Vector Poker Hand

1. Vector Poker Hands
2. Poker Hand Vector Art
3. Vector Poker Cards

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This post works with 5-card Poker hands drawn from a standard deck of 52 cards. The discussion is mostly mathematical, using the Poker hands to illustrate counting techniques and calculation of probabilities

Working with poker hands is an excellent way to illustrate the counting techniques covered previously in this blog – multiplication principle, permutation and combination (also covered here). There are 2,598,960 many possible 5-card Poker hands. Thus the probability of obtaining any one specific hand is 1 in 2,598,960 (roughly 1 in 2.6 million). The probability of obtaining a given type of hands (e.g. three of a kind) is the number of possible hands for that type over 2,598,960. Thus this is primarily a counting exercise.

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Preliminary Calculation

Usually the order in which the cards are dealt is not important (except in the case of stud poker). Thus the following three examples point to the same poker hand. The only difference is the order in which the cards are dealt.

These are the same hand. Order is not important.

The number of possible 5-card poker hands would then be the same as the number of 5-element subsets of 52 objects. The following is the total number of 5-card poker hands drawn from a standard deck of 52 cards.

The notation is called the binomial coefficient and is pronounced “n choose r”, which is identical to the number of -element subsets of a set with objects. Other notations for are , and . Many calculators have a function for . Of course the calculation can also be done by definition by first calculating factorials.

Thus the probability of obtaining a specific hand (say, 2, 6, 10, K, A, all diamond) would be 1 in 2,598,960. If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of all diamond cards? It is

This is definitely a very rare event (less than 0.05% chance of happening). The numerator 1,287 is the number of hands consisting of all diamond cards, which is obtained by the following calculation.

The reasoning for the above calculation is that to draw a 5-card hand consisting of all diamond, we are drawing 5 cards from the 13 diamond cards and drawing zero cards from the other 39 cards. Since (there is only one way to draw nothing), is the number of hands with all diamonds.

If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of cards in one suit? The probability of getting all 5 cards in another suit (say heart) would also be 1287/2598960. So we have the following derivation.

Thus getting a hand with all cards in one suit is 4 times more likely than getting one with all diamond, but is still a rare event (with about a 0.2% chance of happening). Some of the higher ranked poker hands are in one suit but with additional strict requirements. They will be further discussed below.

Another example. What is the probability of obtaining a hand that has 3 diamonds and 2 hearts? The answer is 22308/2598960 = 0.008583433. The number of “3 diamond, 2 heart” hands is calculated as follows:

One theme that emerges is that the multiplication principle is behind the numerator of a poker hand probability. For example, we can think of the process to get a 5-card hand with 3 diamonds and 2 hearts in three steps. The first is to draw 3 cards from the 13 diamond cards, the second is to draw 2 cards from the 13 heart cards, and the third is to draw zero from the remaining 26 cards. The third step can be omitted since the number of ways of choosing zero is 1. In any case, the number of possible ways to carry out that 2-step (or 3-step) process is to multiply all the possibilities together.

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The Poker Hands

Here’s a ranking chart of the Poker hands.

The chart lists the rankings with an example for each ranking. The examples are a good reminder of the definitions. The highest ranking of them all is the royal flush, which consists of 5 consecutive cards in one suit with the highest card being Ace. There is only one such hand in each suit. Thus the chance for getting a royal flush is 4 in 2,598,960.

Royal flush is a specific example of a straight flush, which consists of 5 consecutive cards in one suit. There are 10 such hands in one suit. So there are 40 hands for straight flush in total. A flush is a hand with 5 cards in the same suit but not in consecutive order (or not in sequence). Thus the requirement for flush is considerably more relaxed than a straight flush. A straight is like a straight flush in that the 5 cards are in sequence but the 5 cards in a straight are not of the same suit. For a more in depth discussion on Poker hands, see the Wikipedia entry on Poker hands.

The counting for some of these hands is done in the next section. The definition of the hands can be inferred from the above chart. For the sake of completeness, the following table lists out the definition.

