Important Poker Math

Poker Math Is Easy to Learn Poker math is a vitally important aspect to No Limit Holdem poker, but it is often overlooked or simply not used because many poker players fear it is too difficult to learn. I'm here to tell you it is not. In fact, fundamental poker. The main underpinning of poker is math – it is essential. For every decision you make, while factors such as psychology have a part to play, math is the key element. In this lesson we’re going to give an overview of probability and how it relates to poker.

Essential Poker Math For No Limit Hold'em
Essential Poker Math Pdf

What is Poker Combinatorics?

Combinatorics is the practice of breaking down ranges and counting individual combinations of hands. Generally we won't have enough time during a hand to assign our opponent a specific number of combinations – it's standard practice to think more generally about our opponents range and make estimates. For example we might think something like he has some weak top pairs and second pair type hands along with draws, but he doesn't have that many overpairs because he would have 3bet preflop. We'll refer to this as “category-based thinking”.

This kind of thought process is sufficient in most cases, and is also pretty much all the average human brain is capable of when ranges are very wide. We'd need to be some type of savant to assign our opponent an exact number of combinations at an early stage in the hand. However as the hand progresses, ranges get significantly narrower – to the point where on the river it may be possible to assign our opponent an exact number of combos before we make our decision.

Combinatorics can also be used for off-table analysis. While it is not feasible to list the exact number of flop combos someone has during an actual hand – it should be easy enough afterwards, especially if we use calculation software such as Flopzilla.

Why use Poker Combinatorics?

Combinatorics can be used to increase the accuracy of our standard “category-based thinking”. Previously facing a large river bet we might think along the lines of he has some busted draws and some nutted type hands. With combinatorics we can instead think he has X combos of busted draws and Y combos of nutted hands.

Even if it's an early stage in the hand where it is impractical to count exact combos, having an understanding of how combinatorics works will allow us to assign weights to our “category-based thinking”. For example instead of thinking he has some flush-draws, some top pairs, some underpairs, we can say he has hardly any flush-draws, an average amount of top pair hands, and a large amount of underpairs.

Combinatorics will help us to increase accuracy of our standard 'category based thinking'

With the first type of “category-based thinking”, we could easily be forgiven for assuming that these three categories of hands represent roughly an equal portion of our opponents range. In the second example, we are not quite using full combinatorics, but our knowledge of combinatorics has allowed us to add weightings to our “category-based thinking”.

Preflop Combinations

Let's start with an decent understanding of preflop distribution. This is generally an important step in being able to add weightings to postflop rangeswithout needing to calculate specific combos.

There are 1326 possible combinations of starting hands. These are not distributed evenly between the different hand categories however. Some hands are simply more likely to be dealt than others.

We can divide a hole-cards grid into 3 sections.

Creating a diagonal line from the top left of the grid to the bottom right of the grid are the pocket pairs. There are 6 possible combinations of every pocket pair. There are 13 different pocket pairs ranging from 22 - AA. Therefore the total number of pocket-pair combos is (13*6) = 78

Just by looking at the grid we could easily assume that there is an equal amount of hands either side of that diagonal line. It's true that there are equal amount of hand types, but there is quite a considerable difference between the number of combos. To the right of the diagonal line are the suited combos, and there are way less of these than the offsuit combos which are to the left of the diagonal line. Each type of hand has 12 offsuit combos and 4 suited combos. This means there are 3 times as many offsuit (non-pair) hands compared to suited.

In total there are 154 hand types which are not pocket-pairs. 78 of these are suited, 78 of these are offsuit.

Since there are 4 combos of every suited hand this results in (78 * 4) 312 combos.

The 12 combos of every offsuit hand result in (78 * 12) 936 combos.

So to summarise:

Suited Combos 312

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Postflop Poker Combinations

Let's imagine we are now in a situation where we feel our opponent's range is narrow enough to assign a specific amount of combinations rather than estimate the weightings.

Let's look at some examples.

Example 1 - The board texture is A5272. How many combinations of A-K does our opponent have?

We know he started off with 16 combos of AK, 12 offsuit, and 4 suited. However, he can't have 16 combos of this hand any more since there is already an Ace out there. To calculate the available combinations we multiply the number of available remaining cards which make up the hand. There are 3 Aces left in the deck and 4 Kings. 3 * 4 = 12. There are 12 combos of AK.

Example 2 - The board texture is A527K. How many combinations of A-K does our opponent have?

This should be pretty simple if the last example made sense. There are 3 available Aces, and 3 available Kings. 3 * 3 = 9. 7 of these would be offsuit combos and 2 of them would be suited combos. Our opponent cannot hold AK or AK since the A and A are already on the board.

Example 3 - The board texture is K72. How many combinations of sets does our opponent have?

