In the case of throwing all five cards away there are combin(47,5)= 1533939 possible replacement hands. The total number of hands that must be analyzed to determine the best play of a specific hand is combin(47,5)+5.combin(47,4)+10.combin(47,3)+10.combin(47,2)+5.47+1, which coincidentally also equals 2598960. Example 3 - Two Players Have Two Pairs. When two players have two pairs, it can sometimes be confusing for people to know which poker hand wins. Take this example: Board: K K Q Q 2 2 3 3 2 2. Player 1: A A A A. Player 2: K K Q Q. In this scenario Player 1 wins the entire pot. The highest possible Straight is A-K-Q-J-10 (also called “Broadway”). Straight combinations go all the way down to A-2-3-4-5, which is known as the “Wheel” or “Bicycle”, in poker lingo. A♠K ♥ Q♣J ♥ 10♠ aka BROADWAY. A ♥ 2♣3♠4 ♦ 5 ♥ aka the WHEEL or BICYCLE. When it comes to Straights, the suits aren’t important. What Casino Games Regle Poker Suite 1 2 3 4 5 Can I Play at Mega Slot Casino? Although the name implies that this is a slots Regle Poker Suite 1 2 3 4 5 casino.

This page describes the ranking of poker hands. This applies not only in the game of poker itself, but also in certain other card games such as Chinese Poker, Chicago, Poker Menteur and Pai Gow Poker.

• Low Poker Ranking: A-5, 2-7, A-6
• Hand probabilities and multiple decks - probability tables

Standard Poker Hand Ranking

There are 52 cards in the pack, and the ranking of the individual cards, from high to low, is ace, king, queen, jack, 10, 9, 8, 7, 6, 5, 4, 3, 2. In standard poker - that is to say in the formal casino and tournament game played internationally and the home game as normally played in North America - there is no ranking between the suits for the purpose of comparing hands - so for example the king of hearts and the king of spades are equal. (Note however that suit ranking is sometimes used for other purposes such as allocating seats, deciding who bets first, and allocating the odd chip when splitting a pot that can't be equally divided. See ranking of suits for details.)

A poker hand consists of five cards. The categories of hand, from highest to lowest, are listed below. Any hand in a higher category beats any hand in a lower category (so for example any three of a kind beats any two pairs). Between hands in the same category the rank of the individual cards decides which is better, as described in more detail below.

In games where a player has more than five cards and selects five to form a poker hand, the remaining cards do not play any part in the ranking. Poker ranks are always based on five cards only, and if these cards are equal the hands are equal, irrespective of the ranks of any unused cards.

Some readers may wonder why one would ever need to compare (say) two threes of a kind of equal rank. This obviously cannot arise in basic draw poker, but such comparisons are needed in poker games using shared (community) cards, such as Texas Hold'em, in poker games with wild cards, and in other card games using poker combinations.

1. Straight Flush

If there are no wild cards, this is the highest type of poker hand: five cards of the same suit in sequence - such as J-10-9-8-7. Between two straight flushes, the one containing the higher top card is higher. An ace can be counted as low, so 5-4-3-2-A is a straight flush, but its top card is the five, not the ace, so it is the lowest type of straight flush. The highest type of straight flush, A-K-Q-J-10 of a suit, is known as a Royal Flush. The cards in a straight flush cannot 'turn the corner': 4-3-2-A-K is not valid.

2. Four of a kind

Four cards of the same rank - such as four queens. The fifth card, known as the kicker, can be anything. This combination is sometimes known as 'quads', and in some parts of Europe it is called a 'poker', though this term for it is unknown in English. Between two fours of a kind, the one with the higher set of four cards is higher - so 3-3-3-3-A is beaten by 4-4-4-4-2. If two or more players have four of a kind of the same rank, the rank of the kicker decides. For example in Texas Hold'em with J-J-J-J-9 on the table (available to all players), a player holding K-7 beats a player holding Q-10 since the king beats the queen. If one player holds 8-2 and another holds 6-5 they split the pot, since the 9 kicker makes the best hand for both of them. If one player holds A-2 and another holds A-K they also split the pot because both have an ace kicker.