Definitions of Poker Hands

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Counting Poker Hands

Straight Flush
Counting from A-K-Q-J-10, K-Q-J-10-9, Q-J-10-9-8, …, 6-5-4-3-2 to 5-4-3-2-A, there are 10 hands that are in sequence in a given suit. So there are 40 straight flush hands all together.

Four of a Kind
There is only one way to have a four of a kind for a given rank. The fifth card can be any one of the remaining 48 cards. Thus there are 48 possibilities of a four of a kind in one rank. Thus there are 13 x 48 = 624 many four of a kind in total.

Full House
Let’s fix two ranks, say 2 and 8. How many ways can we have three of 2 and two of 8? We are choosing 3 cards out of the four 2’s and choosing 2 cards out of the four 8’s. That would be = 4 x 6 = 24. But the two ranks can be other ranks too. How many ways can we pick two ranks out of 13? That would be 13 x 12 = 156. So the total number of possibilities for Full House is

Note that the multiplication principle is at work here. When we pick two ranks, the number of ways is 13 x 12 = 156. Why did we not use = 78?

Flush
There are = 1,287 possible hands with all cards in the same suit. Recall that there are only 10 straight flush on a given suit. Thus of all the 5-card hands with all cards in a given suit, there are 1,287-10 = 1,277 hands that are not straight flush. Thus the total number of flush hands is 4 x 1277 = 5,108.

Straight
There are 10 five-consecutive sequences in 13 cards (as shown in the explanation for straight flush in this section). In each such sequence, there are 4 choices for each card (one for each suit). Thus the number of 5-card hands with 5 cards in sequence is . Then we need to subtract the number of straight flushes (40) from this number. Thus the number of straight is 10240 – 10 = 10,200.

Three of a Kind
There are 13 ranks (from A, K, …, to 2). We choose one of them to have 3 cards in that rank and two other ranks to have one card in each of those ranks. The following derivation reflects all the choosing in this process.

Two Pair and One Pair
These two are left as exercises.

High Card
The count is the complement that makes up 2,598,960.

The following table gives the counts of all the poker hands. The probability is the fraction of the 2,598,960 hands that meet the requirement of the type of hands in question. Note that royal flush is not listed. This is because it is included in the count for straight flush. Royal flush is omitted so that he counts add up to 2,598,960.

Probabilities of Poker Hands

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2017 – Dan Ma

How can I speed up my Python poker hand vs. hand equity calculator

First a disclaimer: I am a medical professional who plays with both Python and poker as a hobby. I have no formal training in neither of those and I am not aware of what is in the curriculum of a computer science class.The computer that I use is a desktop i7-4790 3.6 Ghz with 16 GB of RAM along with Jupyter Notebooks.

My goal was to code the equivalent of pokerstrategy.com Equilab or https://www.cardschat.com/poker-odds-calculator.php for me. I would stick to Texas Holdem only.

In order to to that I needed to code an evaluator for any 5 card combination.I did that and it does the job perfectly, considers every card in a hand and produces a tuple as an output, for example:

So it differentiates between A 9 8 7 3 and A 9 8 7 5 flush or high-hand. I checked with theoretical number of royal flushes, quads, full houses etc. over all 2 598 960 card ccombinations and the frequencies check out (https://www.quora.com/How-many-possible-hands-are-there-in-a-five-card-poker-game)

Now I tried to evaluate every possible 5-card combination out of those 2.6 millions and it took disappointing 51 seconds.

I kind of expected that thinking my 5-card evaluator can't be the champion of algorithm competitions and surely there is a better way to do it (I can post it here if it is of relevance), but I thought never mind. Once all 5-card combinations are evaluated I will save them in a dictionary and next time I will load the dictionary and when I have any 5-card combination I will simply look up the result.

Vector Poker Hands

Another disappointment. 10 000 000 (10 million) board searches takes approx. 23-24 seconds.This is a a part that I do not understand!!!I have basically a database that has 2.6 mil. rows x 2 columns and the search is SO PROHIBITIVELY slow. How do then billion record databases get anything accomplished?My whole dictionary when saved to a file takes 88 Mb - is that a HUGE database?

