Poker Hand Combinatorics: Combinatorial ProbabilityLike it or not, poker is math. The more you put it into numbers, the higher your chances of success, and many players consider combinatorics essential. This article will delve into poker hand combinatorics, which isn’t rocket science but has a scientific flavor to it.
Even if mathematics weren’t your thing, don’t worry. Once you get into the swing of things, using combinatorial probability will give you an advantage. But it takes practice.
What Is Poker Combinatorics?
Poker hand combinatorics refers to the exact number of combinations of playing hands under any given circumstance.
Poker Hand Combinations and Reading the Hands
In Texas hold’em, we can immediately determine the number of possible combinations for three types of hands:
- Paired hands
- Unpaired hands
- Unsuited hands
Let’s explore examples.
If you get the bullets (AA), they can be combined in six different ways. Here’s how:
- A♥, A♦
- A♥, A♣
- A♥, A♠
- A♦, A♣
- A♦, A♠
This holds true for every paired hand. But what happens when you get an unpaired hand, such as QK?
In that case, there are 12 preflop unpaired off-suit combinations and 4 unpaired suited combos. Here’s how.
The four paired combos are:
The 12 unpaired combinations are basically all ways your queen and king (in this case) can pair when they don’t have the same suit. We won’t list them all here, but you get the idea.
The playing cards poker deck has 52 cards, and they can make 169 hands total (paired, suited, and unpaired unsuited).
The table above is divided into paired, suited, and off-suite pairs, but each of these pairs can further combine in 4 (suited), 6 (paired), or 12 ways (off-suit, unpaired).
This means we can make a total of 78 paired hands, 312 suited unpaired, and 934 off-suit unpaired hands. Add them all up and get 1,326 possible combinations dealt as hole cards in Texas hold’em.
And this is just the tip of the iceberg.
So far, we’ve been mainly dealing with simple pre-flop combinatorics. But in poker, cards are sometimes removed from the deck, which could affect the number of possible combinations.
How To Calculate Upaired Hands
The formula for calculating unpaired hands is simple:
The number of available cards for the first card is multiplied by the number of available cards for the second card.
If you have Q and K as your hole cards and no further cards are revealed, you have to assume both are available four times in the deck.
Therefore, the number of available combos is pretty simple 4Q * 4K = 16.
But what happens once you see the flop cards, which are Q♥, 3♦, and 2♣?
In that case, you have one less queen to include in your calculations, so the calculation would be 3 * 4 = 12 combinations of QK.
Let’s say we have pocket nines (99). Therefore:
- The number of available nines pre-flop is 4
- The number of available nines – 1 = 3
- 4 * 3 = 12
The last step is to divide it by two, which means there are 6 combos of poker pairs (12/2). That’s actually something we already stated in this article.
Things get interesting when you see the flop, which might be 9, Q, and A in this case.
All of a sudden, there’s one nine out of the deck, so the number of available nines is 3.
3 * 2 = 6 and 6 / 2 = 3
In short, we have three possible pocket nine combinations.
Power of Poker Hand Combinatorics
Poker hand combinatorics can help us narrow down the number of combinations our opponents can have.
Imagine this. You get a pair of aces, and you need to calculate how many strong combinations your opponent will likely have, QQ and stronger or AK.
If we apply that there are six combos for every paired hand, that’s a total of 18 combinations, right? Wrong.
Remember that the two aces you have in your hand are excluded, so only two aces are in the deck. If we apply this formula: number of available cards * (number of available cards – 1) / 2, we get (2 * (2-1)/2 = 1. Therefore, instead of six, there’s just a single card combo now available, which makes perfect sense. If you think about it, only two suits remain, and those can create just one combo.
The number of QQ and KK pairs remains the same, which is six each or 12 combined.
Let’s now focus on the AK combos. There are 16 of those (4 Aces * 4 Kings). Once again, we have to take out the two aces you hold in your hand, so the number of combos would be 8 (2 * 4).
Once you add it all up, the number of available combinations is 21 rather than 34, which was before we removed the two aces.
Combinatorics and Bluffing — The Correlation
Can poker combinatorics help with catching a bluff? Well, it can assist, but there’s no certain way to be 100% the opponent is bluffing.
Let’s say we have a bluff catcher on the river while facing a pot-sized bet. In this case, we might estimate that our opponent’s range includes 40 combinations, 15 being bluffs, while the remaining are valid combos. What’s our next move?
The pot odds suggests that when facing a pot-sized bet, we need to win a minimum of 33% of the time. Is it possible to expect such a win frequently? We can at least estimate how likely we are expected to have a better hand by dividing 40 by 15, which is 0.375 or 37.5%.
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ConclusionWe explained the introductory part of poker hand combinatorics, which should be enough to help you determine the number of combinations whenever needed and act accordingly.
Hopefully, this has opened up a new way to think about poker and ranges, so you’ll continue practicing hand combinatorics and incorporating them into your strategy.