Hi, my local casino offers high hand jackpots ($250+ per 15 minutes) with lots of side action between players betting on whether current high hands hold with varying odds based on strength of current high hand. Both hole cards must play to qualify and there is typically 6-10 full-ring tables eligible (usually half no limit and half limit holdem games).

I would like to consult the math wizards here to determine what are the right odds to lay for each full house+ taking into consideration the number of tables eligible. We can make simplifying assumptions such as every pocket pair and suited connectors see to the river.

Thanks a ton!

I ran a simulation overnight that should provide some useful information. The simulation consisted of 1,000,000 deals of 9-handed NLHE where each player goes to showdown on each deal.

I kept track of the winning hand "category" of each deal and only tallied it if the winning player used both hole cards to make the winning hand (AK on AAKK2 qualifies).

Tally of Winning Hands (both hole cards must play)

Winning Hand	Winning Hand Tally	Cumulative Higher Tally
2's full of x	6,637	97,681
3's full of x	6,807	90,874
4's full of x	6,979	83,895
5's full of x	7,089	76,806
6's full of x	7,071	69,735
7's full of x	7,353	62,382
8's full of x	7,375	55,007
9's full of x	7,372	47,635
T's full of x	7,564	40,071
J's full of x	7,630	32,441
Q's full of x	7,785	24,656
K's full of x	7,892	16,764
A's full of 2's	589	16,175
A's full of 3's	618	15,557
A's full of 4's	648	14,909
A's full of 5's	608	14,301
A's full of 6's	657	13,644
A's full of 7's	648	12,996
A's full of 8's	646	12,350
A's full of 9's	710	11,640
A's full of T's	678	10,962
A's full of J's	677	10,285
A's full of Q's	720	9,565
A's full of K's	689	8,876
Four of a Kind	7,539	1,337
Royal/Straight Flush	1,337	0

Since there were 1,000,000 deals simulated, the tallies are easily converted to probabilities. For example, the probability of Jacks Full being the winning hand in a 9-handed NLHE deal (requiring both hole cards to play) is 7,630 / 1,000,000 = 0.763%.

The second column of the table is of most interest. For example, the table shows that the probability of the winning hand in a 9-handed NLHE deal (requiring both hole cards to play) being better than Jacks Full is 32,441 / 1,000,000 = 3.2441%.

These figures can be used to answer OP's questions.


Edit to add: As I always say, caution should be used before trusting these figures. While I tried to be careful in programming the simulation, errors have been known to creep in. Until someone else can confirm/verify that these figures are at least in the right ballpark, I would treat these results with healthy skepticism.

I re-ran the exact same simulation of 1,000,000 deals of 9-handed NLHE where each player goes to showdown on each deal. This time I kept track of more output levels.

Before I moth-ball my program, I decided to give it one more spin. Someone in either the present or the future may find these results of interest.

I kept track of the winning hand "category" of each deal and only tallied it if the winning player used both hole cards to make the winning hand (AK on AAKK2 qualifies).

Tally of Winning Hands (both hole cards must play)

