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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
Choosing your value bets on a brick turn is the easy part: simply bet with the same strong hands that you bet on the flop. If the turn card is not a brick, you'll have to re-evaluate your value range and bet with your new strongest hands (e.g. You should not continue betting with K ♥ 9 ♥ on a board of 9♠ 8♠ 4 ♥ Q♠). The chart is good if you where at a table with real poker players, but online poker you get people that goes all in on a 7-2 and hands similar to that, so playing poker online you just start out slow and get to know your opponents and how they play before you make any drastic plays. How to Use the Chart The Open Raise row is used if action folds to you and the pot has not been raised. The 4-bet and Call 3-Bet rows are used if you have open raised and been 3-bet. If you do not have a hand that falls into the ranges listed, you fold. The 3-bet and Call Raise rows are used if an opponent open-raises from EP or MP. It shows you when it is profitable to shove a specific hand based on your position and stack depth for the play to be winning you chips even when your opponents are calling perfectly. This poker push/fold chart assumes you only are pushing or folding. Sometimes you can choose to open some hands instead of strictly using push fold strategy.
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. Paragon casino theater marksville la.
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
| Poker Hand | Definition | |
|---|---|---|
| 1 | Royal Flush | A, K, Q, J, 10, all in the same suit |
| 2 | Straight Flush | Five consecutive cards, |
| all in the same suit | ||
| 3 | Four of a Kind | Four cards of the same rank, |
| one card of another rank | ||
| 4 | Full House | Three of a kind with a pair |
| 5 | Flush | Five cards of the same suit, |
| not in consecutive order | ||
| 6 | Straight | Five consecutive cards, |
| not of the same suit | ||
| 7 | Three of a Kind | Three cards of the same rank, |
| 2 cards of two other ranks | ||
| 8 | Two Pair | Two cards of the same rank, |
| two cards of another rank, | ||
| one card of a third rank | ||
| 9 | One Pair | Three cards of the same rank, |
| 3 cards of three other ranks | ||
| 10 | High Card | If no one has any of the above hands, |
| the player with the highest card wins |
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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.
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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.
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Probabilities of Poker Hands
| Poker Hand | Count | Probability | |
|---|---|---|---|
| 2 | Straight Flush | 40 | 0.0000154 |
| 3 | Four of a Kind | 624 | 0.0002401 |
| 4 | Full House | 3,744 | 0.0014406 |
| 5 | Flush | 5,108 | 0.0019654 |
| 6 | Straight | 10,200 | 0.0039246 |
| 7 | Three of a Kind | 54,912 | 0.0211285 |
| 8 | Two Pair | 123,552 | 0.0475390 |
| 9 | One Pair | 1,098,240 | 0.4225690 |
| 10 | High Card | 1,302,540 | 0.5011774 |
| Total | 2,598,960 | 1.0000000 |
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2017 – Dan Ma
Choosing your starting hands wisely can make the difference between winning and losing in poker. This is especially true when you can't depend on your poker skills to help you out after the flop in more difficult situations (yet). Many beginning poker players will come to realize this quite early on in their poker career and they subsequently start to look for a guaranteed formula towards proper starting hand selection: they need the best starting hands chart available to beat the game, or so they think…
Texas hold'em starting hand charts
Starting hand charts offer an overview of common situations regarding your position at the poker table and/or the action in front of you and tell you which starting hands to play and how for every situation. They're easy to read and easy to use.
It isn't anything else but logical that the first time poker player resorts to charts as a quick fix for their leaks. Many beginning poker players have such big leaks in their game caused by improper starting hand selection that the use of a starting hands chart can improve their game significantly.
However, there are some shortcomings associated with starting hands charts. They lead to a very straightforward and predictable game; they don't take into account all of the aspects of the game that are important for starting hand selection; they can't offer a solution for all the different scenarios you will encounter at the poker table and above all, they don't make you think for yourself.
Proper Texas hold'em starting hand selection
Proper starting hand selection goes beyond the use of charts. It is the result of a true understanding of 'starting hand strength'. What factors other than position and the action in front of you influence the strength of your Texas hold'em starting hands and why? What are strengths and weaknesses of the different starting hands? Knowing the answers to these questions will most likely also result in an insight in the best way to play certain hands.
Take for example a starting hand like 6♦7♦. Now, imagine that you're at a full ring game in late position and there's a raise with two callers in front of you. A starting hands chart would probably tell you that you should either fold or call.
And that's it.
If however you would truly understand the strength of a starting hand like 6♦7♦ then you would know that 6♦7♦ is a great hand because it is both connected and suited and therefore has a higher probability of hitting straights and flushes when compared to hands other than suited connectors. You would also realize that, despite it being a suited connector, the chance of really flopping something great with this starting hand is still very slim. You would therefore be looking to see cheap flops; to avoid the possibility of someone raising/re-raising you pre-flop and to be in a position to win a lot of money for when you do hit to make up for the times when you miss and have to fold (high implied odds).
In this case you would not only see that there are already three players in the hand with a full stack, but also that the initial raiser has a very strong range because he is tight and raised from early position. You also know that the players who are still left to act behind you are passive and are therefore unlikely to make a re-raise in which case you would certainly have to fold and lose the initial call. You just know that this is an excellent opportunity to play the hand. Because you realize you are facing a strong range of hands from your opponents you also know what to look for after the flop. You don't want to hit just a top pair or a gutshot straight draw. You are looking for combo draws which give you at least around 40% equity when all the money goes in on the flop. You are looking to hit two-pair or better and you also realize that hitting the flush and getting it all-in in pots with many players will sometimes only result in seeing your opponent show a higher flush. In addition you would also have a betting strategy in mind: you know that if you hit what you are looking to hit, you should bet big to get value and to protect your hand. You would have a plan for the rest of the hand from the moment you see your cards and decide to play them; a plan that takes many more aspects into consideration than just the action in front of you and your position; a plan that goes way beyond the use of a simple starting hands chart and will therefore get you further in the end.
Starting hand selection charts - conclusion
Proper starting hand selection is a very important aspect towards playing winning poker. Starting hand selection is more than just selecting hands based on your position and the action at the table. It is about making a plan for the rest of the hand considering all possible aspects involved. Although starting hands charts can offer a quick solution for beginning players to improve their starting hand selection, taking the time to really learn and understand this aspect of the game will certainly be more beneficial in the long run.
Poker Hand Betting Chart Printable
Many beginning poker players look at a starting hands chart as an easy and guaranteed formula towards proper starting hand selection. Do you?
Further reading at First Time Poker Player:
Further reading across the internet:
- PlayWinningPoker - Training Wheels of Fortune - Poker Starting Hand Charts
Poker Hand Betting Chart Template
