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Frequency of Winning Hands in Poker

October 3rd, 2021 | Posted by pftq in Blabberbox | #
Some interesting stats on how often hands show in Texas Hold'em #poker w/ 8 players after running 1 million games on a simulator I wrote for fun.  It's counterintuitive, since having 1 pair win is less frequent than 2-pair, trips, straight, flush, and full-house.  Winning w/ top card is actually rarer than straight flush with only 40 games.  So pretty much you should expect two-pair or better in most games played to the river and be more suspicious of trying to win with top pair or high card.  It also means if someone is playing with friends at a table, they can end up dominating if they play every game out to the river.

8 Players Played to the River:
Two pair: 260263 (26%)
Straight: 199774 (19%)
Three of a kind: 157152 (15%)
Fullhouse: 133175 (13%)
Flush: 126192 (12%)
One pair: 110008 (11%)
Four of a kind: 12156 (1%)
Straight Flush: 619 (0.0619%)
High card: 40 (0%)

With 4 players, one pair as the winning hand goes up to 24% but it still being beaten majority of the time.
 You still expect to see someone with two-pair or higher in most games, and two-pair being the winning hand actually goes up to 33%.

4 Players Played to the River:
Two pair: 337174 (33.72%)
One pair: 241025 (24.10%)
Straight: 131859 (13.19%)
Three of a kind: 110867 (11.09%)
Flush: 84394 (8.44%)
Fullhouse: 81842 (8.18%)
High card: 6408 (0.64%)
Four of a kind: 6131 (0.61%)
Straight Flush: 300 (0.03%)

One pair finally becomes the best hand only if 2 players are playing heads up.

2 Players Played to the River:
One pair: 370932 (37.09%)
Two pair: 314515 (31.45%)
Straight: 80884 (8.09%)
Three of a kind: 72001 (7.20%)
High card: 59450 (5.95%)
Flush: 52446 (5.24%)
Fullhouse: 46495 (4.65%)
Four of a kind: 3129 (0.31%)
Straight Flush: 148 (0.01%)

One pair looks much better if limited to just the flop (not played to turn or river), but it also means that it's misleading to think you're winning with a pair but continue to drag the game out to the river.

8 Players on the Flop:
One pair: 609024 (60.90%)
Two pair: 194754 (19.48%)
Three of a kind: 131810 (13.18%)
Straight: 29436 (2.94%)
Flush: 14396 (1.44%)
Fullhouse: 10987 (1.10%)
High card: 7650 (0.77%)
Four of a kind: 1912 (0.19%)
Straight Flush: 31 (0.00%)

4 Players on the Flop:
One pair: 660342 (66.03%)
Two pair: 139127 (13.91%)
High card: 98207 (9.82%)
Three of a kind: 72909 (7.29%)
Straight: 15426 (1.54%)
Flush: 7541 (0.75%)
Fullhouse: 5503 (0.55%)
Four of a kind: 927 (0.09%)
Straight Flush: 18 (0.00%)

2 Players On the Flop:
One pair: 563065 (56.31%)
High card: 297558 (29.76%)
Two pair: 85303 (8.53%)
Three of a kind: 39112 (3.91%)
Straight: 7669 (0.77%)
Flush: 3852 (0.39%)
Fullhouse: 2946 (0.29%)
Four of a kind: 488 (0.05%)
Straight Flush: 7 (0.00%)

Confirming this is the below showing the most frequent winning starting cards to be just random high cards  - high cards here meaning not even suited or connected.

Winning Starting Hands for 8 Players:
High card: 498271 (49.83%)
Pocket suited: 228904 (22.89%)
Pocket connected: 122740 (12.27%)
Pocket pair: 97713 (9.77%)
Suited connected: 52372 (5.24%)

Winning Starting Hands for 2 Players:
High card: 566414 (56.64%)
Pocket suited: 199150 (19.92%)
Pocket connected: 113960 (11.40%)
Pocket pair: 80838 (8.08%)
Suited connected: 39638 (3.96%)

And perhaps the most interesting stat of all - what you start with and what you end up winning with.  Notice the top 35% most frequent winning starting hands are just random high cards - high cards here meaning not even suited or connected.  Pocket pairs becoming three-of-a-kind is even more rare than just random cards becoming full house.  An additional stat of interest is that playing pocket pairs only will skip 90% of hands, and of the remaining 10%, you only win 25% of them - so pocket pairs are much weaker than most people treat them.  That percentage only increases to winning 50% of hands played if limited to pocket Aces.

