Combinatorics is Wasting Your Valuable Time at the Poker Table
The main skill in poker is putting your opponent on a range. You can do this by watching your opponent’s betting actions. You also need to know the type of player that you are up against. If you are playing against a tight player, you can use probe bets. These small bets ask your opponent how strong his hand is. If your opponent is tight, he will give you the answer. Tight players are fearful. Therefore, they don’t tend to raise a probe bet as a bluff.
You can’t use probe bets against aggressive players. Aggressive players will usually raise a probe bet. However, aggressive players make it obvious when they have a showdown value hand. They bet heavily with their strong hands, draws and when they miss the board. By contrast, when they have a showdown value hand, they usually want to get the hand to showdown cheaply.
Once you have put your opponent on a range, you still need to figure out how to play the hand. This involves understanding combinatorics.
In this article, I will explain how combinatorics work. At the end of this article, I will tell you why, you should not be using combinatorics at the poker table.
In poker, combinatorics refers to the counting of combinations (or combos) of hands. You will usually be counting the number of combos in your opponent’s hand.
Pre-flop
Starting Hands
Each unpaired hand = 16 combos
For example, a starting hand that isn’t a pair, such as AK, has 16 combos. You can either memorise these numbers or work them out mathematically.
The Formula for Unpaired Hands: There are 4 aces and 4 kings. So, 4 multiplied by 4 equals 16.
Each Pocket Pair = 6 combos
Again, it is simple to work these out in your head. There are 4 aces. Once you have taken one ace out, there are 3 aces left.
The Formula for Paired Hands: You calculate 4 multiplied by 3, which is 12. Next you have to divide by 2.
The reason that you divide by 2 is because
Each Specific Suited Connector = 1 combo
This one is easy enough to remember.
Post-flop
Hands that Connect with the Flop
The number of combos in your opponent’s range can be reduced post-flop. This is because your opponent cannot have the cards that are on the board.
One Pair Hands
For example, when villain has AK and the flop contains a king, the king that is on the board can’t be in his hand. As such, the number of combos of AK that your opponent can have is reduced from 16 pre-flop to 12 post-flop.
Again, this is simple to calculate if you prefer.
The Formula for Counting One Pair Hands: After the flop, there are 4 aces and 3 kings left in the deck. Therefore, 4 multiplied by 3 equals 12.
Sets
There are three 9s left in the deck.
The Formula for Counting Sets: The calculation is 3 multiplied by 2 equals 6. Then, you divide this by 2 to give you 3.
You divide this by 2 because when you reverse the order of the 2 hole cards, the hand is the same.
For example,
In the Tables below, you will find a summary of all the combinatorics that you might use.
Combinations of various hands
| Pre-flop (Starting hand) | Combos |
|---|---|
| Each unpaired hand eg AK | 16 |
| Each paired hand eg QQ | 6 |
| A specific suited connector eg A♦Q♦ | 1 |
| Post-flop | Combos |
|---|---|
| Each unpaired hand that has made 1 pair | 12 |
| Each unpaired hand that has made 2 pair | 9 |
| Each set | 3 |
| Each set when you have one of villain’s outs | 1 |
| Pair when villain has the same hand (eg you both have AK) | 6 |
| 2 pair when villain has the same hand (eg you both have AK) | 4 |
A Hand Example Using Combinatorics
So, let’s look at a hand.
Villain raised pre-flop and hero 3-bet. The flop came up K♥8♦2♠. Hero has a backdoor flush and a backdoor straight draw. However, hero shouldn’t bank on those helping.
The regular checks to hero. What should hero do?
The questions are:
1. Should hero bet on the flop?
2. If villain call and checks on the turn, should hero fire a second barrel?
Hero should start by putting villain on a range. He can take AA and KK out of villain’s range because most players would 4-bet with these hands.
Villain will probably have AK, QQ and JJ in his range. He may have AQ and TT. You might be asking whether villain would 4-bet hands like AK and QQ. At microstakes, players have a tiny 4-bet range. Usually, it consists of just
The more pocket pairs that villain has in his range, the better for hero. However, let’s be conservative and assume that villain called the 3-bet with a range that includes AK, QQ and JJ.
Since we have just learnt combinatorics, we know that villain has 12 combos of AK, 6 combos of QQ and 6 combos of JJ in his range. A regular will probably call a flop bet with all of these combos. However, if hero fires a second barrel on the turn, the regular will fold QQ and JJ. This is fairly typical at the micros.
Hero should bet ½ pot on the flop and ¾ pot on the turn. Why should hero choose these bet sizes? When hero bets ¾ pot on the turn, he needs this bet to win over 43% of the time to make a profit. He doesn’t want QQ or JJ to fold on the flop. If QQ and JJ call a flop bet, hero will get 50% folds on the turn.
If, JJ folds on the flop, hero is only getting 33.3% of folds on the turn. Therefore, he needs to bet small on the flop.
What if, instead of a regular, hero is playing a player with a wider 3-bet calling range? Let’s say that villain calls the 3-bet with a range comprising AK-AT/KQ-KJ/QQ-TT.
In this case, hero should bet 2/3 pot on the flop. If this bet is called, he should give up on the hand.
Why?
The hands that will fold on the flop include AQ (16 combos), AJ (16 combos) and AT (16 combos). The hands that will definitely call are AK (12 combos), KQ (12 combos) and KJ (12 combos). Villain might call a bet with QQ-TT (18 combos).
That works out as 47% folds, if we assume villain will fold his unpaired hands. A 2/3 pot bet needs to win 40% of the time. If some of villain’s pairs fold, this will increase the profit.
On the turn, villain will be left with 36 combos that have a paired king (AK – KJ) and possibly, 18 combos of pairs (QQ-TT). If hero bets ½ pot, he needs to win 33.3% of the time to break even. Even if villain folds QQ-TT, hero can only break even. In addition, if villain has called a flop bet of 2/3 pot with TT, he will probably call a turn bet with QQ and possibly, JJ.
Therefore, hero isn’t getting the odds to bet.
Why You Should NOT Be Using Combinatorics at the Poker Table
Combinatorics isn’t difficult. You can just memorise the numbers or you can use the formulae.
When you are playing online, you don’t have a lot of time to think.
The best strategy is to create a database of reference hands.
This is off the table work. You study the most common hands that come up. You work out how you should play them in every scenario that you can think of. Then, you memorise these hands.
Creating a database of reference hands, working out how they should be played and memorising them, is the way to go!