Poker Hand Vector Art

And finally I made a full hand vs. hand evaluator that in pseudo-code does this:

• Given 2 hands, for example AhAs vs. 6d6h

• list all boards that can be dealt with those 2 'dead' cards, which is 1 712 304 boards

• list all 21 combinations of hand1 with board 1,

• search the ranked_hands dictionary with those 21 combinations and return the best possible outcome (21 combinations because in texas holdem you can use one, two or no cards from your hand with any of the 5 community cards on board)

• do the same with hand2 and board1

• compare best outcome of hand1 with best outcome of hand2

• count if the outcome favours hand1, hand2 or if it is a tie

• go to next board

This algorithm does approximately 71 million dictionary lookups - each of 1.7 million boards x 42 (every hand's 21 combinations twice).

Now, THIS IS A DISASTER. Approximately 80 seconds per hand vs. hand matchup.With those speeds there is nothing I can begin.So, any input would be appreciated as to how I can make this better?

Is it me and my lack of proper computer science and algorithm knowledge?

Is it Python?Is it Jupyter Notebooks inside of Chrome?

Any other suggestions?

Code as requested:

Maybe one print(total) to many but was originally in Jupyter Notebook

[Python 3] Holdem Hand v Hand Equity Calculator :confused , This simple method misses draws but is still accurate enough for my applications. If I (or anyone here) can speed this up substantially, in the� Once again, I’ll use an equity calculator to determine Hero’s equity. In this case, Hero’s hand has 35.84% equity vs. Villain’s range. Now you can create variables for EstFoldPercent and Equity. Hand2_Fold_Percent = .10 Hand2_Equity = .3584 Now plug the variables for this hand into the AllinExpectedValue function, and print the results.

You could use the dbm module (see https://docs.python.org/3/library/dbm.html) , or bsddb in python 2.x, to store your entire lookup table in a database file, if it's too large to fit in memory in a dict. Then it might take some time to fill out the table, but you'd only need to do that once.

Vector Poker Cards

#1 Poker Odds Calculator Online 2020, The #1 Ranked Poker Odds Calculator by CardsChat™ - Easy & FREE tool for As a professional poker player it is important to review hands and be aware of equity calculations. You can fill in your own hand and calculate the chances of you winning To get you up to speed, here's a quick rundown of hand order. The hackerrank question asked me to write a program that would determine the best poker hand possible in five-card draw poker. We are given 10 cards, the first 5 are the current hand, and the second 5 are the next five cards in the deck. We assume that we can see the next five cards (they are not hidden).

ktseng/holdem_calc: Texas Hold'em Odds Calculator, GitHub is home to over 50 million developers working together to host and review code, also shows how likely each set of hole cards is to make a certain poker hand. python holdem_calc.py Ad Kd Qc Qs Winning Percentages: (Ad, Kd) hands in a single run, the user has the choice to allow Holdem Calculator to read � To calculate your poker equity - or how often you should win a hand, you can use a simple formula. Count how many outs you have. For example, if you're drawing to a flush, you have 13 suited cards, two in your hand, two on the board - leaves 9 outs.

julianandrews/pyeval7: Python Texas Hold'em hand evaluation and , Python Texas Hold'em hand evaluation and equity calculation library - julianandrews/pyeval7. poker hand evaluator and range equity calculator with a clean native python interface and all performance critical parts implemented in Cython. If there isn't a wheel for your package, feel free to open an issue on GitHub. Calculate your equity with the PokerStrategy.com Equilab. Our software tool gives you the ability to analyse the equity against a specific hand or against a range of hands. On top of that, you can also check your equity in any given scenario, thanks to the various filters you can use. What is equity?