3. Full House

This combination, sometimes known as a boat, consists of three cards of one rank and two cards of another rank - for example three sevens and two tens (colloquially known as 'sevens full of tens' or 'sevens on tens'). When comparing full houses, the rank of the three cards determines which is higher. For example 9-9-9-4-4 beats 8-8-8-A-A. If the threes of a kind are equal, the rank of the pairs decides.

4. Flush

Five cards of the same suit. When comparing two flushes, the highest card determines which is higher. If the highest cards are equal then the second highest card is compared; if those are equal too, then the third highest card, and so on. For example K-J-9-3-2 beats K-J-7-6-5 because the nine beats the seven.If all five cards are equal, the flushes are equal.

5. Straight

Five cards of mixed suits in sequence - for example Q-J-10-9-8. When comparing two sequences, the one with the higher ranking top card is better. Ace can count high or low in a straight, but not both at once, so A-K-Q-J-10 and 5-4-3-2-A are valid straights, but 2-A-K-Q-J is not. 5-4-3-2-A, known as a wheel, is the lowest kind of straight, the top card being the five.

6. Three of a Kind

Three cards of the same rank plus two unequal cards. This combination is also known as Triplets or Trips. When comparing two threes of a kind the rank of the three equal cards determines which is higher. If the sets of three are of equal rank, then the higher of the two remaining cards in each hand are compared, and if those are equal, the lower odd card is compared.So for example 5-5-5-3-2 beats 4-4-4-K-5, which beats 4-4-4-Q-9, which beats 4-4-4-Q-8.

7. Two Pairs

A pair consists of two cards of equal rank. In a hand with two pairs, the two pairs are of different ranks (otherwise you would have four of a kind), and there is an odd card to make the hand up to five cards. When comparing hands with two pairs, the hand with the highest pair wins, irrespective of the rank of the other cards - so J-J-2-2-4 beats 10-10-9-9-8 because the jacks beat the tens. If the higher pairs are equal, the lower pairs are compared, so that for example 8-8-6-6-3 beats 8-8-5-5-K. Finally, if both pairs are the same, the odd cards are compared, so Q-Q-5-5-8 beats Q-Q-5-5-4.

8. Pair

A hand with two cards of equal rank and three cards which are different from these and from each other. When comparing two such hands, the hand with the higher pair is better - so for example 6-6-4-3-2 beats 5-5-A-K-Q. If the pairs are equal, compare the highest ranking odd cards from each hand; if these are equal compare the second highest odd card, and if these are equal too compare the lowest odd cards. So J-J-A-9-3 beats J-J-A-8-7 because the 9 beats the 8.

9. Nothing

Five cards which do not form any of the combinations listed above. This combination is often called High Card and sometimes No Pair. The cards must all be of different ranks, not consecutive, and contain at least two different suits. When comparing two such hands, the one with the better highest card wins. If the highest cards are equal the second cards are compared; if they are equal too the third cards are compared, and so on. So A-J-9-5-3 beats A-10-9-6-4 because the jack beats the ten.

Hand Ranking in Low Poker

There are several poker variations in which the lowest hand wins: these are sometimes known as Lowball. There are also 'high-low' variants in which the pot is split between the highest and the lowest hand. A low hand with no combination is normally described by naming its highest card - for example 8-6-5-4-2 would be described as '8-down' or '8-low'.

It first sight it might be assumed that in low poker the hands rank in the reverse order to their ranking in normal (high) poker, but this is not quite the case. There are several different ways to rank low hands, depending on how aces are treated and whether straights and flushes are counted.

Ace to Five

This seems to be the most popular system. Straights and flushes do not count, and Aces are always low. The best hand is therefore 5-4-3-2-A, even if the cards are all in one suit. Then comes 6-4-3-2-A, 6-5-3-2-A, 6-5-4-2-A, 6-5-4-3-A, 6-5-4-3-2, 7-4-3-2-A and so on. Note that when comparing hands, the highest card is compared first, just as in standard poker. So for example 6-5-4-3-2 is better than 7-4-3-2-A because the 6 is lower than the 7. The best hand containing a pair is A-A-4-3-2. This version is sometimes called 'California Lowball'.

