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10-02-2017, 12:06 PM   #1
whosnext
Pooh-Bah

Join Date: Mar 2009
Location: California
Posts: 4,837
winning hands distribution for 2-10 player nlhe

Everyone knows that heads-up poker is different from full-ring poker. Clearly, in general, you should play tighter the more players are at the table.

It is fairly obvious that it would typically take a better hand to win a pot in a 10-handed game than in a 2-handed game.

Over the weekend I ran a set of simulations to try to answer how large this effect is.

I simulated 1,000,000 deals for each of 2-handed up to 10-handed NLHE and kept track of the category (royal flush, straight flush, four of a kind, full house, etc.) of the winning hands. Of course, the simulation had all players going to showdown on each deal.

Distribution of Winning Hands

Here is a table of the percentages of the distribution over the ten poker categories of the 1 million winning hands.

Category2-Handed3-Handed4-Handed5-Handed6-Handed7-Handed8-Handed9-Handed10-Handed
ROYAL FLUSH
0.007%
0.008%
0.014%
0.015%
0.017%
0.022%
0.024%
0.026%
0.028%
STRAIGHT FLUSH
0.053%
0.077%
0.110%
0.131%
0.160%
0.185%
0.205%
0.225%
0.261%
FOUR OF A KIND
0.312%
0.458%
0.611%
0.747%
0.909%
1.031%
1.166%
1.318%
1.494%
FULL HOUSE
4.703%
6.556%
8.178%
9.581%
10.904%
12.135%
13.227%
14.245%
15.177%
FLUSH
5.248%
7.019%
8.530%
9.732%
10.805%
11.775%
12.657%
13.490%
14.215%
STRAIGHT
8.013%
10.820%
13.039%
14.977%
16.559%
17.917%
19.059%
20.013%
20.847%
THREE OF A KIND
7.195%
9.268%
11.054%
12.589%
13.893%
15.081%
15.985%
16.719%
17.337%
TWO PAIR
31.407%
33.895%
33.702%
32.507%
30.534%
28.474%
26.465%
24.560%
22.731%
ONE PAIR
37.133%
29.939%
24.140%
19.527%
16.167%
13.369%
11.208%
9.402%
7.911%
HIGH CARD
5.930%
1.961%
0.622%
0.194%
0.054%
0.012%
0.003%
0.001%
0.000%

You can see that the "quality" of the winning hand significantly improves moving from 2-handed tables to 10-handed tables.

Cumulative Distribution of Winning Hands

To make it even easier to see the effect, here is the same table in terms of the "cumulative" percentages. That is, what is the percentage of deals that was won by a hand in this category or a higher category.

Category2-Handed3-Handed4-Handed5-Handed6-Handed7-Handed8-Handed9-Handed10-Handed
ROYAL FLUSH
0.007%
0.008%
0.014%
0.015%
0.017%
0.022%
0.024%
0.026%
0.028%
STRAIGHT FLUSH
0.060%
0.084%
0.123%
0.146%
0.178%
0.207%
0.229%
0.251%
0.289%
FOUR OF A KIND
0.372%
0.542%
0.734%
0.893%
1.086%
1.238%
1.395%
1.569%
1.783%
FULL HOUSE
5.075%
7.098%
8.912%
10.474%
11.990%
13.373%
14.622%
15.814%
16.960%
FLUSH
10.323%
14.117%
17.442%
20.206%
22.795%
25.148%
27.280%
29.304%
31.174%
STRAIGHT
18.335%
24.938%
30.481%
35.183%
39.354%
43.065%
46.339%
49.318%
52.021%
THREE OF A KIND
25.530%
34.206%
41.535%
47.772%
53.246%
58.146%
62.324%
66.037%
69.358%
TWO PAIR
56.937%
68.101%
75.237%
80.279%
83.780%
86.620%
88.789%
90.597%
92.089%
ONE PAIR
94.070%
98.040%
99.378%
99.806%
99.947%
99.988%
99.997%
99.999%
100.000%
HIGH CARD
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%

 10-03-2017, 08:45 AM #2 heehaww Pooh-Bah     Join Date: Aug 2011 Location: Tacooos!!!! Posts: 4,419 Re: winning hands distribution for 2-10 player nlhe Thanks, good idea generating those. I'm too lazy to calculate now, but I think most of the two-pairs use a pair on the board, even when 9-handed. So one pair is stronger than the chart might suggest.

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