in 33 0.03030303
499422 1 in 28 0.03571428
582659 1 in 24 0.04166666
607992 1 in 23 0.04347826
635628 1 in 22 0.04545454
665896 1 in 21 0.04761904
998844 1 in 14 0.07142857
1165318 1 in 12 0.08333333
1271256 1 in 11 0.09090909
1747977 1 in 8 0.12500000
1997688 1 in 7 0.14285720
2330636 1 in 6 0.16666670
3495954 1 in 4 0.25000000
4661272 1 in 3 0.33333330
6991908 1 in 2 0.50000000
13983816 1 in 1 1.00000000

Colin Fairbrother

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