Understanding how poker hands are ranked is the most fundamental concept in poker and should be the first thing you learn. Most casino poker games use a standard 52 card deck. There are some exceptions to this, as there are games that use a deck with a joker added or a deck that has cards stripped away.
A K Q J 10 9 8 7 6 5 4 3 2 (A when used in a Straight) Random chance: Medium. The Education of a Poker Player by Herbert O. Yardley, a former U.S. Government code breaker, was published in 1957. Interest in hold 'em outside of Nevada began to grow in the 1980s as well. Question 241523: A poker hand consists of five cards from a standard deck of 52. Find the number of different poker hands of straight (five cards of consecutive.
The Deck
A standard fifty-two card deck consists of thirteen sequential cards in four different suits.
2♣ 3♣ 4♣ 5♣ 6♣ 7♣ 8♣ 9♣ 10♣ J♣ Q♣ K♣ A♣
2♦ 3♦ 4♦ 5♦ 6♦ 7♦ 8♦ 9♦ 10♦ J♦ Q♦ K♦ A♦2♥ 3♥ 4♥ 5♥ 6♥ 7♥ 8♥ 9♥ 10♥ J♥ Q♥ K♥ A♥
2♠ 3♠ 4♠ 5♠ 6♠ 7♠ 8♠ 9♠ 10♠ J♠ Q♠ K♠ A♠
Depending on the type of poker game, the aces (A♣ A♦ A♥ A♠) can play as a high card (sequentially higher than the king {K}), a low card (sequentially lower than the 2), or most commonly, either.
In some card games there is a natural ranking of the suits, which is, from low to high, clubs ♣ diamonds ♦ hearts ♥ spades ♠. An easy way to remember this is that the first letter of each suit is in alphabetical order from low to high. Now that you know this, forget it. Generally, in casino poker games, this ranking is not used; the suits are all considered of equal value. Four players making exactly the same hand, each in a different suit, would each receive an equal share of the pot.Poker Hands
Standard poker hands consists of five cards. There are many different types of poker games, with various numbers of cards dealt out, but ultimately you will be considering your best five cards in most games. This means that at the end of the hand, you will play the highest ranking five card combination possible from the cards you have been dealt. The player with the most highly ranked hand, relative to those of their opponents, will be awarded the pot (pool of wagers). In the event of a tie, the pot will be split equally.
Now we will rank the five card poker hands from low to high. Hands are counted from the top down.
High Cards Only – These are poker hands that contain no pair, no straight, and no flush. It is the worst poker hand. If you were dealt seven cards: K♠ J♥ 10♦ 9♦ 4♣ 3♣ 2♠, your best five card hand would be “king high” (K♠ J♥ 10♦ 9♦ 4♣).
One Pair – This is a hand that contains one pair only, with no straight or flush. The higher the pair, the higher the hand ranks. If two hands have the same pair, the other high cards are considered for ranking purposes. Any one pair hand beats any high card only hand.
Question: If player “A” is dealt J♠ J♣ K♠ 5♦ 4♦ 3♥ 2♦, and player “B” is dealt J♦ J♥ K♣ 10♠ 8♠ 7♣ 5♥, who has the winning hand?
Answer: Player “A” has a five card hand of J♠ J♣ K♠ 5♦ 4♦ and player “B” has J♦ J♥ K♣ 10♠ 8♠. Both players have a pair of Jacks, so we go to the next highest card for a tie breaker. They both have a king as their next highest card, so we have to go to the next highest card for a tie breaker. Player “B” has a ten and player “A” has a five. Ten is ranked higher than five, so player “B” wins. This is what is counting the hand from the top down means.
Two Pair – This is a hand that contains two pairs of different rankings, but no straight or flush. Any two pair hand beats any one pair hand. Remember that hands are counted from the top down.
Question: Who has the winning hand in each of the following three scenarios?