When this form of low poker is played as part of a high-low split variant, there is sometimes a condition that a hand must be 'eight or better' to qualify to win the low part of the pot. In this case a hand must consist of five unequal cards, all 8 or lower, to qualify for low. The worst such hand is 8-7-6-5-4.

Deuce to Seven

The hands rank in almost the same order as in standard poker, with straights and flushes counting and the lowest hand wins. The difference from normal poker is that Aces are always high , so that A-2-3-4-5 is not a straight, but ranks between K-Q-J-10-8 and A-6-4-3-2. The best hand in this form is 7-5-4-3-2 in mixed suits, hence the name 'deuce to seven'. The next best is 7-6-4-3-2, then 7-6-5-3-2, 7-6-5-4-2, 8-5-4-3-2, 8-6-4-3-2, 8-6-5-3-2, 8-6-5-4-2, 8-6-5-4-3, 8-7-4-3-2, etc. The highest card is always compared first, so for example 8-6-5-4-3 is better than 8-7-4-3-2 even though the latter contains a 2, because the 6 is lower than the 7. The best hand containing a pair is 2-2-5-4-3, but this would be beaten by A-K-Q-J-9 - the worst 'high card' hand. This version is sometimes called 'Kansas City Lowball'.

Ace to Six

Many home poker players play that straights and flushes count, but that aces can be counted as low. In this version 5-4-3-2-A is a bad hand because it is a straight, so the best low hand is 6-4-3-2-A. There are a couple of issues around the treatment of aces in this variant.

• First, what about A-K-Q-J-10? Since aces are low, this should not count as a straight. It is a king-down, and is lower and therefore better than K-Q-J-10-2.
• Second, a pair of aces is the lowest and therefore the best pair, beating a pair of twos.

It is likely that some players would disagree with both the above rulings, preferring to count A-K-Q-J-10 as a straight and in some cases considering A-A to be the highest pair rather than the lowest. It would be wise to check that you agree on these details before playing ace-to-six low poker with unfamiliar opponents.

Selecting from more than five cards

Note that in games where more than five cards are available, the player is free to select whichever cards make the lowest hand. For example a player in Seven Card Stud Hi-Lo 8 or Better whose cards are 10-8-6-6-3-2-A can omit the 10 and one of the 6's to create a qualifying hand for low.

Regle Poker Suite 1 2 3 4 5 4z

Poker Hand Ranking with Wild Cards

A wild card card that can be used to substitute for a card that the holder needs to make up a hand. In some variants one or more jokers are added to the pack to act as wild cards. In others, one or more cards of the 52-card pack may be designated as wild - for example all the twos ('deuces wild') or the jacks of hearts and spades ('one-eyed jacks wild', since these are the only two jacks shown in profile in Anglo-American decks).

The most usual rule is that a wild card can be used either

1. to represent any card not already present in the hand, or
2. to make the special combination of 'five of a kind'.

This approach is not entirely consistent, since five of a kind - five cards of equal rank - must necessarily include one duplicate card, since there are only four suits. The only practical effect of the rule against duplicates is to prevent the formation of a 'double ace flush'. So for example in the hand A-9-8-5-joker, the joker counts as a K, not a second ace, and this hand is therefore beaten by A-K-10-4-3, the 10 beating the 9.

Five of a Kind

When playing with wild cards, five of a kind becomes the highest type of hand, beating a royal flush. Between fives of a kind, the higher beats the lower, five aces being highest of all.

The Bug

Some games, especially five card draw, are often played with a bug. This is a joker added to the pack which acts as a limited wild card. It can either be used as an ace, or to complete a straight or a flush. Thus the highest hand is five aces (A-A-A-A-joker), but other fives of a kind are impossible - for example 6-6-6-6-joker would count as four sixes with an ace kicker and a straight flush would beat this hand. Also a hand like 8-8-5-5-joker counts as two pairs with the joker representing an ace, not as a full house.