For pocket pairs we need to use a slightly different method. First count how many available cards there are to make the pocket pairs. For example, to make a set of Sevens there are 3 Sevens remaining in the deck. The formula (where X is number of available cards is) (X * (X-1)) / 2

It looks complicated but is actually very simple. There are 3 cards. We multiply it by 2 to make 6. Then we divide by 2. There are 3 combos of each set (Kings, Sevens, Twos). There are 3 possible sets meaning there are 9 combos of sets in total here

So if we wanted to know how many possible combinations of 88 there are here (no set), we know there are 4 available cards so we can calculate (4*3)/2 = 6 combos.

It looks complicated but is actually very simple. There are 3 cards. We multiply it by 2 to make 6. Then we divide by 2. There are 3 combos of each set (Kings, Sevens, Twos).

Or imagine there are 2 sevens out there and we want to know how often someone has made Quads. (2*1)/2 = 1 combo. Multiply the part in brackets first and then divide by two.

Some calculations are a little more complex. For example working out how many flush-draws opponent has. On a two-tone board we know he will have 1 combo of each suited hand that can make a flush-draw. But we obviously don't just count all possible flush-draws; we have to think about which of these are actually in his range. So in most situations we can discount holdings like 27.

Final Note on Poker Combinatorics

Keep in mind that combinatorics can not be used independent of frequency. A common mistake when first beginning to look at combinatorics is simply looking at which hands opponent might have rather than the frequency with which he continues.

For example, imagine a typical bluff-catch situation where we want to work out whether opponent has enough bluffs for us to make a profitable call. We might calculate that he has 100 combinations of possible bluffs and only 10 combinations of possible value-hands. Easy call? Not necessarily.

A common mistake when first beginning to look at combinatorics is simply looking at which hands opponent might have rather than the frequency with which he continues.

We can't just assume he is firing all of his bluffs. Some guys just never bluff – so just because they reach the river with a ton of possible bluff combos does not automatically mean that calling down is correct. We calculate the possible bluff combos and then we assign a bluff frequency before making our decision.

Introduction

Derivations for Five Card Stud

I have been asked so many times how I derived the probabilities of drawing each poker hand that I have created this section to explain the calculation. This assumes some level mathematical proficiency; anyone comfortable with high school math should be able to work through this explanation. The skills used here can be applied to a wide range of probability problems.

The Factorial Function

If you already know about the factorial function you can skip ahead. If you think 5! means to yell the number five then keep reading.

The instructions for your living room couch will probably recommend that you rearrange the cushions on a regular basis. Let's assume your couch has four cushions. How many combinations can you arrange them in? The answer is 4!, or 24. There are obviously 4 positions to put the first cushion, then there will be 3 positions left to put the second, 2 positions for the third, and only 1 for the last one, or 4*3*2*1 = 24. If you had n cushions there would be n*(n-1)*(n-2)* ... * 1 = n! ways to arrange them. Any scientific calculator should have a factorial button, usually denoted as x!, and the fact(x) function in Excel will give the factorial of x. The total number of ways to arrange 52 cards would be 52! = 8.065818 * 10⁶⁷.

The Combinatorial Function

Assume you want to form a committee of 4 people out of a pool of 10 people in your office. How many different combinations of people are there to choose from? The answer is 10!/(4!*(10-4)!) = 210. The general case is if you have to form a committee of y people out of a pool of x then there are x!/(y!*(x-y)!) combinations to choose from. Why? For the example given there would be 10! = 3,628,800 ways to put the 10 people in your office in order. You could consider the first four as the committee and the other six as the lucky ones. However you don't have to establish an order of the people in the committee or those who aren't in the committee. There are 4! = 24 ways to arrange the people in the committee and 6! = 720 ways to arrange the others. By dividing 10! by the product of 4! and 6! you will divide out the order of people in an out of the committee and be left with only the number of combinations, specifically (1*2*3*4*5*6*7*8*9*10)/((1*2*3*4)*(1*2*3*4*5*6)) = 210. The combin(x,y) function in Excel will tell you the number of ways you can arrange a group of y out of x.

Now we can determine the number of possible five card hands out of a 52 card deck. The answer is combin(52,5), or 52!/(5!*47!) = 2,598,960. If you're doing this by hand because your calculator doesn't have a factorial button and you don't have a copy of Excel, then realize that all the factors of 47! cancel out those in 52! leaving (52*51*50*49*48)/(1*2*3*4*5). The probability of forming any given hand is the number of ways it can be arranged divided by the total number of combinations of 2,598.960. Below are the number of combinations for each hand. Just divide by 2,598,960 to get the probability.

Poker Math

The next section shows how to derive the number of combinations of each poker hand in five card stud.

