Short Deck Poker Math 101: Board Probabilities

Short Deck Poker Strategy is one of the newest domains in which I currently conduct research at the Cardquant Institute. During the course of my research I have identified some essential concepts that enable a player new to Triton Holdem to familiarize himself with the structure of the game.

This article is the first in a series that demonstrates how the new probability structures inherent to Short Deck dictate sound strategy. Before continuing, I recommend that you read my key article on Short Deck Poker Strategy Fundamentals if you haven’t already.

When you transition from No-Limit Hold’em to PLO the hand probabilities change but the board probabilities stay the same. The only thing that changes as regards your in-game evaluation of board probabilities is that in PLO you hold two extra cards which gives you additional information. However, when you transition from either game to Short Deck Hold’em the hand probabilities change AND the board probabilities change. This means that, when you start to learn Short Deck, you should largely disregard any intuition you have about relative hand value and treat it as the entirely new game that it is.

By way of illustration consider this: when you see a flop in a Triton Holdem game you can count 5 cards, leaving 31 cards between your hand and your opponents’ hands. In the same scenario in No Limit Holdem there would still be 47 cards remaining. As a result of this difference, any one card is more likely to hit in Short Deck than in No Limit, for both you and your opponents. This has led some writers to hastily invent a ‘Rule of 6 and 3’ as a way of simplifying in-game calculation. We shall see the folly of this rule shortly! (pun intended)

Let us continue our analysis below with some important practical examples, starting with an exposition of the impact of back-door draws.

As we discussed in the introduction, the short deck makes any one category of ‘outs’ worth more in equity. The key challenge, common to other poker games where multi-way pots are common, is being able to recognize when a particular hand type qualifies as an ‘out’ in a given context.

Consider the following example: we call a raise from seat #3 as the button and join a heads-up pot holding the T♥9♥ on this flop:

Were we playing NLHE, then the T♥9♥ would have only 23% equity against A♠K♣. If you attempt to naively use “The rule of 6 and 3” that has been lazily rolled out as the new Short Deck dogma on mainstream poker blogs, then you will focus excessively on the gut-shot draw here. This would lead to a quick calculation of 6×4= 24% as your in-game estimate of the strength of the T♥9♥ on this flop in Short Deck.

In fact, the Cardquant Short Deck Calculator shows that T♥9♥ has 35% equity against A♠K♣. Good thing we don’t play by mainstream rules!

A deeper examination of the Equity Breakdown by turn transition will show you where this extra equity comes from.

Whilst there are several turn cards which are a near-lock for one of the two players, there are 8 cards which keep the equity of T♥9♥ at or above 33%. The K♥ transition brings both additional straight and flush draws, as does the Q♥, and all of the tens and nines bring additional trip or two pair outs. This demonstrates an essential feature of heads-up pots in Short Deck, namely that your hand is often more likely to win by hitting any one of its back-door draws than by hitting a flopped gut-shot straight draw.

The significance of this in practical play cannot be overstated. In case you thought I was being uncharitable in my criticism of “The Rule of 6 and 3”, consider the equities for two other gut-shot hands facing A♠K♣:

What accounts for the large gap between the equities of these hands facing A♠K♣ and the equity of the T♥9♥?

It is obvious that the King blocker in our opponent’s hand hurts the Q♥T♥, but the large drop in equity when we hold K♥Q♥ is counter-intuitive for the NLHE player. The reason for this drop is that our opponent’s King also prevents runner-runner two pair or trip draws which involve the King from promoting our hand. In NLHE the K♥Q♥ has 20% equity against A♠K♣, significantly closer to the strength of the T♥9♥.

It’s not just subtle features like back-door draws that are affected by the change in board probabilities. The relative hand strength on the flop changes dramatically in Short Deck Holdem, lending a false familiarity to many No Limit players that leads to catastrophic failures in judgement. We will take a look at one of the more pronounced examples in the next section.

In No Limit Holdem you have to be unlucky to run a flopped set into a flopped straight on the board below, but when you do you have only 30% equity.

In Triton Holdem, pocket tens have just over 50% equity against J♦9♦. Does this mean that all sets and straights are equally valuable on this flop?

No, because middle and bottom set can be dominated by top set and the low straight is dominated by the high straight. As a consequence of the short deck the probability of any one of these strong hands occurring is much higher and so your play with any given set or straight will vary according to the number of players in the pot and the SPR.

Let’s say that an opponent bets pot into you with a marked J♦9♦ at an SPR of 4. You hold the TT and there are no other players left to act. What should your action be? Take a quick look at the equity breakdown by turn card for J9 versus TT below and see if you can figure it out:

The answer is that you should shove every time.

You should shove because, even though your equity is 50-50 on the flop, there is an asymmetry on the turn in your opponent’s favour as to the number of cards which favour him over you. When you hit you have 100% equity, and so you need to hit fewer turn cards to ‘fill up’ your flop allocation of 50% equity. Since Short Deck is played with a No-Limit structure, if you call then your opponent can shove on every non-pairing turn card and just check-fold the pairing cards, reducing the Expected Value of your flop decision.

In case you think that such a situation is unlikely, consider the common situation of playing a 6-way flop facing a pot-sized lead from a tight player. A tight player will know that only TT and J9 are the indomitable hands here, and so it stands to reason that that is his leading range. If you yourself hold the TT then it follows that he must hold exactly J9. There certainly can be reasons to smooth call in the multiway spot, not least when a player behind may shove another J9- giving you the chance to have more than 50% equity in a 3-handed all-in confrontation. Nevertheless, situations like these are commonplace in Short Deck games and slowplaying to ‘avoid variance’ can actually be a grave mathematical mistake.

Our final example in this analysis features a quick reality check as to the power of flush draws in a game where the relative value of making a flush has increased in power.

In a full-deck NLHE game, your chances of flopping a flush draw with your suited hand is 11%, a chance that drops to 9.5% in the short deck game. Even when you do flop a flush draw, the chances of completing that flush draw by the river are 34.5% in NLHE, but only 30% in Short Deck.

Set against this, the increase in relative hand value means that completing a flush gives you a virtual lock on the hand- a rare commodity in the Short Deck game. Unlike in PLO, where flush draw domination is a very real threat in multi-way pots, in Short Deck you usually have been unfortunate to run into a higher made flush when you complete it on the board. For obvious reasons, running into a higher flush draw on a four flush board is somewhat more likely.

The Hand v Hand Equity curve above is for K♠9♠ on a flop of J♠T♠6♠ facing a 15% range heads-up. As you can see, it is extremely rare to run into a higher flush even against a tight range and the best chance an opponent has of a redraw is around 14% with the bare A♠. The strength of flushes means that, contrary to the opinions of some commentators, suitedness is valuable in Triton Holdem, but it needs to be accompanied by high card value and/or connection to qualify as a solid starting hand.

The unifying idea holding today’s examples together has been the fact that board probabilities change when we move from No Limit Holdem to Short Deck Holdem. We have seen that flopped sets are the equal of flopped straights, that flushes hit less often, and that back-door draws are a significant consideration that makes conventional hand categories inadequate.

My aim with this series of Short Deck articles is to give a gifted poker player everything he needs to hit the ground running and start winning in Short Deck games. Those with the best combination of talent, discipline and ambition will then be eligible for personal coaching to take them to the highest level.

To that end I have provided some links to additional resources below:

• Be sure to check out the Cardquant Short Deck Holdem Calculator
• If you want the best possible preparation for a Triton Holdem event this year then you can apply for Short Deck coaching with Phil here