Wild Cards in Low Poker

In Low Poker, a wild card can be used to represent a card of a rank not already present in the player's hand. It is then sometimes known as a 'fitter'. For example 6-5-4-2-joker would count as a pair of sixes in normal poker with the joker wild, but in ace-to-five low poker the joker could be used as an ace, and in deuce-to-seven low poker it could be used as a seven to complete a low hand.

Lowest Card Wild

Some home poker variants are played with the player's lowest card (or lowest concealed card) wild. In this case the rule applies to the lowest ranked card held at the time of the showdown, using the normal order ace (high) to two (low). Aces cannot be counted as low to make them wild.

Double Ace Flush

Some people play with the house rule that a wild card can represent any card, including a duplicate of a card already held. It then becomes possible to have a flush containing two or more aces. Flushes with more than one ace are not allowed unless specifically agreed as a house rule.

Natural versus Wild

Some play with the house rule that a natural hand beats an equal hand in which one or more of the cards are represented by wild cards. This can be extended to specify that a hand with more wild cards beats an otherwise equal hand with fewer wild cards. This must be agreed in advance: in the absence of any agreement, wild cards are as good as the natural cards they represent.

Incomplete Hands

In some poker variants, such as No Peek, it is necessary to compare hands that have fewer than five cards. With fewer than five cards, you cannot have a straight, flush or full house. You can make a four of a kind or two pairs with only four cards, triplets with three cards, a pair with two cards and a 'high card' hand with just one card.

The process of comparing first the combination and then the kickers in descending order is the same as when comparing five-card hands. In hands with unequal numbers of cards any kicker that is present in the hand beats a missing kicker. So for example 8-8-K beats 8-8-6-2 because the king beats the 6, but 8-8-6-2 beats 8-8-6 because a 2 is better than a missing fourth card. Similarly a 10 by itself beats 9-5, which beats 9-3-2, which beats 9-3, which beats a 9 by itself.

Ranking of suits

In standard poker there is no ranking of suits for the purpose of comparing hands. If two hands are identical apart from the suits of the cards then they count as equal. In standard poker, if there are two highest equal hands in a showdown, the pot is split between them. Standard poker rules do, however, specify a hierarchy of suits: spades (highest), hearts, diamonds, clubs (lowest) (as in Contract Bridge), which is used to break ties for special purposes such as:

• drawing cards to allocate players to seats or tables;
• deciding who bets first in stud poker according to the highest or lowest upcard;
• allocating a chip that is left over when a pot cannot be shared exactly between two or more players.

I have, however, heard from several home poker players who play by house rules that use this same ranking of suits to break ties between otherwise equal hands. For some reason, players most often think of this as a way to break ties between royal flushes, which would be most relevant in a game with many wild cards, where such hands might become commonplace. However, if you want to introduce a suit ranking it is important also to agree how it will apply to other, lower types of hand. If one player A has 8-8-J-9-3 and player B has 8-8-J-9-3, who will win? Does player A win by having the highest card within the pair of eights, or does player B win because her highest single card, the jack, is in a higher suit? What about K-Q-7-6-2 against K-Q-7-6-2 ? So far as I know there is no universally accepted answer to these questions: this is non-standard poker, and your house rules are whatever you agree that they are. Three different rules that I have come across, when hands are equal apart from suit are:

1. Compare the suit of the highest card in the hand.
2. Compare the suit of the highest paired card - for example if two people have J-J-7-7-K the highest jack wins.
3. Compare the suit of the highest unpaired card - for example if two people have K-K-7-5-4 compare the 7's.

Although the order spades, hearts, diamonds, clubs may seem natural to Bridge players and English speakers, other suit orders are common, especially in some European countries. Up to now, I have come across:

• spades (high), hearts, clubs, diamonds (low)
• spades (high), diamonds, clubs, hearts (low)
• hearts (high), spades, diamonds, clubs (low) (in Greece and in Turkey)
• hearts (high), diamonds, spades, clubs (low) (in Austria and in Sweden)
• hearts (high), diamonds, clubs, spades (low) (in Italy)
• diamonds (high), spades, hearts, clubs (low) (in Brazil)
• diamonds (high), hearts, spades, clubs (low) (in Brazil)
• clubs (high), spades, hearts, diamonds (low) (in Germany)

As with all house rules, it would be wise to make sure you have a common understanding before starting to play, especially when the group contains people with whom you have not played before.

