I let my poor computer run all weekend since I was a trifle dissatisfied with the "lumpiness" of the results of the simulation over 1,000,000 deals. The following results are from a simulation over 10,000,000 deals.
Some of the lumpiness gets smoothed out with the larger sample size. Of course, interested readers can use the standard formulas to determine the statistical range for the true values given these results. I will come back to this at the end of this post.
Tally of Winning Hands (both hole cards must play)
Winning Hand | Winning Hand Tally | Cumulative Higher Tally |
---|
2's full of 3's | 5,329 | 1,036,752 |
2's full of 4's | 5,442 | 1,031,310 |
2's full of 5's | 5,406 | 1,025,904 |
2's full of 6's | 5,378 | 1,020,526 |
2's full of 7's | 5,409 | 1,015,117 |
2's full of 8's | 5,704 | 1,009,413 |
2's full of 9's | 5,672 | 1,003,741 |
2's full of T's | 5,731 | 998,010 |
2's full of J's | 5,799 | 992,211 |
2's full of Q's | 5,950 | 986,261 |
2's full of K's | 5,900 | 980,361 |
2's full of A's | 6,095 | 974,266 |
3's full of 2's | 5,889 | 968,377 |
3's full of 4's | 5,463 | 962,914 |
3's full of 5's | 5,400 | 957,514 |
3's full of 6's | 5,435 | 952,079 |
3's full of 7's | 5,449 | 946,630 |
3's full of 8's | 5,775 | 940,855 |
3's full of 9's | 5,628 | 935,227 |
3's full of T's | 5,733 | 929,494 |
3's full of J's | 5,752 | 923,742 |
3's full of Q's | 5,955 | 917,787 |
3's full of K's | 6,018 | 911,769 |
3's full of A's | 5,936 | 905,833 |
4's full of 2's | 5,712 | 900,121 |
4's full of 3's | 5,852 | 894,269 |
4's full of 5's | 5,315 | 888,954 |
4's full of 6's | 5,431 | 883,523 |
4's full of 7's | 5,599 | 877,924 |
4's full of 8's | 5,587 | 872,337 |
4's full of 9's | 5,779 | 866,558 |
4's full of T's | 5,772 | 860,786 |
4's full of J's | 5,805 | 854,981 |
4's full of Q's | 6,031 | 848,950 |
4's full of K's | 6,225 | 842,725 |
4's full of A's | 6,171 | 836,554 |
5's full of 2's | 5,710 | 830,844 |
5's full of 3's | 5,821 | 825,023 |
5's full of 4's | 6,058 | 818,965 |
5's full of 6's | 5,615 | 813,350 |
5's full of 7's | 5,742 | 807,608 |
5's full of 8's | 5,721 | 801,887 |
5's full of 9's | 5,910 | 795,977 |
5's full of T's | 5,911 | 790,066 |
5's full of J's | 5,928 | 784,138 |
5's full of Q's | 5,988 | 778,150 |
5's full of K's | 6,141 | 772,009 |
5's full of A's | 6,131 | 765,878 |
6's full of 2's | 5,957 | 759,921 |
6's full of 3's | 5,884 | 754,037 |
6's full of 4's | 6,117 | 747,920 |
6's full of 5's | 6,082 | 741,838 |
6's full of 7's | 5,740 | 736,098 |
6's full of 8's | 5,622 | 730,476 |
6's full of 9's | 5,824 | 724,652 |
6's full of T's | 6,008 | 718,644 |
6's full of J's | 6,061 | 712,583 |
6's full of Q's | 6,006 | 706,577 |
6's full of K's | 6,090 | 700,487 |
6's full of A's | 6,520 | 693,967 |
7's full of 2's | 5,868 | 688,099 |
7's full of 3's | 5,808 | 682,291 |
7's full of 4's | 6,029 | 676,262 |
