Mathematics (Jackpot) - Match m numbers out of n

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lottoarchitect
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Mathematics (Jackpot) - Match m numbers out of n

Post by lottoarchitect » Mon Jan 28, 2008 4:51 am

TRANSFERED FROM OLD FORUM

There is a simple equation to determine the chance to match m numbers out of n. It works only for fields that don't allow same numbers to be drawn (no Pick style games).

Suppose you have a lottery game a of b (eg 5 of 45, a=5 b=45). You want to pick n numbers (eg 10 numbers, n=10) and you want to find out the chance to have m matches (eg 3 matches, m=3) in your selection of n numbers.

Then, chance %= [ nCm * (b-n)C(a-m) / bCa ] * 100%

where nCk =n!/ (k!*(n-k)!) [The combination formula]

Example: game 6/49, we pick 14 numbers and we want to find the chance to have 3 numbers correct out of the 14 picked ones. Then,
a=6, b=49, n=14, m=3

%=14C3 * (49-14)C(6-3) / 49C6 * 100% = 364 * 35C3 / 13983816 * 100% =
= 364 * 6545 / 13983816 * 100% = 17.03%

The following tables are the results of the above equation. They describe the chance to match m numbers out of n picked ones for several types of lotteries.

Horizontal axis - m parameter
Vertical axis - n parameter

Game type 5 of 45 (a=5, b=45)

Code: Select all

      0       1       2       3       4       5
01  88,89%  11,11%  ------  ------  ------  ------
02  78,79%  20,20%  01,01%  ------  ------  ------  
03  69,63%  27,48%  02,82%  00,07%  ------  ------  
04  61,34%  33,16%  05,24%  00,27%  00,00%  ------  
05  53,86%  37,40%  08,09%  00,64%  00,02%  00,00%
06  47,13%  40,39%  11,22%  01,21%  00,05%  00,00%
07  41,08%  42,29%  14,50%  02,01%  00,11%  00,00%
08  35,68%  43,25%  17,81%  03,05%  00,21%  00,00%
09  30,86%  43,39%  21,04%  04,33%  00,37%  00,01%
10  26,57%  42,86%  24,11%  05,84%  00,60%  00,02%
11  22,78%  41,75%  26,94%  07,58%  00,92%  00,04%
12  19,43%  40,19%  29,47%  09,51%  01,34%  00,06%
13  16,48%  38,26%  31,67%  11,61%  01,87%  00,11%
14  13,91%  36,06%  33,48%  13,85%  02,54%  00,16%
15  11,66%  33,65%  34,89%  16,20%  03,35%  00,25%
16  09,72%  31,10%  35,89%  18,61%  04,32%  00,36%
17  08,04%  28,49%  36,47%  21,04%  05,45%  00,51%
18  06,61%  25,86%  36,63%  23,44%  06,76%  00,70%
19  05,38%  23,25%  36,39%  25,78%  08,25%  00,95%
20  04,35%  20,71%  35,77%  27,99%  09,91%  01,27%
21  03,48%  18,26%  34,79%  30,05%  11,76%  01,67%
22  02,75%  15,95%  33,48%  31,89%  13,77%  02,16%
23  02,16%  13,77%  31,89%  33,48%  15,95%  02,75%
24  01,67%  11,76%  30,05%  34,79%  18,26%  03,48%
25  01,27%  09,91%  27,99%  35,77%  20,71%  04,35%
26  00,95%  08,25%  25,78%  36,39%  23,25%  05,38%
27  00,70%  06,76%  23,44%  36,63%  25,86%  06,61%
28  00,51%  05,45%  21,04%  36,47%  28,49%  08,04%
29  00,36%  04,32%  18,61%  35,89%  31,10%  09,72%
30  00,25%  03,35%  16,20%  34,89%  33,65%  11,66%
31  00,16%  02,54%  13,85%  33,48%  36,06%  13,91%
32  00,11%  01,87%  11,61%  31,67%  38,26%  16,48%
33  00,06%  01,34%  09,51%  29,47%  40,19%  19,43%
34  00,04%  00,92%  07,58%  26,94%  41,75%  22,78%
35  00,02%  00,60%  05,84%  24,11%  42,86%  26,57%
36  00,01%  00,37%  04,33%  21,04%  43,39%  30,86%
37  00,00%  00,21%  03,05%  17,81%  43,25%  35,68%
38  00,00%  00,11%  02,01%  14,50%  42,29%  41,08%
39  00,00%  00,05%  01,21%  11,22%  40,39%  47,13%
40  00,00%  00,02%  00,64%  08,09%  37,40%  53,86%
41  ------  00,00%  00,27%  05,24%  33,16%  61,34%
42  ------  ------  00,07%  02,82%  27,48%  69,63%
43  ------  ------  ------  01,01%  20,20%  78,79%
44  ------  ------  ------  ------  11,11%  88,89%
45  ------  ------  ------  ------  ------ 100,00%
Game type 6 of 49 (a=6, b=49)