Stripped Decks

In some places, especially in continental Europe, poker is sometimes played with a deck of less than 52 cards, the low cards being omitted. Italian Poker is an example. As the pack is reduced, a Flush becomes more difficult to make, and for this reason a Flush is sometimes ranked above a Full House in such games. In a stripped deck game, the ace is considered to be adjacent to the lowest card present in the deck, so for example when using a 36-card deck with 6's low, A-6-7-8-9 is a low straight.

Playing poker with fewer than 52 cards is not a new idea. In the first half of the 19th century, the earliest form of poker was played with just 20 cards - the ace, king, queen, jack and ten of each suit - with five cards dealt to each of four players. The only hand types recognised were, in descending order, four of a kind, full house, three of a kind, two pairs, one pair, no pair.

No Unbeatable Hand

Regle Poker Suite 1 2 3 4 5

In standard poker a Royal Flush (A-K-Q-J-10 of one suit) cannot be beaten. Even if you introduce suit ranking, the Royal Flush in the highest suit is unbeatable. In some regions, it is considered unsatisfactory to have any hand that is guaranteed to be unbeaten - there should always be a risk. There are several solutions to this.

In Italy this is achieved by the rule 'La minima batte la massima, la massima batte la media e la media batte la minima' ('the minimum beats the maximum, the maximum beats the medium and the medium beats the minimum'). A minimum straight flush is the lowest that can be made with the deck in use. Normally they play with a stripped deck so for example with 40 cards the minimum straight flush would be A-5-6-7-8 of a suit. A maximum straight flush is 10-J-Q-K-A of a suit. All other straight flushes are medium. If two players have medium straight flushes then the one with higher ranked cards wins as usual. Also as usual a maximum straight flush beats a medium one, and a medium straight flush beats a minimum one. But if a minimum straight flush comes up against a maximum straight flush, the minimum beats the maximum. In the very rare case where three players hold a straight flush, one minimum, one medium and one maximum, the pot is split between them. See for example Italian Poker.

In Greece, where hearts is the highest suit, A-K-Q-J-10 is called an Imperial Flush, and it is beaten only by four of a kind of the lowest rank in the deck - for example 6-6-6-6 if playing with 36 cards. Again, in very rare cases there could also be a hand in the showdown that beats the four of a kind but is lower than the Imperial Flush, in which case the pot would be split.

Hand probabilities and multiple decks

The ranking order of poker hands corresponds to their probability of occurring in straight poker, where five cards are dealt from a 52-card deck, with no wild cards and no opportunity to use extra cards to improve a hand. The rarer a hand the higher it ranks.

This is neither an essential nor an original feature of poker, and it ceases to be true when wild cards are introduced. In fact, with a large number of wild cards, it is almost inevitable that the higher hand types will be the commoner, not rarer, since wild cards will be used to help make the most valuable type of hand from the available cards.

Regle Poker Suite 1 2 3 4 5 12 Fraction

Mark Brader has provided probability tables showing the frequency of each poker hand type when five cards are dealt from a 52-card deck, and also showing how these probabilities would change if multiple decks were used.

For a great training video on poker combinatorics, check out this poker combos video.

'Combinatorics' is a big word for something that isn’t all that difficult to understand. In this article, I will go through the basics of working out hand combinations or 'combos' in poker and give a few examples to help show you why it is useful.

Oh, and as you’ve probably noticed, 'combinatorics', 'hand combinations' and 'combos' refer to the same thing in poker. Don’t get confused if I use them interchangeably, which I probably will.

What is poker combinatorics?

Poker combinatorics involves working out how many different combinations of a hand exists in a certain situation.

For example:

• How many ways can you be dealt AK?
• How many ways can you be dealt 66?
• How combinations of T9 are there on a flop of T32?
• How many straight draw combinations are there on a flop of AT7?

