Combinatorics in Poker – Poker Hand Combinations

When thinking about poker strategy often we are forced to make tough decisions, particularly when facing bets on the river. Professional players need to be able to analyze their play objectively to maximize profit. In this article, I will explain what is meant by the term combinatorics, its strengths, weaknesses and why it is an invaluable part of any professional’s arsenal.

An Introduction to Combinatorics

The term combinatorics in the context of poker refers solely to the number of hand combinations possible. For example, there are 16 different ways of being dealt AK since there are both an ace and king of each suit in the deck. This is, of course, simple and when considered abstractly it ‘s hard to understand how it would make you a better poker player. The application of this, however, is what makes it such a powerful tool in the decision-making process.

Thinking Logically

Identifying the correct logic path to take as a poker player is the first step to improving your game. Although looking at a situation in isolation may provide you a solution, in real time you do not have the advantage of retrospect. This means that to correctly problem solve you must be trying to think concerning ranges. To play an optimal poker strategy, our decision should be correct versus all possible hands our opponent may have. Although this means we sometimes take an incorrect action against opponents holding the decision we make makes the most money in the long term. This is where combinatorics comes into its own. If we think of our competitors range in terms of combinations of bluffs and value hands it can aid our decision making.

Worked Example: Combinatorics on the River

Let’s look at an example of when combinatorics may be helpful in aiding a decision on the river:

We’re playing in a $0.5/$1 cash game with 100bb effective stacks. We raise the Button $2 with QhJh after it folds to us and the small blind 3 Bets to $7. The big blind folds, we decide to call and see a flop.

Pot: $15

Board: Qd 7c 3h

Action: Villain continuation bets $10 and we call.

Pot: $35

Board: Qd 7c 3h Ts

Action: Villain fires a second barrel of $25 and we call

Pot: $85

Board: Qd 7c 3h Ts 2d

Action: Villian moves all-in for his remaining $58 and we have a decision to make.

Problem Solving using Combinatorics

In this hand, our opponent puts us all in on the river when we hold a top pair and a weak kicker. We have a pure bluff catcher since our opponent will not value bet a worse hand. However, we do beat all of his bluffs.

Step 1: To decide whether to call we should evaluate the pot odds we are getting. Opponent bets $58 into $85 making the pot $143. We must call $58 to win $143. Hence our odds are ~2.5:1. This means we must win 1 in every 3.5 times to break even – roughly 28% of the time.

Step 2: Count the number of possible value combo’s the opponent has on the river. We can safely assume that our opponent will have QQ, AQ, KK, AA, QTs, and TT for value. To do this correctly, we must account for card removal i.e the cards we have in our hands and the ones on the board. This is equivalent to:

QQ = 1 combo             AQ = 8 combos            KK = 6 combos

AA = 6 combos            QTs = 1 combo            TT = 2 combos

Total value combos = 24

Step 3: Now we have the number of value hands we expect our opponent to have, we need to calculate the number of bluffs he must have for us to call profitably. Above we see that we have to be correct ~1/3.5 of the time, so we need to see 24/3.5=~7 possible bluffs to call.

Step 4: Count possible bluffs. Generally speaking, people do not bluff hands without additional equity, so we should look at possible flush draws and straight draws that miss as the most frequent bluffs. Watching the board the turn brings all AK, AJ and KJs combinations a straight draw that misses on the river. These are all possible hands our opponent will 3 Bet preflop, bet flop, bluff turn and could feasibly bluff on the river.

AK = 16 combos

AJ = 12 combos

KJs = 3 combos

Total possible bluff combos = 27 combos

Solution: Theoretically, the correct play here is to call the river bet.

Combinatorics and Calculating Expectation

In the worked example, we win more than the 7/28 times needed to make the call profitable. If we wish to calculate the overall expectation, we need to make an equation for the weighted average losses and wins such we have a net profit for the hand.

Profit = [($143*(27/51)) + (-$58*(24/51))] = $48.41

N.B: 51 is the total combinations in our opponent’s range and is derived from value + bluff combos (27+24)

Combinatorics, Assumptions, and Adjustments

Unfortunately, the above example is a slightly idealistic view of how the scenario will be in real life. Firstly, we assume that our opponent will always take this exact line of betting 3 streets with all his value combinations and all of his bluffs. Additionally, we assume that all of his bluffs will bluff on the river.

In reality, opponents will check the turn with some value hands such as QQ and possibly AQ, and he may decide not to bluff the turn with hands like AJ and AK since they have some showdown value and poor equity. The potential differences made at these decision nodes will significantly influence the profitability of calling a river bet. As poker players working with imperfect information, we must try and make the most realistic assumptions possible. Using information from previous hands is a good way of doing this.

For example:

  • Have we seen our opponent triple barrel bluff before?
  • Was the bluff a reasonable one to make?
  • How frequently does our opponent like to trap preflop and postflop?
  • Is our opponent chasing losses in the game?
  • Are we perceived to call light or tight postflop?

Making the correct adjustments based upon information available sets the best poker players out from the crowd.

Combinatorics Conclusion

As with anything in life, it is not quite as simple as it may first appear; combinatorics is no exception! What begins with counting the number of hands preflop turns into pot odds and then further transforms into expectation equations. After this analysis, we then question the basis for our assumptions and then adjust as necessary. Needless to say, this can take some time however if done correctly will improve you leaps and bounds as a player. After some time calculating pot odds in your head will be second nature, as will counting combinations. I hope that this will make the next tough river decision you face significantly easier! Good luck at the tables.

Patrick Sekinger is from the UK and is an avid poker enthusiast, currently playing both live and online as a professional poker player. Patrick plays all formats of NLHE, however specialises in 100bb cash games and you will find him regularly playing in the small stakes games on Pokerstars.

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