Code: Select all

      0       1       2       3       4       5       6
01  87,76%  12,24%  ------  ------  ------  ------  ------
02  76,79%  21,94%  01,28%  ------  ------  ------  ------
03  66,98%  29,41%  03,50%  00,11%  ------  ------  ------
04  58,25%  34,95%  06,39%  00,41%  00,01%  ------  ------
05  50,48%  38,83%  09,71%  00,95%  00,03%  00,00%  ------
06  43,60%  41,30%  13,24%  01,77%  00,10%  00,00%  00,00%
07  37,51%  42,58%  16,81%  02,87%  00,22%  00,01%  00,00%
08  32,15%  42,87%  20,28%  04,27%  00,41%  00,02%  00,00%
09  27,45%  42,35%  23,53%  05,93%  00,70%  00,04%  00,00%
10  23,33%  41,17%  26,47%  07,84%  01,11%  00,07%  00,00%
11  19,74%  39,48%  29,03%  09,95%  01,66%  00,13%  00,00%
12  16,62%  37,41%  31,17%  12,22%  02,36%  00,21%  00,01%
13  13,93%  35,05%  32,86%  14,60%  03,22%  00,33%  00,01%
14  11,61%  32,50%  34,07%  17,04%  04,26%  00,50%  00,02%
15  09,62%  29,85%  34,82%  19,47%  05,48%  00,73%  00,04%
16  07,92%  27,16%  35,11%  21,85%  06,87%  01,03%  00,06%
17  06,48%  24,48%  34,97%  24,12%  08,44%  01,42%  00,09%
18  05,27%  21,87%  34,43%  26,23%  10,18%  01,90%  00,13%
19  04,25%  19,36%  33,51%  28,13%  12,06%  02,49%  00,19%
20  03,40%  16,98%  32,27%  29,79%  14,07%  03,22%  00,28%
21  02,69%  14,76%  30,75%  31,16%  16,18%  04,07%  00,39%
22  02,12%  12,70%  28,99%  32,21%  18,36%  05,08%  00,53%
23  01,65%  10,82%  27,05%  32,93%  20,58%  06,26%  00,72%
24  01,27%  09,12%  24,97%  33,29%  22,80%  07,60%  00,96%
25  00,96%  07,60%  22,80%  33,29%  24,97%  09,12%  01,27%
26  00,72%  06,26%  20,58%  32,93%  27,05%  10,82%  01,65%
27  00,53%  05,08%  18,36%  32,21%  28,99%  12,70%  02,12%
28  00,39%  04,07%  16,18%  31,16%  30,75%  14,76%  02,69%
29  00,28%  03,22%  14,07%  29,79%  32,27%  16,98%  03,40%
30  00,19%  02,49%  12,06%  28,13%  33,51%  19,36%  04,25%
31  00,13%  01,90%  10,18%  26,23%  34,43%  21,87%  05,27%
32  00,09%  01,42%  08,44%  24,12%  34,97%  24,48%  06,48%
33  00,06%  01,03%  06,87%  21,85%  35,11%  27,16%  07,92%
34  00,04%  00,73%  05,48%  19,47%  34,82%  29,85%  09,62%
35  00,02%  00,50%  04,26%  17,04%  34,07%  32,50%  11,61%
36  00,01%  00,33%  03,22%  14,60%  32,86%  35,05%  13,93%
37  00,01%  00,21%  02,36%  12,22%  31,17%  37,41%  16,62%
38  00,00%  00,13%  01,66%  09,95%  29,03%  39,48%  19,74%
39  00,00%  00,07%  01,11%  07,84%  26,47%  41,17%  23,33%
40  00,00%  00,04%  00,70%  05,93%  23,53%  42,35%  27,45%
41  00,00%  00,02%  00,41%  04,27%  20,28%  42,87%  32,15%
42  00,00%  00,01%  00,22%  02,87%  16,81%  42,58%  37,51%
43  00,00%  00,00%  00,10%  01,77%  13,24%  41,30%  43,60%
44  ------  00,00%  00,03%  00,95%  09,71%  38,83%  50,48%
45  ------  ------  00,01%  00,41%  06,39%  34,95%  58,25%
46  ------  ------  ------  00,11%  03,50%  29,41%  66,98%
47  ------  ------  ------  ------  01,28%  21,94%  76,79%
48  ------  ------  ------  ------  ------  12,24%  87,76%
49  ------  ------  ------  ------  ------  ------ 100,00%

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