Winning Hand	Winning Hand Tally	Cumulative Higher Tally
2's full of 3's	508	103,810
2's full of 4's	535	103,275
2's full of 5's	563	102,712
2's full of 6's	568	102,144
2's full of 7's	568	101,576
2's full of 8's	520	101,056
2's full of 9's	555	100,501
2's full of T's	551	99,950
2's full of J's	543	99,407
2's full of Q's	556	98,851
2's full of K's	572	98,279
2's full of A's	598	97,681
3's full of 2's	566	97,115
3's full of 4's	529	96,586
3's full of 5's	512	96,074
3's full of 6's	569	95,505
3's full of 7's	532	94,973
3's full of 8's	579	94,394
3's full of 9's	593	93,801
3's full of T's	558	93,243
3's full of J's	588	92,655
3's full of Q's	590	92,065
3's full of K's	577	91,488
3's full of A's	614	90,874
4's full of 2's	536	90,338
4's full of 3's	606	89,732
4's full of 5's	529	89,203
4's full of 6's	593	88,610
4's full of 7's	534	88,076
4's full of 8's	594	87,482
4's full of 9's	595	86,887
4's full of T's	579	86,308
4's full of J's	575	85,733
4's full of Q's	602	85,131
4's full of K's	615	84,516
4's full of A's	621	83,895
5's full of 2's	590	83,305
5's full of 3's	589	82,716
5's full of 4's	609	82,107
5's full of 6's	581	81,526
5's full of 7's	528	80,998
5's full of 8's	605	80,393
5's full of 9's	574	79,819
5's full of T's	560	79,259
5's full of J's	586	78,673
5's full of Q's	623	78,050
5's full of K's	568	77,482
5's full of A's	676	76,806
6's full of 2's	596	76,210
6's full of 3's	591	75,619
6's full of 4's	567	75,052
6's full of 5's	586	74,466
6's full of 7's	580	73,886
6's full of 8's	578	73,308
6's full of 9's	565	72,743
6's full of T's	575	72,168
6's full of J's	597	71,571
6's full of Q's	631	70,940
6's full of K's	600	70,340
6's full of A's	605	69,735
7's full of 2's	574	69,161
7's full of 3's	612	68,549
7's full of 4's	628	67,921
7's full of 5's	611	67,310
7's full of 6's	672	66,638
7's full of 8's	574	66,064
7's full of 9's	542	65,522
7's full of T's	623	64,899
7's full of J's	579	64,320
7's full of Q's	627	63,693
7's full of K's	630	63,063
7's full of A's	681	62,382
8's full of 2's	594	61,788
8's full of 3's	653	61,135
8's full of 4's	585	60,550
8's full of 5's	592	59,958
8's full of 6's	648	59,310
8's full of 7's	617	58,693
8's full of 9's	605	58,088
8's full of T's	588	57,500
8's full of J's	592	56,908
8's full of Q's	650	56,258
8's full of K's	635	55,623
8's full of A's	616	55,007
9's full of 2's	535	54,472
9's full of 3's	607	53,865
9's full of 4's	610	53,255
9's full of 5's	609	52,646
9's full of 6's	640	52,006
9's full of 7's	634	51,372
9's full of 8's	669	50,703
9's full of T's	567	50,136
9's full of J's	610	49,526
9's full of Q's	625	48,901
9's full of K's	648	48,253
9's full of A's	618	47,635
T's full of 2's	588	47,047
T's full of 3's	621	46,426
T's full of 4's	636	45,790
T's full of 5's	566	45,224
T's full of 6's	650	44,574
T's full of 7's	652	43,922
T's full of 8's	665	43,257
T's full of 9's	649	42,608
T's full of J's	614	41,994
T's full of Q's	602	41,392
T's full of K's	651	40,741
T's full of A's	670	40,071
J's full of 2's	602	39,469
J's full of 3's	584	38,885
J's full of 4's	637	38,248
J's full of 5's	630	37,618
J's full of 6's	609	37,009
J's full of 7's	617	36,392
J's full of 8's	632	35,760
J's full of 9's	690	35,070
J's full of T's	668	34,402
J's full of Q's	675	33,727
J's full of K's	657	33,070
J's full of A's	629	32,441
Q's full of 2's	616	31,825
Q's full of 3's	615	31,210
Q's full of 4's	624	30,586
Q's full of 5's	643	29,943
Q's full of 6's	665	29,278
Q's full of 7's	635	28,643
Q's full of 8's	623	28,020
Q's full of 9's	670	27,350
Q's full of T's	677	26,673
Q's full of J's	747	25,926
Q's full of K's	642	25,284
Q's full of A's	628	24,656
K's full of 2's	606	24,050
K's full of 3's	609	23,441
K's full of 4's	631	22,810
K's full of 5's	640	22,170
K's full of 6's	644	21,526
K's full of 7's	673	20,853
K's full of 8's	632	20,221
K's full of 9's	715	19,506
K's full of T's	700	18,806
K's full of J's	680	18,126
K's full of Q's	671	17,455
K's full of A's	691	16,764
A's full of 2's	589	16,175
A's full of 3's	618	15,557
A's full of 4's	648	14,909
A's full of 5's	608	14,301
A's full of 6's	657	13,644
A's full of 7's	648	12,996
A's full of 8's	646	12,350
A's full of 9's	710	11,640
A's full of T's	678	10,962
A's full of J's	677	10,285
A's full of Q's	720	9,565
A's full of K's	689	8,876
Quad 2's	584	8,292
Quad 3's	595	7,697
Quad 4's	559	7,138
Quad 5's	607	6,531
Quad 6's	562	5,969
Quad 7's	535	5,434
Quad 8's	607	4,827
Quad 9's	593	4,234
Quad T's	577	3,657
Quad J's	578	3,079
Quad Q's	587	2,492
Quad K's	564	1,928
Quad A's	591	1,337
StrFlush 5-high	115	1,222
StrFlush 6-high	123	1,099
StrFlush 7-high	141	958
StrFlush 8-high	136	822
StrFlush 9-high	146	676
StrFlush T-high	137	539
StrFlush J-high	153	386
StrFlush Q-high	131	255
StrFlush K-high	119	136
RoyalFlush	136	0

Since there were 1,000,000 deals simulated, the tallies are easily converted to probabilities. For example, the probability of Queens Full Of Tens being the winning hand in a 9-handed NLHE deal (requiring both hole cards to play) is 677 / 1,000,000 = 0.0677%.

The second column of the table is of most interest. For example, the table shows that the probability of the winning hand in a 9-handed NLHE deal (requiring both hole cards to play) being better than Queens Full Of Tens is 26,673 / 1,000,000 = 2.6673%.

These figures can be used to answer OP's questions.


Repeat of caution: As I always say, caution should be used before trusting these figures. While I tried to be careful in programming the simulation, errors have been known to creep in. Until someone else can confirm/verify that these figures are at least in the right ballpark, I would treat these results with healthy skepticism.

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.