Welcome to the world of probabilities!

So as you can tell, the answer will be a measure of cases to consider, among all possible cases , in a frequency format given by the You can test by yourself this Python code if you have access to an interpreter:. But such qualities depend on your question! Examples of usage Amount of possible standard deck shufflings? All the possible initial states again: If you want to add more possible hands in to the range, just work out their individual probability and add them in.

Overview of working out hand probabilities.

Poker Hand Odds Charts

In Hold'Em, this just means dividing by 2. In 7-stud this means dividing by 6 3! If you are a programmer, you can install a python interpreter linux distribution have one by default in most cases and test these functions with ipython , against counting the generated items by itertools. If you are not a programmer, this site helps you calculating permutations and combinations.

It is time to put this in practice. The actual card value I ask is irrelevant here! Asking about AA or KK or 22 will bring the same value. If you can notice, the spaces being analyzed is the same everytime: Since they use all the same space, you do not need to convert them to wider spaces in this case, the widest space not involving more cards is 52p5. For this question, which restricts to just pairs, the cards in? For the second question example exactly one pair, no better hand , you have three different analysis to make:.

So always try to cross-check whether you used the right formula and assumptions. But such qualities depend on your question!

However, in a static analysis at the river, not counting the timing but asking about the overall table state at the end, you will have only two spaces to consider: If you are going to write any type of equity calculator, game, or bot this is where I recommend you start. Any simulation needs to be able to identify hands.

Monte Carlo is popular for running random suffles but this just runs all the hands. Computers are getting fast enough that you can run all the hands. This runs the 2,, possible 5 card combinations in 1 second. By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service , privacy policy and cookie policy , and that your continued use of the website is subject to these policies. Home Questions Tags Users Unanswered.

How do I calculate poker hand probabilities? Luis Masuelli 3 Should all of the italicised text really be in the question? Seems like a preface to the answer to me.

Welcome to the world of probabilities! Questions, Answers, Frequencies Probabilities calculation is about a question requiring an answer.

It is about a question involving: This is the configuration of the world you want to analyze. This world is just a narrowed view of what you want to analyze. When analyzing poker hands and showdowns, the world you analyze is the shuffled deck.

A condition or predicate you expect to be true over the current configuration of said world. The world of analysis was just your hand, and the community cards. The initial state is how the deck, having being shuffled, was dealt to make your hand and the community cards. The condition was to get 4 of a kind among the seven cards. The answer was true or false, given the whole known data. There are two types of domains for this: Discrete domains is when the amount of initial states is enumerable.

This means that, for any given initial state, you can know which one would be the next initial state you'd ask, if trying the question one-by-one. Continuous domains is when the amount of initial states is not enumerable. Most of the times this means involving real numbers, integrals , and a plot area to analyze.

You can test by yourself this Python code if you have access to an interpreter: Poker hand analysis belongs to the discrete probabilities domain. You will get something like this: All the possible initial states again: All the expected cases i. This is the value you want. It will be a value between 0 and 1 both bounds can occur! Shuffles, Permutations, Combinations Now we will tell the formulas, the reason of being, and how and when you apply them. Shuffles When to use: The ultimate case of analysis is when you have the deck entirely shuffled.

In games like 7 card stud with 8 players playing up to the last card, or 5 card draw with all the players switching their entire hands, you will notice that the initial deck shuffling matters. An exceptional case is 0! Factorials are not defined on negative numbers. This has a direct relationship with amount of possible shuffles, because: The amount of possible shuffles of a deck of N elements is N possible cards in the top of the deck, multiplied by the amount of possible shuffles of a remaining deck of N - 1 elements, where having a 0-card deck has only one possible shuffle: Permutations Calculating possible distinct permutations involves shuffling a deck and taking a certain number R of cards from it.

Your deck of N cards will be shuffled like this: D1 D2 D3 D DN remaining You will notice that the Python function I copied is already named permutations.

The actual cases I care about is my two cards being like this: Notice that ignoring duplicates could be done here because the number of duplicates or copies is always the same for each possible case , so asking AB would mean both asking: However, in the implementation details, the formula: In practice, the calculation becomes: It is important to consider this: This operation is the building block of most of the poker probabilities calculation except in the cases I stated.

We reached this operation by dividing on shufflings we did not care about. When you study cases of R-sized possible distinct set cases and it happens that you want to consider shuffling on the R elements, you must multiply the amount of cases by R!

When you study cases of R-sized possible distinct sequence cases and it hannd you suddenly don't care anymore about the shuffling on the R elements, you have to collapse every possible shuffling in one case, which involves dividing by R! A sequence is a shuffled set case. A set case is the set of elements of a sequence, disregarding the order. Asking for a sequence in the domain of set cases implies converting the set cases into sequences, multiplying the amount of cases by R!

Which leads on the calculation of Combinations We already showed why we divide by the factorial of N-R: Assume I want to tell the cases of haing a royal flush of spades in 5 card draw yep, the Maverick's finale hand: The calculations for combinations become: Python implementations could be like this: Examples of usage Amount of possible standard deck shufflings?

Amount of possible suit combinations for any pair? Amount of possible 4-of-a-kind hands? This one is tricky.

Involves every possible case for the 4 equal cards, order doesn't matter, and every possible remaining card 48 remaining for each possible case of 4 equal cards. Amount of possible 4-of-a-kind values i. This one is like a permutation among the 13 possible values: It is all about asking the right question! The best way to put it in practice is by example. Since I only care about my hole cards, I take two cards out of the whole deck.

It doesn't matter if I am the first or last player receiving cards. As long as I don't have more information of the initial state, I am still taking two cards out of Additionally, I don't care in which order I receive those cards, so it is not a permutation but a combination of two cards.

I expect two aces. Among the possible cards combinations, there are certain combinations that will be AA. HC and CH is the same so we use combinations. The formula to apply here is the combinatorial with 4 taking two: What is the probability of getting a pair and not a better hand!

The world configuration I want, in this case, must be split in three cases: The expected world configuration responds to the pattern XX??? The expected world configuration responds to the pattern?? The expected world configuration responds to the pattern X? The overall initial conditions amount is: Add the three scenarios!: Add the two scenarios if I don't want the in-table pair: Making a constructive analysis.

For the second question example exactly one pair, no better hand , you have three different analysis to make: Consider that we stole one value for the X from the 13 available values A , and have 12 remaining values for the? I can make six 4c2 combinations of suits for my XX cards, which would work, among 52c2 combinations in my hand. If you get one, there's a 4. A pair against two overcards is often called a coin-flip or race, because they each win about half the time.

People talk about middle suited connectors being better than aces, because of the straight and flush possibilities. There's a reason for folding low hands. If you do get two pair on the flop, the chances of making at least a full house by the river is But if you flop three-of-a-kind, the odds of making a full house or better by the river go up to You bluffed with random cards to steal the blinds and were called by A-K.

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