Scenario 1: Player “A” is dealt J♠ J♣ 10♣ 10♦ 4♦ 3♥ 2♦, player “B” is dealt J♦ J♥ 9♦ 9♣ A♠ Q♣ 8♠.
Scenario 2: Player “A” is dealt J♠ J♣ 10♣ 10♦ A♥ 3♥ 2♦, player “B” is dealt J♦ J♥ 10♥ 10♠ K♠ Q♣ 8♠.
Scenario 3: Player “A” is dealt 10♥ 10♠ 2♣ 2♦ Q♥ 4♥ 5♥, player “B” is dealt 9♦ 9♣ 8♠ 8♣ A♥ Q♣ 7♠.
Answer:
Scenario 1: Player “A” has jacks and tens with a four. Player “B” has jacks and nines with an ace. Because they both have the same high pair, we go to the second pair for a tie breaker. Player “A” has tens and player “B” has nines. Player “A” wins.
Scenario 2: Player “A” has jacks and tens with an ace. Player “B” has jacks and tens with a king. Because the both have the same high two pair we must go to the fifth card for a tiebreaker. Player “A” has an ace, and player “B” has a king. Player A wins.
Scenario 3: Player “A” has tens and deuces (twos) with a queen. Player “B” has eights and nines with an ace. Remember, we count from the top down until we have a winner. Player “A” has tens as the highest pair of the two pair. Player “B” has nines as the highest pair of the two pair. Tens beat nines, so we do not have to go any further. Player “A” wins.
Three of a kind – This is a hand that contains three cards of the same rank, but no straight or flush. Any three of a kind hand beats any two pair hand.
Question: If player “A” is dealt 5♣ 5♠ 5♦ K♠ Q♣ 8♠ 2♣ and player “B” is dealt A♠ A♦ K♥ K♦ Q♥ Q♦ 5♥, who has the winning hand?
Answer: Player “A” has three fives with a king, queen. Remember that we may only play our best five cards. Player “B” has aces and kings with a queen. Player “A” wins.
Straight – This is a poker hand that contains five sequentially ranked cards, but no flush. Any straight beats any three of a kind.
Question: If player “A” is dealt 5♣ 4♣ 3♠ 2♥ A♥ A♦ A♣ and player “B” is dealt A♠ K♣ Q♥ J♦ 10♥ 9♣ 8♠, who has the winning hand?
Answer: Player “A” has a five high straight. In this case, the ace plays as a low card, below the two, to start the string of five sequential cards needed for a straight. Notice that the hand also contains three aces, but they do not play. A straight beats three of a kind, so the best five card hand for player “A” is 5♣ 4♣ 3♠ 2♥ A♥. Player “B” has seven sequential cards. The highest five sequenced cards will play, which are A♠ K♣ Q♥ J♦ 10♥, or an ace high straight. Notice that in the case of a straight, the ace can play as either the highest ranking card or the lowest ranking card, depending on the situation. Player “B” wins.
Flush – This is a hand that contains five cards of the same suit. Any flush beats any straight.
Question: If player “A” is dealt A♠ 8♠ 4♠ 3♠ 2♠ 5♥ 4♥ and player “B” is dealt K♣ J♣ 10♣ 6♣ 4♣ 3♣ 2♣, who has the winning hand?
Answer: Notice that player “A” has both a five high straight and a flush. Because we must play our best five cards, and a flush beats a straight, player “A” plays the flush. Furthermore, because the ace has the option to be played as a high card, it is always counted as high when used in a flush (except in the case of a five high straight flush). Player “A” has A♠ 8♠ 4♠ 3♠ 2♠, an ace high flush. Player “B” also has a flush and must play the best five flush cards, K♣ J♣ 10♣ 6♣ 4♣, a king high flush. Player “A” wins.
Full House – This is a five card hand that contains three of a kind plus a pair. Any full house beats any flush, except a straight flush.