Using combinatorics, you will be able to quickly work these numbers out and use them to help you make better decisions based on the probability of certain hands showing up.

Poker starting hand combinations basics.

• Any two (e.g. AK or T5) = 16 combinations
• Pairs (e.g. AA or TT) = 6 combinations

If you were take a hand like AK and write down all the possible ways you could be dealt this hand from a deck of cards (e.g. A K, A K, A K etc.), you would find that there are 16 possible combinations.

See all 16 AK hand combinations:

Similarly, if you wrote down all the possible combinations of a pocket pair like JJ (e.g. JJ, JJ, JJ etc.), you would find that there are just 6 possible combinations.

See all 6 JJ pocket pair hand combinations:

So as you can see from these basic starting hand combinations in poker, you’re almost 3 times as likely to be dealt a non-paired hand like AK than a paired hand. That’s pretty interesting in itself, but you can do a lot more than this…

Note: two extra starting hand combinations.

As mentioned above, there are 16 combinations of any two non-paired cards. Therefore, this includes the suited and non-suited combinations.

Here are 2 extra stats that give you the total combinations of any two suited and any two unsuited cards specifically.

• Any two (e.g. AK or 67 suited or unsuited) = 16 combinations
• Any two suited (AKs) = 4 combinations
• Any two unsuited (AKo) = 12 combinations
• Pairs (e.g. AA or TT) = 6 combinations

You won’t use these extra starting hand combinations nearly as much as the first two, but I thought I would include them here for your interest anyway.

It’s easy to work out how there are only 4 suited combinations of any two cards, as there are only 4 suits in the deck. If you then take these 4 suited hands away from the total of 16 'any two' hand combinations (which include both the suited and unsuited hands), you are left with the 12 unsuited hand combinations. Easy.

Fact: There are 1,326 combinations of starting hands in Texas Hold’em in total.

Working out hand combinations using 'known' cards.

Let’s say we hold KQ on a flop of KT4 (suits do not matter). How many possible combinations of AK and TT are out there that our opponent could hold?

Unpaired hands (e.g. AK).

How to work out the total number of hand combinations for an unpaired hand like AK, JT, or Q3.

Method: Multiply the numbers of available cards for each of the two cards.
Word equation: (1st card available cards) x (2nd card available cards) = total combinations

Example.

If we hold KQ on a KT4 flop, how many possible combinations of AK are there?

There are 4 Aces and 2 Kings (4 minus the 1 on the flop and minus the 1 in our hand) available in the deck.

C = 8, so there are 8 possible combinations of AK if we hold KQ on a flop of KT4.

Paired hands (e.g. TT).

How to work out the total number of hand combinations for an paired hand like AA, JJ, or 44.

Method: Multiply the number of available cards by the number of available cards minus 1, then divide by two.
Word equation: [(available cards) x (available cards - 1)] / 2 = total combinations

Example.

How many combinations of TT are there on a KT4 flop?

Well, on a flop of KT4 here are 3 Tens left in the deck, so…

C = 3, which means there are 3 possible combinations of TT.

Thoughts on working out hand combinations.

Working out the number of possible combinations of unpaired hands is easy enough; just multiply the two numbers of available cards.

Working out the combinations for paired hands looks awkward at first, but it’s not that tricky when you actually try it out. Just find the number of available cards, take 1 away from that number, multiply those two numbers together then half it.

Note: You’ll also notice that this method works for working out the preflop starting hand combinations mentioned earlier on. For example, if you’re working out the number of AK combinations as a starting hand, there are 4 Aces and 4 Kings available, so 4 x 4 = 16 AK combinations.

Why is combinatorics useful?

Because by working out hand combinations, you can find out more useful information about a player’s range.

For example, let’s say that an opponents 3betting range is roughly 2%. This means that they are only ever 3betting AA, KK and AK. That’s a very tight range indeed.

