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12 rows  Frequency of 5-card poker hands. Cumulative probability refers to the probability of drawing. May 19, 2006 Instead it is the probability of getting a 2 of a kind, a 3 of a kind, a 4 of a kind, 2 pair, or a full house. And if you add the counts (provided in the link above) of each of these together and divide by the total number of hands, you will get. Which is exactly your number above.

The odds of flopping Four of a Kind or better with a pocket pair is 0.24% or 1 in 416

Definition of Four of a Kind (also known as Quads) –

We hold four cards of equal rank.

Example – AdAhAsAcKh

Four of a Kind Aces is the strongest Four of a Kind hand in poker and is also referred to as “Quad Aces”.

Odds of Making a Four of a Kind (Quads) on the Flop

Making Four of a Kind on the flop is an extremely rare occurrence.

We’ll focus solely on the odds of making Four of a Kind or better.

Odds of flopping Four of a Kind or better with any starting hand = 0.03%

Odds of flopping Four of a Kind or better with a pocket pair = 0.24%

Odds of flopping Four of a Kind or better with AKo = 0.01%

To put this into context, assuming any starting hand, we’ll flop Quads or better roughly once every 3,333 flops.

Assuming we play roughly 25% of our starting hands, we can expect to flop Quads or better approximately once every 13,332 hands.

Odds of Making Four of a Kind on the Later Streets

The most common draw to make Four of a Kind is where we already hold Three of a Kind on the current street. Since there are only 4 cards of each rank in the deck, it means that we only have one out to make Quads on each street.

Odds of hitting Quads on the turn = 1/47 = 0.0213 or roughly 2.1%

Odds of hitting Quads on the river = 1/46 = 0.0217 or roughly 2.2%

Poker Four Of A Kind Probability Calculator

To calculate the odds of hitting Quads on either the turn or river, we can use a simple trick.

We’ll calculate the probability of not hitting and then subtract from 100.

Odds of not hitting Quads on the turn = 46/47

Odds of not hitting Quads on the river = 45/46

Odds of not hitting Quads on either the turn or the river = 46/47 * 45/46 = 0.9574 or roughly 95.7%

Hence, the chance of hitting Quads by the river after flopping trips is (100 – 95.7) = roughly 4.3%

So even after making Three of a Kind, we are statistically unlikely to make Four of a Kind by the river.

Implied Odds Analysis of Four of a Kind

Four of a Kind always carries excellent implied odds provided we use at least one of our hole cards.

Poker Four Of A Kind Probability Meaning

If the four cards of equal rank are on the board, then every player at the table has Quads, so the implied odds are much less relevant. Any player with an Ace in the hole will win, once there are four cards of the same rank on the board.

Unfortunately, it’s not that likely our opponent will invest a large amount of his stack with the King kicker, so we can’t expect excellent implied odds.

Using one of our hole-cards to make Quads (i.e. three cards of equal rank on the board) carries excellent implied odds. If our opponent has a big pair in the hole, he’ll often assume he has the nuts with his full house and never fold.

He might be aware of the fact that we could hold Quads, but it will rarely be correct for him to make the laydown unless we are playing with very deep effective stacks.

Using two of our hole-cards to make Quads (i.e. two cards of identical rank on the board) is also an extremely valuable configuration in terms of implied odds since our Quads will be relatively disguised and we can frequently “cooler” any full houses our opponent may have.

It does, however, vary by board texture. One issue is that there will be two cards of identical rank on the board, and our opponent may give us credit for having trips even if he never suspects that we might have Quads.

For example, we are holding 55 on 5-5-4-2, and our opponent makes some laydowns, concerned we might hold a 5x. We also know that our opponent can never have a 5x himself due to card removal effects.

Holding 55 on a board texture like 5-5-6-6 is extremely valuable, however, since our opponent will never fold the 6x overfull.

In other words, the exact implied odds of Quads made with a pocket pair depends on the board structure.

Basic Strategy Advice

Quads are basically the nuts. Never fold Quads when 100bb deep (provided we use at least one of our hole cards in formulating the four of a kind).

If the four cards of equal rank are on the board itself, then we should proceed cautiously unless we specifically have the Ace. In such circumstances, we should usually look to continue betting for value even though it’s unlikely we’ll get paid off by worse.

Since this is an uncommon scenario, newer players may make call downs after incorrectly determining the strength of their hand. (For example, they might not realise that their QQ has been counterfeited on the KKKKx board and that any Ace will now win the pot.)

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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.

Probability Of Four Of A Kind In Poker

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.

• 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:

Three Of A Kind Probability

• 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:

• 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:

• 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:

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

Probabilities In Poker

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