20 Card Question
December 23rd, 2011 at 2:00:40 PM
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If you take the 10 cards dealt in a Hold'em game preflop, lumped them all together, what would be the average number of pairs, sets, and quads be ? If you give me a math formula, it probably won't help.
December 23rd, 2011 at 4:00:44 PM
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I know there are 13 ranks so by the 14th card there must be at least one pair. But am at a loss to figure out how often pairs, sets and quads occur and in relationship to each other. Like how often will there be 3 pair and 2 sets on the same hand, etc?
December 23rd, 2011 at 7:16:07 PM
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| Singles | Pairs | Trips | Quads | Combinations | Probability |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 5 | 1,287 | 0.00000000001 |
| 0 | 0 | 4 | 2 | 6,589,440 | 0.00000005230 |
| 0 | 1 | 2 | 3 | 9,884,160 | 0.00000007845 |
| 0 | 1 | 6 | 0 | 295,206,912 | 0.00000234301 |
| 0 | 2 | 0 | 4 | 926,640 | 0.00000000735 |
| 0 | 2 | 4 | 1 | 1,660,538,880 | 0.00001317944 |
| 0 | 3 | 2 | 2 | 1,245,404,160 | 0.00000988458 |
| 0 | 4 | 0 | 3 | 77,837,760 | 0.00000061779 |
| 0 | 4 | 4 | 0 | 29,889,699,840 | 0.00023722995 |
| 0 | 5 | 2 | 1 | 26,900,729,856 | 0.00021350696 |
| 0 | 6 | 0 | 2 | 1,681,295,616 | 0.00001334418 |
| 0 | 7 | 2 | 0 | 115,288,842,240 | 0.00091502982 |
| 0 | 8 | 0 | 1 | 10,808,328,960 | 0.00008578405 |
| 0 | 10 | 0 | 0 | 17,293,326,336 | 0.00013725447 |
| 1 | 0 | 1 | 4 | 823,680 | 0.00000000654 |
| 1 | 0 | 5 | 1 | 295,206,912 | 0.00000234301 |
| 1 | 1 | 3 | 2 | 1,107,025,920 | 0.00000878629 |
| 1 | 2 | 1 | 3 | 415,134,720 | 0.00000329486 |
| 1 | 2 | 5 | 0 | 31,882,346,496 | 0.00025304528 |
| 1 | 3 | 3 | 1 | 79,705,866,240 | 0.00063261321 |
| 1 | 4 | 1 | 2 | 22,417,274,880 | 0.00017792247 |
| 1 | 5 | 3 | 0 | 717,352,796,160 | 0.00569351891 |
| 1 | 6 | 1 | 1 | 269,007,298,560 | 0.00213506959 |
| 1 | 8 | 1 | 0 | 691,733,053,440 | 0.00549017895 |
| 2 | 0 | 2 | 3 | 92,252,160 | 0.00000073219 |
| 2 | 0 | 6 | 0 | 2,361,655,296 | 0.00001874410 |
| 2 | 1 | 0 | 4 | 17,297,280 | 0.00000013729 |
| 2 | 1 | 4 | 1 | 26,568,622,080 | 0.00021087107 |
| 2 | 2 | 2 | 2 | 29,889,699,840 | 0.00023722995 |
| 2 | 3 | 0 | 3 | 2,490,808,320 | 0.00001976916 |
| 2 | 3 | 4 | 0 | 797,058,662,400 | 0.00632613212 |
| 2 | 4 | 2 | 1 | 896,690,995,200 | 0.00711689863 |
| 2 | 5 | 0 | 2 | 67,251,824,640 | 0.00053376740 |
| 2 | 6 | 2 | 0 | 4,304,116,776,960 | 0.03416111345 |
| 2 | 7 | 0 | 1 | 461,155,368,960 | 0.00366011930 |
| 2 | 9 | 0 | 0 | 691,733,053,440 | 0.00549017895 |