How to Calculate Poker Probabilities in Python, hand to construct my hand. I do not remember exactly what happened pre-flop and what my position was. However, I do� How can I speed up my Python poker hand vs. hand equity calculator First a disclaimer: I am a medical professional who plays with both Python and poker as a hobby. I have no formal training in neither of those and I am not aware of what is in the curriculum of a

Approximating Poker Probabilities with Deep Learning, Many poker systems, whether created with heuristics or machine learning, rely 2 22We implemented a na�ve hand evaluator in python. to calculate all the features in the input vector which is a speed-up of ∼600x holding your hole cards (two cards in your hand) and the current board cards constant. Both when determining your final poker hand and the poker hand that wins the pot. Figuring Out Your Hand: In Texas Hold'em your final best 5-card hand can use both your hole cards (the two cards dealt to you face down at the beginning of the hand), one of your hole cards or none of your hole cards to make up your final hand.

Comments

• Where's the code?
• Perhaps better suited to Code Review although they'll require code to review.
• I'd be interested to see and review the code but might be better on Code Review as pointed out by @Sayse as this is very broad for a SO question. If you choose to post on CR please link here and I'll take a look. Using a dictionary is often a good idea for these kind of combinatorics problems but it sounds like in your case you have a dictionary which is not especially usefully structured.
• Sorry. I wasn't aware that there are different platforms for asking different questions, lilke StackOverflow and Code Review. As said, a hobbyist. Code added.
• @ rioZg I added a comment on my answer with some run times for this algorithm using multiprocessing Python and Go.
• That was my initial idea but if you look at the link in the original question to Quora statitistics page you will see that a very small minority of hands constitute anything else but 'nothing' and 'one pair' and for those hands you need complete evaluation down to last kicker. So I thought any 'shortcut' would be more work than it was worth.
• @rioZg having so few playable hands out the set of all possible hands is exactly the thing that makes it advantageous to do this computation on-the-fly. All that is needed is to code some functions that will determine if their input matches a certain pattern, and then return the details (probably suit/high card) needed to assign a numerical score to the input. Even better: there are only 8 cases that all playable hands will fall into, and they are all easy to express in Python code. I think if you try this approach you'll find it is a very straightforward solution.
• No need to evaluate any given hand past those 8 special cases: if a hand doesn't fall into one of them, it is automatically a zero score.
• Not true Z4-tier. The board is 9 8 7 3 2 , one player holds A6 the other A4, the best possible hand for player 1 is A 9 8 7 6, for player 2 A 9 8 7 4, player 1 wins on the last kicker. The board is As 3s 4s 5s Qd , player1 is holding 2s2d, player 2 is holding KsKd, player 3 is holding 8s7s, you need more than just player 2 has flush, player 3 has flush because player 2 beats player 3 but loses to player one who has straight flus. So knowing that hand is zero score is not sufficient.
• I just updated this answer with a working example that should work for any hand. It first assigns a hand to a category (ie. SF, FH, 3K, etc...) and then maintains a list of the cards that can be used to determine the relative order of 2 hands within the same category. This takes advantage of the fact that tie breakers follow the same algorithm regardless of the hand category; the only thing that changes are the specific cards that are compared. Variants of this problem are frequently used in programming competitions.
• ...hand vs. hand or range equity is the essential tool in every players toolbox. You are doing it almost in every hand you get involved in.
• Ok, I don't really know anything about poker but I thought as your hand is only visible to you then hand Vs hand is more of an academic exercise than an in-play tool.
• Thanks for the link. Very useful. But it seems to be above my competence level especially given that all are in programming languages not familiar to me so I can not easily evaluate how they did it. But if they managed 15-16 million evaluations per second on old computers in 2006. that would be very acceptable to me. Full evaluation of hand vs. hand on all possible boards is around 72 million evaluations.
• I have to admit that I haven't got my head round it yet either but it's enough to convince me that we're not going to get easy improvements in your existing algorithm. It's looks like a very elaborate system of lookups optimised to take advantage of the exact characteristics of poker hands. Having said that you could probably get your existing algorithm quite a bit faster if you optimised it for multiprocessing (Python only uses one CPU thread by default) and/or used a faster language. I love Python but if you are looking for fastest possible execution it's far from the best choice.
• Thanks SimonN. It occured to me through looking for the solution that leaving Python for 'greener pastures' might be the only solution.