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Poker Odds
On this page we cover the topic of poker odds.
Poker Odds - the probability of each type of 5 card hand in poker can be calculated by computing the proportion of hands of that type among all possible hands.
So what are the odds of getting the best possible hand in poker? Well, let's see.
The frequencies given in the poker odds chart below are exact and the probabilities and odds are approximate.
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| Hand |
Frequency |
Probability |
Odds Against |
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| Straight Flush |
40 |
.00154 % |
64,973 : 1 |
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| Four of a Kind |
624 |
.0240 % |
4,164 : 1 |
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| Full House |
3,744 |
.144 % |
693 : 1 |
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| Flush |
5,108 |
.197 % |
508 : 1 |
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| Straight |
10,200 |
.392 % |
254 : 1 |
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| Three of a Kind |
54,912 |
2.11 % |
46.3 : 1 |
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| Two Pair |
123,552 |
4.75 % |
20.0 : 1 |
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| Pair |
1,098,240 |
42.3 % |
1.37 : 1 |
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| High Card |
1,302,540 |
50.1 % |
0.995 : 1 |
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| Total |
2,598,960 |
100 % |
0 : 1 |
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The Royal Flush, also known as an Ace-high Straight Flush is included in the Straight Flush calculations above.
Calculated alone, the Royal Flush can be formed 4 ways, once in each suit, giving it a probability of .00000154 and odds of 649,739 : 1.
When Ace-Low Straights and Straight Flushes are not counted, the probabilities of each are reduced and Straights and Straight Flushes become 9/10 as common as they otherwise would be.
Calculating Poker Odds
The above chart shows the frequency of each poker hand, given all combinations of 5 cards randomly drawn from a full deck of 52 cards without the use of wild cards.
The probability is calculated based on the total number of 5 card combinations, which is 2,598,960. The probability is the frequency of the hand divided by the total number of 5 card hands (2,598,960).
The odds are defined by (1/p) - 1 : 1 where p is the probability.
The following calculations show how the above frequencies were determined.
Familiarity with the basic properties of the binomial coefficients and their interpretation as the number of ways of choosing elements from a given set is essencial to fully understand these derivations.
For some more information related this subject you can also reffer to sample space and event (probability theory).
 Straight Flush - Each Straight Flush is uniquely determined by its highest ranking card.
In each of the 4 suits, these rankings go from 5 (A-2-3-4-5) up to
an Ace (10-J-Q-K-A).
Therefore, the total number of Straight Flushes is 40.
Four of a Kind - Any one of the thirteen ranks can form Four of a Kind.
This leaves 52-4=48 possibilities for the final card.
Therefore, the total number of Four of a Kind is 624.
Full House - The Full House consists of Three of a Kind and a Pair.
The Three of a Kind can contain any one of the thirteen ranks, in any three of the four suits.
The Pair can be any one of the remaining twelve ranks, and any two of the four suits.
Therefore, the total number of Full Houses is 3, 744.
Flush - The Flush comprises of any five of the thirteen ranks, in one of the four suits, minus the 40 Straight Flushes.
Therefore, the total number of Flushes is 5, 108.
Straight - Any one of the ten possible sequences of five consecutive cards, from (5-4-3-2-A) to (A-K-Q-J-10) can form a Straight. Each one of these 5 cards can come from any one of the four suits. The 40 Straight Flushes must be excluded just like in the Flush calculation above.
Therefore, the total number of Straights is 10, 200.
Three of a Kind - The Three of a Kind can consist of any three cards out of the thirteen ranks, and can contains any three of the four suits. The other two cards can be any two cards of the remaining twelve ranks. Each card can come from any one of the four suits.
Therefore, the total number of Three of a Kinds is 54, 912.
Two Pair - Each Pair in the Two Pair hand can have any two cards of the thirteen ranks, and each Pair can come out of any two of the four suits. The final card can be any card out of eleven remaining ranks, in any one of the four suits.
Therefore, the total number of Two Pairs is 123, 552.
Pair - The Pair can be formed out of any one of the thirteen ranks, in any two of the four suits.
The remaining three cards can have any three of the remaining twelve ranks, and each can come from any one of the four suits.
Therefore, the total number of Pairs is 1, 098, 240.
High Card - The High Card hand consists of any five cards out of the thirteen ranks, in any one of the four suits, discounting the ten possible Straights and the four possible Flushes.
Alternatively, the High Card hand is any hand that does does not fall into one of the two categories mentioned above. High Card can have 5 cards that are chosen in any way out of the 52 card deck, discounting all of the above hands. . Therefore, the total number of High Card hands is 1, 302, 540.
Other related topic that may be of interest to you is our poker pot odds section and you can access it by following this link.
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