Royal Flush

There are four different ways to draw a royal flush (one for each suit).

Straight Flush

The highest card in a straight flush can be 5,6,7,8,9,10,Jack,Queen, or King. Thus there are 9 possible high cards, and 4 possible suits, creating 9 * 4 = 36 different possible straight flushes.

Four of a Kind

There are 13 different possible ranks of the 4 of a kind. The fifth card could be anything of the remaining 48. Thus there are 13 * 48 = 624 different four of a kinds.

Full House

There are 13 different possible ranks for the three of a kind, and 12 left for the two of a kind. There are 4 ways to arrange three cards of one rank (4 different cards to leave out), and combin(4,2) = 6 ways to arrange two cards of one rank. Thus there are 13 * 12 * 4 * 6 = 3,744 ways to create a full house.

Flush

There are 4 suits to choose from and combin(13,5) = 1,287 ways to arrange five cards in the same suit. From 1,287 subtract 10 for the ten high cards that can lead a straight, resulting in a straight flush, leaving 1,277. Then multiply for 4 for the four suits, resulting in 5,108 ways to form a flush.

Straight

The highest card in a straight can be 5,6,7,8,9,10,Jack,Queen,King, or Ace. Thus there are 10 possible high cards. Each card may be of four different suits. The number of ways to arrange five cards of four different suits is 4⁵ = 1024. Next subtract 4 from 1024 for the four ways to form a flush, resulting in a straight flush, leaving 1020. The total number of ways to form a straight is 10*1020=10,200.

Three of a Kind

There are 13 ranks to choose from for the three of a kind and 4 ways to arrange 3 cards among the four to choose from. There are combin(12,2) = 66 ways to arrange the other two ranks to choose from for the other two cards. In each of the two ranks there are four cards to choose from. Thus the number of ways to arrange a three of a kind is 13 * 4 * 66 * 4² = 54,912.

Two Pair

There are (13:2) = 78 ways to arrange the two ranks represented. In both ranks there are (4:2) = 6 ways to arrange two cards. There are 44 cards left for the fifth card. Thus there are 78 * 6² * 44 = 123,552 ways to arrange a two pair.

One Pair

There are 13 ranks to choose from for the pair and combin(4,2) = 6 ways to arrange the two cards in the pair. There are combin(12,3) = 220 ways to arrange the other three ranks of the singletons, and four cards to choose from in each rank. Thus there are 13 * 6 * 220 * 4³ = 1,098,240 ways to arrange a pair.

Nothing

First find the number of ways to choose five different ranks out of 13, which is combin(13,5) = 1287. Then subtract 10 for the 10 different high cards that can lead a straight, leaving you with 1277. Each card can be of 1 of 4 suits so there are 4⁵=1024 different ways to arrange the suits in each of the 1277 combinations. However we must subtract 4 from the 1024 for the four ways to form a flush, leaving 1020. So the final number of ways to arrange a high card hand is 1277*1020=1,302,540.

Specific High Card

For example, let's find the probability of drawing a jack-high. There must be four different cards in the hand all less than a jack, of which there are 9 to choose from. The number of ways to arrange 4 ranks out of 9 is combin(9,4) = 126. We must then subtract 1 for the 10-9-8-7 combination which would form a straight, leaving 125. From above we know there are 1020 ways to arrange the suits. Multiplying 125 by 1020 yields 127,500 which the number of ways to form a jack-high hand. For ace-high remember to subtract 2 rather than 1 from the total number of ways to arrange the ranks since A-K-Q-J-10 and 5-4-3-2-A are both valid straights. Here is a good site that also explains how to calculate poker probabilities.

Essential Poker Math For No Limit Hold'em

Five Card Draw — High Card Hands

Hand	Combinations	Probability
Ace high	502,860	0.19341583
King high	335,580	0.12912088
Queen high	213,180	0.08202512
Jack high	127,500	0.04905808
10 high	70,380	0.02708006
9 high	34,680	0.01334380
8 high	14,280	0.00549451
7 high	4,080	0.00156986
Total	1,302,540	0.501177394

Ace/King High

For the benefit of those interested in Caribbean Stud Poker I will calculate the probability of drawing ace high with a second highest card of a king. The other three cards must all be different and range in rank from queen to two. The number of ways to arrange 3 out of 11 ranks is (11:3) = 165. Subtract one for Q-J-10, which would form a straight, and you are left with 164 combinations. As above there 1020 ways to arrange the suits and avoid a flush. The final number of ways to arrange ace/king is 164*1020=167,280.

Essential Poker Math Pdf

Internal Links

For lots of other probabilities in poker, please see my section on Probabilities in Poker.

Written by:Michael Shackleford

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