Now, just looking at this range of hands you might think that whenever this player 3bets, they are more likely to have a big pocket pair. After all, both AA and KK are in his range, compared to the single unpaired hand of AK. So without considering combinatorics for this 2% range, you might think that the probability break-up of each hand looks like this:

• AA = 33%
• KK = 33%
• AK = 33%

…with the two big pairs making up the majority of this 2% 3betting range (roughly 66% in total).

However, let’s look at these hands by comparing the total combinations for each hand:

• AA = 6 combinations (21.5%)
• KK = 6 combinations (21.5%)
• AK = 16 combinations (57%)

So out of 28 possible combinations made up from AA, KK and AK, 16 of them come from AK. This means that when our opponent 3bets, the majority of the time he is holding AK and not a big pocket pair.

Now obviously if you’re holding a hand like 75o this is hardly comforting. However, the point is that it’s useful to realise that the probabilities of certain types of hands in a range will vary. Just because a player either has AA or AK, it doesn’t mean that they’re both equally probable holdings - they will actually be holding AK more often than not.

Analogy: If a fruit bowl contains 100 oranges, 1 apple, 1 pear and 1 grape, there is a decent range of fruit (the 'hands'). However, the the fruits are heavily weighted toward oranges, so there is a greater chance of randomly selecting an orange from the bowl than any of the 3 other possible fruits ('AK' in the example above).

This same method applies when you’re trying to work out the probabilities of a range of possible made hands on the flop by looking at the number of hand combinations. For example, if your opponent could have either a straight draw or a set, which of the two is more likely?

Poker combinatorics example hand.

You have 66 on a board of A J 6 8 2. The pot is $12 and you bet $10. Your opponent moves all in for $60, which means you have to call $50 to win a pot of $82.

You are confident that your opponent either has a set or two pair with an Ace (i.e. AJ, A8, A6 or A2). Don’t worry about how you know this or why you’re in this situation, you just are.

According to pot odds, you need to have at least a 38% chance of having the best hand to call. You can now use combinatorics / hand combinations here to help you decide whether or not to call.

Poker combinatorics example hand solution.

First of all, let’s split our opponent’s hands in to hands you beat and hands you don’t beat, working out the number of hand combinations for each.

Adding them all up…

Seeing as you have the best hand 79% of the time (or 79% 'equity') and the pot odds indicate that you only need to have the best hand 38% of the time, it makes it +EV to call.

So whereas you might have initially thought that the number of hands we beat compared to the number of hands we didn’t beat was close to 50/50 (making it likely -EV to call), after looking at the hand combinations we can see that it is actually much closer to 80/20, making calling a profitable play.

Being able to assign a range to your opponent is good, but understanding the different likelihoods of the hands within that range is better.

Poker combinatorics conclusion.

Working out hand combinations in poker is simple:

• Unpaired hands: Multiply the number of available cards. (e.g. AK on an AT2 flop = [3 x 4] = 12 AK combinations).
• Paired hands: Find the number of available cards. Take 1 away from that number, multiply those two numbers together and divide by 2. (e.g. TT on a AT2 flop = [3 x 2] / 2 = 3 TT combinations).

By working out hand combinations you can gain a much better understanding about opponent’s hand ranges. If you only ever deal in ranges and ignore hand combinations, you are missing out on useful information.

It’s unrealistic to think that you’re going to work out all these hand combinations on the fly whilst you’re sat at the table. However, a lot of value comes from simply familiarising yourself with the varying probabilities of different types of hands for future reference.

For example, after a while you’ll start to realise that straight draws are a lot more common than you think, and that flush draws are far less common than you think. Insights like these will help you when you’re faced with similar decisions in the future.

The next time you’re doing some post session analysis, spend some time thinking about combinatorics and noting down what you find.

Poker combinatorics further reading.

Hand combinations in poker all stem from statistics. So if you’re interested in finding out more about the math side of things, here are a few links that I found helpful:

• Combinations video - Youtube (all the stuff on this channel is awesome)

If you’re more interested in finding out more about combinations in poker only, here are a few interesting reads:

Regle Poker Suite 1 2 3 4 5 8 Equals

Go back to the awesome Texas Hold'em Strategy.

1 2 3 4 Chi Dan

Comments