The Average Winning Hold’em Hand - Gambling and Probability

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The Average Winning Hold’em Hand

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02-29-2024 , 01:28 PM

statmanhal

Pooh-Bah

Join Date: Jan 2009 Posts: 4,987

It is commonly said that the average winning Hold’em hand is 2 pair, which, to me, seems reasonable. I have not seen any analytical basis for this so I attempted one.

I assigned ranks to the 9 hand types, 1 for a straight flush, 2 for quads, … , 9 for high card. I then used Wiki showdown probabilities and determined the win probability for each hand type. These were then used as weights to calculate the average win rank.

For two players, the average rank was 6.75, fairly close to the 7 ranking for 2 pair. For 3 players, however, the average rank was 6.30, closer to the 6 ranking for 3 of a kind. The results lend support to the logical supposition that the greater the number of players in showdown, the more likely a higher ranked hand is the winner.

Yes, the analysis has issues. It assumes independence, which is not the case and more importantly, it does not consider folds. Perhaps the best way to determine the average winning hand type is analyses of very large samples for various game situations. If anyone knows of such analysis, please post a reference.

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02-29-2024 , 02:43 PM

whosnext

Carpal \'Tunnel

Join Date: Mar 2009 Posts: 6,733

Here is the winning hand distribution in 6-max (all hands go to showdown) I found in a 10 million deal simulation. I did this as a comparison to the distribution in Short Deck.

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Originally Posted by whosnext

The Triton poker series has popularized short deck hold-em. I am sure that everyone knows by now that Short Deck is played with a 36-card deck since all the 2's, 3's, 4's, and 5's have been removed (of course, this is why Short Deck is also called Six-Plus poker).

Short Deck is a wildly fast-moving game with frequent large pots and tons of action. Pre-flop equities run much tighter than in Long Deck (regular 52-card NLHE poker). Similar to PLO, the in-hand swings in Short Deck are wild it being common for each card (flop, turn, river) to greatly change who is ahead in the hand.

In previous threads, I looked at how often does a Flush occur in Short Deck vs Long Deck, how often do Quads occur in Short Deck vs Long Deck, and how often do Full Houses occur in Short Deck vs Long Deck.

Here I want to take a more "high level" view. I want to investigate the distribution of winning hands in Short Deck vs Long Deck. As in the previous threads, I looked at 6-max Short Deck vs 6-max Long Deck. Unlike the previous analyses, this one does not seem amenable to analytical (combinatoric) solutions so I ran 10,000,000 deals of each poker variant. As was previously assumed, on each deal all hands go to showdown.

The table below presents the respective distributions of the category of winning hands. Following the adjusted rules of Short Deck, the table reflects that a Flush beats a Full House in Short Deck (in this and all other of my recent Short Deck threads a Straight beats Three of a Kind).

Long Deck Category	Six-Max Win Percentage	__________	Short Deck Category	Six-Max Win Percentage
Royal Flush	0.02%	.	Royal Flush	0.12%
Straight Flush	0.15%	.	Straight Flush	0.57%
Four of a Kind	0.89%	.	Four of a Kind	2.84%
Full House	10.95%	.	Flush	8.52%
Flush	10.83%	.	Full House	25.37%
Straight	16.56%	.	Straight	35.25%
Three of a Kind	13.89%	.	Three of a Kind	11.68%
Two Pair	30.50%	.	Two Pair	14.46%
One Pair	16.15%	.	One Pair	1.19%
High Card	0.06%	.	High Card	0.00%
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TOTAL	100.00%	.	TOTAL	100.00%

You will see immediately the winning hands are generally better in Short Deck compared to Long Deck due to the number of ranks being compressed from 13 to 9. The other salient findings are that Full House and Flush have dramatically different frequencies in the two variants. A flush and full house have very similar winning frequencies in 6-max Long Deck (both around 11%). However, due to the rank compression, full houses are much more common than flushes in Short Deck. Of course, for the same reason Straights are much more common in Short Deck than in Long Deck as well. Note that the results in the table depend upon the hand rankings utilized (i.e., the Short Deck percentages would change if a different hand ranking was used).

One final peculiarity. In the 10 million short deck 6-max deals, High Card was never the winning hand! I wonder if this result was guaranteed or if High Card winning is so rare that 10 million trials was not sufficient for this phenomenon to occur.

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