Last in the series ...

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).

It 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) hold-em. Similar to PLO, the in-hand swings 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, how often does a Full House occur in Short Deck vs Long Deck, and how often does a Straight occur in Short Deck vs Long Deck.

Here I want to do the same thing for Three of a Kind. Since there are fewer ranks in each suit (rank compression), we would expect that three of a kind is more common in Short Deck than in Long Deck. We have already seen in a previous thread that rank compression does result in more quads in short deck than in long deck.

I want to look into the differential frequency of three of a kind in Short Deck vs Long Deck. To keep the comparison "apples-to-apples", I will focus exclusively on three of a kind. I will not count the deal as making three of a kind if anyone makes quads or a full house on the deal. Also, since I want to focus exclusively on the rank compression phenomenon itself, straights and flushes will be ignored (that is, if one player makes three of a kind and another player makes a flush, I will count this as a three of kind deal).

In particular, I will look at how often three of a kind is made by at least one player at a six-max table if each hand goes to showdown on each deal.

First a couple of preliminary results. I did a brute force exploration of each variant with only one player and found that one player makes three of a kind in Short Deck in 645,120 out of 8,347,680 possible deals, which is 7.728%.

One player makes three of a kind in Long Deck in 6,589,440 out of 133,784,560 possible deals, which is 4.925%. Thus, these brute force calculations show that three of a kind is inherently more likely in one-player Short Deck than in one-player Long Deck.

For these brute force explorations no distinction was made between board cards and hole cards since with only one player that distinction is irrelevant. I would welcome anyone who can confirm/deny either of those one-player results.

On to the 6-max results.


Long Deck (regular 52-card hold-em):

The combinatoric approach appears a bit daunting, so I resorted to our old friend the simulation. I simulated 100,000,000 deals of 6-max Long Deck NLHE where every player goes to showdown on every deal.

The simulation found that the percent of deals with at least one player making three of a kind was:

PERCENT = 18.202%


Short Deck (36-card Six Plus hold-em):

As above, the combinatoric approach appears a bit daunting, so I resorted to our old friend the simulation. I simulated 100,000,000 deals of 6-max Short Deck NLHE where every player goes to showdown on every deal.

The simulation found that the percent of deals with at least one player making three of a kind was:

PERCENT = 24.368%


Summary

We see that three of a kind is inherently more common in 6-max Short Deck than in 6-max Long Deck (24.368% vs 18.202%) when all hands go to showdown on all deals, and ignoring straights and flushes.

The gap between these two frequencies is much smaller than the other results we have found. I believe this is due to the much higher frequency of full houses (and quads, to some degree) in Short Deck compared to Long Deck. That is, a great many deals in Short Deck that seemingly have three of a kind are "disqualified" because their three of a kind becomes a full house (or quads).

Here is a summary of the findings from previous threads of the inherent frequency of poker hands in 6-Max Short Deck vs 6-Max Long Deck NLHE. See the respective threads for the precise definition of "inherent frequency" in each case (i.e., what is included or excluded in each tally).


Table 1: Inherent Frequencies of Poker Categories in 6-Max Short Deck vs 6-Max Long Deck

Category	Short Deck Percentage	Long Deck Percentage
Quads	2.82%	0.89%
Full House	27.16%	10.93%
Flush	9.33%	11.85%
Straight	48.12%	20.04%
Three of a Kind	24.37%	18.20%

The biggest take-away from this table is that every category is more frequent (some much more frequent) in Short Deck compared to Long Deck, with the important lone exception of a flush. Rank compression is good for Quads, Full House, Straight, and Three of a Kind. But rank compression is bad for a Flush since you still need to have five of a suit despite there being fewer in each suit in the deck to begin with.

To repeat from above and the other threads, the frequencies in this table are the "inherent" frequencies of each category on its own due to the rank compression (9 ranks instead of 13) of Short Deck vs Long Deck NLHE.

Another thread presented the overall category frequencies of winning hands of 6-Max Short Deck vs 6-Max Long Deck NLHE when the respective hand rankings of the two variants are utilized. Here is the category table and remarks from that thread.

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).


Table 2: Frequencies of Poker Categories of Winning Hands in 6-Max Short Deck vs 6-Max Long Deck

Long Deck Category	Six-Max Long Deck Win Pct	__________	Short Deck Category	Six-Max Short Deck Win Pct
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%
.
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).

The other thread also presented interesting information on the frequency of winning hands (such as JTs) in 6-Max Short Deck vs 6-Max Long Deck NLHE.