Start vs Winning Hand 8 Players:
High card => Two pair: 155598 (15.56%)
High card => Straight: 102997 (10.30%)
High card => Three of a kind: 80587 (8.06%)
Pocket suited => Flush: 77076 (7.71%)
High card => Fullhouse: 64188 (6.42%)
High card => One pair: 62288 (6.23%)
Pocket suited => Two pair: 50139 (5.01%)
Pocket connected => Straight: 40680 (4.07%)
Pocket suited => Straight: 32052 (3.21%)
Pocket connected => Two pair: 30672 (3.07%)
Pocket pair => Three of a kind: 30672 (3.07%)
Pocket pair => Fullhouse: 29681 (2.97%)
High card => Flush: 26863 (2.69%)
Pocket suited => Three of a kind: 26501 (2.65%)
Pocket suited => Fullhouse: 21342 (2.13%)
Pocket suited => One pair: 19731 (1.97%)
Pocket pair => Two pair: 18090 (1.81%)
Pocket connected => Three of a kind: 16864 (1.69%)
Pocket connected => One pair: 15647 (1.56%)
Suited connected => Flush: 14629 (1.46%)
Pocket connected => Fullhouse: 12891 (1.29%)
Suited connected => Straight: 12831 (1.28%)
Suited connected => Two pair: 9776 (0.98%)
Pocket pair => One pair: 9373 (0.94%)
Pocket connected => Flush: 5426 (0.54%)
Suited connected => Three of a kind: 5416 (0.54%)
High card => Four of a kind: 4954 (0.50%)
Suited connected => One pair: 4903 (0.49%)
Suited connected => Fullhouse: 4307 (0.43%)
Pocket pair => Four of a kind: 3930 (0.39%)
Pocket pair => Straight: 3730 (0.37%)
Pocket pair => Flush: 2642 (0.26%)
Pocket suited => Four of a kind: 1601 (0.16%)
Pocket connected => Four of a kind: 995 (0.10%)
Suited connected => Four of a kind: 344 (0.03%)
Pocket suited => Straight Flush: 220 (0.02%)
High card => Straight Flush: 164 (0.02%)
Suited connected => Straight Flush: 115 (0.01%)
High card => High card: 29 (0.00%)
Pocket connected => Straight Flush: 27 (0.00%)
Pocket pair => Straight Flush: 18 (0.00%)
Pocket suited => High card: 7 (0.00%)
Pocket connected => High card: 4 (0.00%)

Starting vs Winning Hand 2 Players:
High card => One pair: 222319 (22.23%)
High card => Two pair: 179271 (17.93%)
Pocket suited => One pair: 70562 (7.06%)
Pocket suited => Two pair: 58099 (5.81%)
High card => Straight: 44905 (4.49%)
Pocket connected => One pair: 41397 (4.14%)
High card => High card: 39429 (3.94%)
High card => Three of a kind: 37954 (3.80%)
Pocket connected => Two pair: 34630 (3.46%)
Pocket pair => Two pair: 30542 (3.05%)
Pocket pair => One pair: 23985 (2.40%)
Pocket suited => Flush: 23793 (2.38%)
High card => Fullhouse: 23334 (2.33%)
High card => Flush: 17999 (1.80%)
Pocket connected => Straight: 15458 (1.55%)
Pocket suited => Straight: 14056 (1.41%)
Suited connected => One pair: 13199 (1.32%)
Pocket pair => Three of a kind: 12120 (1.21%)
Pocket suited => Three of a kind: 12091 (1.21%)
Pocket suited => High card: 11988 (1.20%)
Suited connected => Two pair: 11384 (1.14%)
Pocket pair => Fullhouse: 9424 (0.94%)
Pocket suited => Fullhouse: 7827 (0.78%)
Pocket connected => Three of a kind: 7318 (0.73%)
Pocket connected => High card: 6049 (0.60%)
Suited connected => Straight: 4866 (0.49%)
Suited connected => Flush: 4826 (0.48%)
Pocket connected => Fullhouse: 4672 (0.47%)
Pocket connected => Flush: 3611 (0.36%)
Suited connected => Three of a kind: 2383 (0.24%)
Pocket pair => Straight: 1877 (0.19%)
Pocket pair => Flush: 1865 (0.19%)
Suited connected => High card: 1864 (0.19%)
Suited connected => Fullhouse: 1575 (0.16%)
High card => Four of a kind: 1368 (0.14%)
Pocket pair => Four of a kind: 987 (0.10%)
Pocket suited => Four of a kind: 470 (0.05%)
Pocket connected => Four of a kind: 280 (0.03%)
Suited connected => Four of a kind: 78 (0.01%)
Pocket suited => Straight Flush: 51 (0.01%)
High card => Straight Flush: 41 (0.00%)
Suited connected => Straight Flush: 27 (0.00%)
Pocket pair => Straight Flush: 14 (0.00%)
Pocket connected => Straight Flush: 12 (0.00%)

Some further misc stats that are useful to know for getting to second place out of 8 players using normal expected value strategies:
- As expected, only playing pocket pairs above ten (but not necessarily going all in) gets you to second place 85% of the time.
- Only playing suited high cards above ten also get you similar results of about 85%.
- Pocket pairs below ten get you to second place only 70% of the time.
- Only playing unsuited low cards under five does surprisingly well at 75%.  This upps slightly to 80% if suited.
- Counterintuitively, only playing unsuited high cards above ten is worse than playing unsuited low cards, getting to second place only 50% of the time.  It's most surprising, since this is a huge drop from 85% if the cards were just suited.
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