Question: If player “A” is dealt 7♣ 7♠ 7♥ A♥ A♦ K♦ K♥, and player “B” is dealt Q♠ Q♣ Q♥ 2♠ 2♥ 5♥ 3♦, who has the winning hand?
Answer: Again, we count from the top down. Player “A” has 7♣ 7♠ 7♥ A♥ A♦, player “B” has Q♠ Q♣ Q♥ 2♠ 2♥. Queens are higher than sevens, so we can stop right there. Player “B” wins.
Four of a Kind – This hand contains all four cards of the same rank. Any four of a kind beats any full house. An example of a four of a kind hand would be 2♣ 2♠ 2♦ 2♥ A♥ K♦ Q♥, or four deuces with an ace.
Straight Flush – This hand contains five sequential suited cards. Any straight flush beats any four of a kind.
Question: If player “A” is dealt 5♣ 4♣ 3♣ 2♣ A♣ 7♠ 6♠, and player “B” is dealt 6♥ 5♥ 4♥ 3♥ 2♥ 9♦ 8♦, who has the winning hand?
Answer: Player “A” has a five high straight flush. The ace must play for low in this situation. Player “B” has a six high straight flush. Player “B” wins.
Royal Flush – This hand consists of an ace high straight flush. It is the highest possible hand in a game with no wild cards (in a game with wild cards five of a kind beats a royal flush). A royal flush in clubs would be A♣ K♣ Q♣ J♣ 10♣.
Now that you have familiarized yourself with the deck and the poker hand rankings order, you can begin learning how to play the different types of poker games. A good place to start is by learning the basics of Limit Holdem, which is one of the simpler, more popular games.
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5 Card Poker probabilities
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Ranking of poker hands
In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands.
Frequency of 5-card poker hands
The following enumerates the (absolute) frequency of each hand, given all combinations of 5 cards randomly drawn from a full deck of 52 without replacement. Wild cards are not considered. The probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand by the total number of 5-card hands (the sample space, five-card hands). The odds are defined as the ratio (1/p) - 1 : 1, where p is the probability. Note that the cumulative column contains the probability of being dealt that hand or any of the hands ranked higher than it. (The frequencies given are exact; the probabilities and odds are approximate.)
The nCr function on most scientific calculators can be used to calculate hand frequencies; entering nCr with 52 and 5, for example, yields as above.
| Hand | Frequency | Approx. Probability | Approx. Cumulative | Approx. Odds | Mathematical expression of absolute frequency |
|---|---|---|---|---|---|
| Royal flush | 4 | 0.000154% | 0.000154% | 649,739 : 1 | |
| Straight flush (excluding royal flush) | 36 | 0.00139% | 0.00154% | 72,192.33 : 1 | |
| Four of a kind | 624 | 0.0240% | 0.0256% | 4,164 : 1 | |
| Full house | 3,744 | 0.144% | 0.170% | 693.2 : 1 | |
| Flush (excluding royal flush and straight flush) | 5,108 | 0.197% | 0.367% | 507.8 : 1 | |
| Straight (excluding royal flush and straight flush) | 10,200 | 0.392% | 0.76% | 253.8 : 1 | |
| Three of a kind | 54,912 | 2.11% | 2.87% | 46.3 : 1 | |
| Two pair | 123,552 | 4.75% | 7.62% | 20.03 : 1 | |
| One pair | 1,098,240 | 42.3% | 49.9% | 1.36 : 1 | |
| No pair / High card | 1,302,540 | 50.1% | 100% | .995 : 1 | |
| Total | 2,598,960 | 100% | 100% | 1 : 1 |
The royal flush is a case of the straight flush. It can be formed 4 ways (one for each suit), giving it a probability of 0.000154% and odds of 649,739 : 1.
When ace-low straights and ace-low straight flushes are not counted, the probabilities of each are reduced: straights and straight flushes each become 9/10 as common as they otherwise would be. The 4 missed straight flushes become flushes and the 1,020 missed straights become no pair.