| 3 | 0 | 3 | 2 | 2,952,069,120 | 0.00002343012 |
| 3 | 1 | 1 | 3 | 2,214,051,840 | 0.00001757259 |
| 3 | 1 | 5 | 0 | 141,699,317,760 | 0.00112464571 |
| 3 | 2 | 3 | 1 | 531,372,441,600 | 0.00421742141 |
| 3 | 3 | 1 | 2 | 199,264,665,600 | 0.00158153303 |
| 3 | 4 | 3 | 0 | 6,376,469,299,200 | 0.05060905696 |
| 3 | 5 | 1 | 1 | 2,869,411,184,640 | 0.02277407563 |
| 3 | 7 | 1 | 0 | 7,378,485,903,360 | 0.05856190876 |
| 4 | 0 | 0 | 4 | 23,063,040 | 0.00000018305 |
| 4 | 0 | 4 | 1 | 29,520,691,200 | 0.00023430119 |
| 4 | 1 | 2 | 2 | 66,421,555,200 | 0.00052717768 |
| 4 | 2 | 0 | 3 | 8,302,694,400 | 0.00006589721 |
| 4 | 2 | 4 | 0 | 2,125,489,766,400 | 0.01686968565 |
| 4 | 3 | 2 | 1 | 3,188,234,649,600 | 0.02530452848 |
| 4 | 4 | 0 | 2 | 298,896,998,400 | 0.00237229954 |
| 4 | 5 | 2 | 0 | 17,216,467,107,840 | 0.13664445378 |
| 4 | 6 | 0 | 1 | 2,152,058,388,480 | 0.01708055672 |
| 4 | 8 | 0 | 0 | 2,766,932,213,760 | 0.02196071579 |
| 5 | 0 | 1 | 3 | 1,476,034,560 | 0.00001171506 |
| 5 | 0 | 5 | 0 | 75,572,969,472 | 0.00059981105 |
| 5 | 1 | 3 | 1 | 566,797,271,040 | 0.00449858284 |
| 5 | 2 | 1 | 2 | 318,823,464,960 | 0.00253045285 |
| 5 | 3 | 3 | 0 | 10,202,350,878,720 | 0.08097449113 |
| 5 | 4 | 1 | 1 | 5,738,822,369,280 | 0.04554815126 |
| 5 | 6 | 1 | 0 | 13,773,173,686,272 | 0.10931556302 |
| 6 | 0 | 2 | 2 | 23,616,552,960 | 0.00018744095 |
| 6 | 1 | 0 | 3 | 5,904,138,240 | 0.00004686024 |
| 6 | 1 | 4 | 0 | 1,133,594,542,080 | 0.00899716568 |
| 6 | 2 | 2 | 1 | 2,550,587,719,680 | 0.02024362278 |
| 6 | 3 | 0 | 2 | 318,823,464,960 | 0.00253045285 |
| 6 | 4 | 2 | 0 | 15,303,526,318,080 | 0.12146173669 |
| 6 | 5 | 0 | 1 | 2,295,528,947,712 | 0.01821926050 |
| 6 | 7 | 0 | 0 | 1,967,596,240,896 | 0.01561650900 |
| 7 | 0 | 3 | 1 | 107,961,384,960 | 0.00085687292 |
| 7 | 1 | 1 | 2 | 121,456,558,080 | 0.00096398204 |
| 7 | 2 | 3 | 0 | 3,886,609,858,560 | 0.03084742519 |
| 7 | 3 | 1 | 1 | 2,914,957,393,920 | 0.02313556889 |
| 7 | 5 | 1 | 0 | 5,246,923,309,056 | 0.04164402401 |
| 8 | 0 | 0 | 3 | 843,448,320 | 0.00000669432 |
| 8 | 0 | 4 | 0 | 107,961,384,960 | 0.00085687292 |
| 8 | 1 | 2 | 1 | 485,826,232,320 | 0.00385592815 |
| 8 | 2 | 0 | 2 | 91,092,418,560 | 0.00072298653 |
| 8 | 3 | 2 | 0 | 2,914,957,393,920 | 0.02313556889 |
| 8 | 4 | 0 | 1 | 546,554,511,360 | 0.00433791917 |
| 9 | 0 | 1 | 2 | 8,996,782,080 | 0.00007140608 |
| 9 | 1 | 3 | 0 | 287,897,026,560 | 0.00228499446 |
| 9 | 2 | 1 | 1 | 323,884,154,880 | 0.00257061877 |
| 10 | 0 | 2 | 1 | 14,394,851,328 | 0.00011424972 |
| 10 | 1 | 0 | 2 | 5,398,069,248 | 0.00004284365 |
| Totals | 125,994,627,894,135 | 1.00000000000 | |||
