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 and how often do Quads occur in Short Deck vs Long Deck. Here I want to do the same thing for Full Houses. Since there are fewer ranks in each suit, we would expect that full houses are more common in Short Deck than in Long Deck.

I want to look into the differential frequency of full houses in Short Deck vs Long Deck. In particular, I will look at how often a full house is made by at least one player at a six-max table if each hand goes to showdown on each deal. Also, to keep the comparison "apples-to-apples", I will ignore the possibility of flushes in both variants, including straight flushes (i.e., both a full house and a flush being made on the same deal). Finally, quads are excluded in both cases (i.e., if one player makes quads and another player makes a full house on the same deal, this does not count as a full house).


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

I will take the total number of deals of 6-Max Long Deck to be:
= C(52,5)*C(47,12)*(1*3*5*7*9*11)
= 1,411,633,731,355,657,009,200

There are hundreds of cases/sub-cases in which a full house can be made in six-max hold-em. Rather than itemize all the tallies, I will here simply present the tally of all the cases/sub-cases of full houses in six-max long deck.

GRAND TOTAL = 154,330,362,337,149,741,600

PERCENT = 10.933%


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

I will take the total number of deals of 6-Max Short Deck to be:
= C(36,5)*C(31,12)*(1*3*5*7*9*11)
= 553,027,606,647,516,000

As above, there are hundreds of cases/sub-cases in which a full house can be made in six-max hold-em. Rather than itemize all the tallies, I will here simply present the tally of all the cases/sub-cases of full houses in six-max short deck.

GRAND TOTAL = 150,191,017,260,217,920

PERCENT = 27.158%


Summary

We see that full houses are much more common in 6-max Short Deck than in 6-max Long Deck (27.158% vs 10.933%) when all hands go to showdown on all deals.

Remember that we earlier found that flushes are slightly less common in 6-max Short Deck compared to 6-max Long Deck (9.327% vs 11.851%). Thus, we can see that full houses and flushes "flip-flop" in relative frequency in the two variants.

Full houses are inherently rarer than flushes in 6-max Long Deck (10.933% vs 11.851%) whereas full houses are inherently more common than flushes in 6-max Short Deck (27.158% vs 9.327%). Accordingly, flushes are elevated above full houses in the hand rankings of Short Deck poker.