Note that since suits have no relative value in poker, two hands can be considered identical if one hand can be transformed into the other by swapping suits. For example, the hand 3♣ 7♣ 8♣ Q♠ A♠ is identical to 3♦ 7♦ 8♦ Q♥ A♥ because replacing all of the clubs in the first hand with diamonds and all of the spades with hearts produces the second hand. So eliminating identical hands that ignore relative suit values, there are only 134,459 distinct hands.
The number of distinct poker hands is even smaller. For example, 3♣ 7♣ 8♣ Q♠ A♠ and 3♦ 7♣ 8♦ Q♥ A♥ are not identical hands when just ignoring suit assignments because one hand has three suits, while the other hand has only two—that difference could affect the relative value of each hand when there are more cards to come. However, even though the hands are not identical from that perspective, they still form equivalent poker hands because each hand is an A-Q-8-7-3 high card hand. There are 7,462 distinct poker hands.
Derivation of frequencies of 5-card poker hands
of the binomial coefficients and their interpretation as the number of ways of choosing elements from a given set. See also: sample space and event (probability theory).
- Straight flush — Each straight flush is uniquely determined by its highest ranking card; and these ranks go from 5 (A-2-3-4-5) up to A (10-J-Q-K-A) in each of the 4 suits. Thus, the total number of straight flushes is:
- Royal straight flush — A royal straight flush is a subset of all straight flushes in which the ace is the highest card (ie 10-J-Q-K-A in any of the four suits). Thus, the total number of royal straight flushes is
- or simply . Note: this means that the total number of non-Royal straight flushes is 36.
- Royal straight flush — A royal straight flush is a subset of all straight flushes in which the ace is the highest card (ie 10-J-Q-K-A in any of the four suits). Thus, the total number of royal straight flushes is
- Four of a kind — Any one of the thirteen ranks can form the four of a kind by selecting all four of the suits in that rank. The final card can have any one of the twelve remaining ranks, and any suit. Thus, the total number of four-of-a-kinds is:
- Full house — The full house comprises a triple (three of a kind) and a pair. The triple can be any one of the thirteen ranks, and consists of three of the four suits. The pair can be any one of the remaining twelve ranks, and consists of two of the four suits. Thus, the total number of full houses is:
- Flush — The flush contains any five of the thirteen ranks, all of which belong to one of the four suits, minus the 40 straight flushes. Thus, the total number of flushes is:
- Straight — The straight consists of any one of the ten possible sequences of five consecutive cards, from 5-4-3-2-A to A-K-Q-J-10. Each of these five cards can have any one of the four suits. Finally, as with the flush, the 40 straight flushes must be excluded, giving:
- Three of a kind — Any of the thirteen ranks can form the three of a kind, which can contain any three of the four suits. The remaining two cards can have any two of the remaining twelve ranks, and each can have any of the four suits. Thus, the total number of three-of-a-kinds is:
- Two pair — The pairs can have any two of the thirteen ranks, and each pair can have two of the four suits. The final card can have any one of the eleven remaining ranks, and any suit. Thus, the total number of two-pairs is:
Poker Escalera K A 2 3 4
- Pair — The pair can have any one of the thirteen ranks, and any two of the four suits. The remaining three cards can have any three of the remaining twelve ranks, and each can have any of the four suits. Thus, the total number of pair hands is:
- No pair — A no-pair hand contains five of the thirteen ranks, discounting the ten possible straights, and each card can have any of the four suits, discounting the four possible flushes. Alternatively, a no-pair hand is any hand that does not fall into one of the above categories; that is, any way to choose five out of 52 cards, discounting all of the above hands. Thus, the total number of no-pair hands is:
Sequencia Poker K A 2 3 4
- Any five card poker hand — The total number of five card hands that can be drawn from a deck of cards is found using a combination selecting five cards, in any order where n refers to the number of items that can be selected and r to the sample size; the '!' is the factorial operator:
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