7's full of 5's | 6,208 | 670,054 |
7's full of 6's | 6,217 | 663,837 |
7's full of 8's | 5,590 | 658,247 |
7's full of 9's | 5,674 | 652,573 |
7's full of T's | 5,911 | 646,662 |
7's full of J's | 6,070 | 640,592 |
7's full of Q's | 6,196 | 634,396 |
7's full of K's | 6,340 | 628,056 |
7's full of A's | 6,230 | 621,826 |
8's full of 2's | 5,994 | 615,832 |
8's full of 3's | 6,030 | 609,802 |
8's full of 4's | 6,164 | 603,638 |
8's full of 5's | 6,181 | 597,457 |
8's full of 6's | 6,286 | 591,171 |
8's full of 7's | 6,396 | 584,775 |
8's full of 9's | 5,884 | 578,891 |
8's full of T's | 5,953 | 572,938 |
8's full of J's | 6,003 | 566,935 |
8's full of Q's | 6,047 | 560,888 |
8's full of K's | 6,234 | 554,654 |
8's full of A's | 6,327 | 548,327 |
9's full of 2's | 6,007 | 542,320 |
9's full of 3's | 5,902 | 536,418 |
9's full of 4's | 6,023 | 530,395 |
9's full of 5's | 6,131 | 524,264 |
9's full of 6's | 6,272 | 517,992 |
9's full of 7's | 6,430 | 511,562 |
9's full of 8's | 6,448 | 505,114 |
9's full of T's | 6,115 | 498,999 |
9's full of J's | 6,135 | 492,864 |
9's full of Q's | 6,035 | 486,829 |
9's full of K's | 6,177 | 480,652 |
9's full of A's | 6,414 | 474,238 |
T's full of 2's | 5,865 | 468,373 |
T's full of 3's | 6,141 | 462,232 |
T's full of 4's | 6,158 | 456,074 |
T's full of 5's | 6,216 | 449,858 |
T's full of 6's | 6,342 | 443,516 |
T's full of 7's | 6,367 | 437,149 |
T's full of 8's | 6,568 | 430,581 |
T's full of 9's | 6,602 | 423,979 |
T's full of J's | 6,300 | 417,679 |
T's full of Q's | 6,267 | 411,412 |
T's full of K's | 6,122 | 405,290 |
T's full of A's | 6,429 | 398,861 |
J's full of 2's | 6,196 | 392,665 |
J's full of 3's | 5,950 | 386,715 |
J's full of 4's | 6,058 | 380,657 |
J's full of 5's | 6,172 | 374,485 |
J's full of 6's | 6,264 | 368,221 |
J's full of 7's | 6,397 | 361,824 |
J's full of 8's | 6,559 | 355,265 |
J's full of 9's | 6,514 | 348,751 |
J's full of T's | 6,778 | 341,973 |
J's full of Q's | 6,287 | 335,686 |
J's full of K's | 6,400 | 329,286 |
J's full of A's | 6,563 | 322,723 |
Q's full of 2's | 6,044 | 316,679 |
Q's full of 3's | 6,200 | 310,479 |
Q's full of 4's | 6,033 | 304,446 |
Q's full of 5's | 6,066 | 298,380 |
Q's full of 6's | 6,267 | 292,113 |
Q's full of 7's | 6,364 | 285,749 |
Q's full of 8's | 6,611 | 279,138 |
Q's full of 9's | 6,584 | 272,554 |
Q's full of T's | 6,769 | 265,785 |
Q's full of J's | 7,149 | 258,636 |
Q's full of K's | 6,475 | 252,161 |
Q's full of A's | 6,559 | 245,602 |
K's full of 2's | 6,060 | 239,542 |
K's full of 3's | 6,196 | 233,346 |
K's full of 4's | 6,050 | 227,296 |
K's full of 5's | 6,285 | 221,011 |
K's full of 6's | 6,432 | 214,579 |
K's full of 7's | 6,547 | 208,032 |
K's full of 8's | 6,648 | 201,384 |
K's full of 9's | 6,594 | 194,790 |
K's full of T's | 6,781 | 188,009 |
K's full of J's | 6,901 | 181,108 |
K's full of Q's | 7,116 | 173,992 |
K's full of A's | 6,491 | 167,501 |
A's full of 2's | 6,064 | 161,437 |
A's full of 3's | 6,191 | 155,246 |
A's full of 4's | 6,185 | 149,061 |
A's full of 5's | 6,299 | 142,762 |
A's full of 6's | 6,378 | 136,384 |
A's full of 7's | 6,498 | 129,886 |
A's full of 8's | 6,511 | 123,375 |
A's full of 9's | 6,761 | 116,614 |
A's full of T's | 6,979 | 109,635 |
A's full of J's | 7,079 | 102,556 |
A's full of Q's | 7,113 | 95,443 |
A's full of K's | 7,305 | 88,138 |
Quad 2's | 5,770 | 82,368 |
Quad 3's | 5,747 | 76,621 |
Quad 4's | 5,725 | 70,896 |
Quad 5's | 5,712 | 65,184 |
Quad 6's | 5,736 | 59,448 |
Quad 7's | 5,749 | 53,699 |
Quad 8's | 5,695 | 48,004 |
Quad 9's | 5,718 | 42,286 |
Quad T's | 5,718 | 36,568 |
Quad J's | 5,835 | 30,733 |
Quad Q's | 5,791 | 24,942 |
Quad K's | 5,826 | 19,116 |
Quad A's | 5,738 | 13,378 |
StrFlush 5-high | 1,311 | 12,067 |
StrFlush 6-high | 1,371 | 10,696 |
StrFlush 7-high | 1,319 | 9,377 |
StrFlush 8-high | 1,330 | 8,047 |
StrFlush 9-high | 1,313 | 6,734 |
StrFlush T-high | 1,347 | 5,387 |
StrFlush J-high | 1,332 | 4,055 |
StrFlush Q-high | 1,337 | 2,718 |
StrFlush K-high | 1,309 | 1,409 |
RoyalFlush | 1,409 | 0 |
Some of these probabilities seem amenable to direct calculation. Let me try what are likely to be the two easiest.
Direct Calculation of Probability of Royal Flush
Clearly, exactly three cards of a royal flush must appear on the board and the other two cards of that royal flush must appear in one of the nine players' hands (hole cards). Using combinatorial notation, and writing it out the long way, I think this probability is:
= [ C(4,1)*C(5,3)*C(47,2)*C(9,1)*C(2,2)*C(45,16)*C(16, 2)*C(14,2)*C(12,2)*C(10,2)*C(8,2)*C(6,2)*C(4,2)*C( 2,2) ] /
[ C(52,23)*C(23,5)*C(18,2)*C(16,2)*C(14,2)*C(12,2)*C (10,2)*C(8,2)*C(6,2)*C(4,2)*C(2,2) ]
= 3 / 21,658 (if I did all the simplifications correctly)
= 1,385 out of 10,000,000 (approximately)
Direct Calculation of Probability of King-High Straight-Flush
Clearly, exactly three cards of a King-high straight-flush must appear on the board (and the Ace of the suit cannot appear on the board) and the other two cards of that King-high straight-flush must appear in one of the nine players' hands (hole cards). Using combinatorial notation, and writing it out the long way, I think this probability is:
= [ C(4,1)*C(5,3)*C(46,2)*C(9,1)*C(2,2)*C(45,16)*C(16, 2)*C(14,2)*C(12,2)*C(10,2)*C(8,2)*C(6,2)*C(4,2)*C( 2,2) ] /
[ C(52,23)*C(23,5)*C(18,2)*C(16,2)*C(14,2)*C(12,2)*C (10,2)*C(8,2)*C(6,2)*C(4,2)*C(2,2) ]
= 135 / 1,017,926 (if I did all the simplifications correctly)
= 1,326 out of 10,000,000 (approximately)
Interested readers can double-check but I think the results of the 10-million deal simulation for both of these probabilities fall within one standard deviation of these purportedly (ostensibly) true values.
Last edited by whosnext; 08-20-2018 at 03:11 PM.