1. Poker Odds - Calculating poker. Probability can be calculated easily for a single event, like the flipping of the River card from the Turn.
2. 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.
3. Estimating probabilities in poker is very simple. You just have to count the number of outs (the cards that will help you) and calculate the probability of winning using the above rules of thumb. Any player who is serious about playing poker should at least know those two rules of thumb.

1. Poker Hand Probability Formula
2. Poker Hand Winning Probability Formula

The probability of forming any given hand is the number of ways it can be arranged divided by the total number of combinations of 2,598.960. Below are the number of combinations for each hand. Just divide by 2,598,960 to get the probability.

POKER PROBABILITIES

• Texas Hold'em Poker
  Texas Hold'em Poker probabilities
• Omaha Poker
  Omaha Poker probabilities
• 5 Card Poker
  5 Card Poker probabilities

POKER CALCULATOR

• Poker calculator
  Poker odds calculator

POKER INFORMATION

• Poker hand rankings
  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.

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:

Poker Hand Probability Formula

• 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:

• 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.

Home > 5 Card Poker probabilities

Some people may argue that poker is a game of chance, instead of a skills game. And you know what? These people are actually right, since most of the time you will not be able to predict the outcome of the next card which will determine if you’ll win – or lose a hand.

But knowing how to calculate odds and probabilities in poker will help you ‘guessing’ the next card based on a few simple formulas which are described below. If you play more than just a few hands of poker you will learn to understand that skill is actually a huge part of the game, and luck becomes a lot less significant.

Poker Hand Winning Probability Formula

Knowing this, poker indeed is a game of skill. If poker is played seriously, skill simply dominates luck.

Here are some related pages and posts:

Probabilities

The following calculates the frequency of each poker hand for all combinations that can be made out of 5 cards drawn at random from a full deck of 52 playing cards, without the use of wild cards.

The probability in poker is determined based on the number 2,598,960, which represents the total number of five card combinations that can be created. In this case, the probability is the frequency of a hand divided by the total number of five card hands, and the odds are defined by the formula ((1 / p) – 1 : 1), where p corresponds to the probability.

Frequency of Five Card Poker hands

In the following table, the frequencies are precise, while the probabilities and the odds given are approximate.

Also, the royal flush is included in the table as a straight flush, since the royal flush can be made in four different ways (hearts, spades, clubs and diamonds). This calculates a probability of 0.000001539077169 and odds of 649,739 : 1.

The probability of a straight flush or an ace low straight will be reduced in case they won’t be counted. They will become 9/10 as common as they usually would be.

This pages will tell you more about card outs:

Pot odds

In poker, pot odds are used to determine the expected value of a play. Poker players use pot odds to calculate the profitability over the long term. Generally, odds are expressed as a win:loss ratio. The following formula can be used to calculate the percentage probability:

win:loss odds = win / (win + loss) % probability

For instance, this means that 1:4 odds equals 1 / (1 + 4) = 20% probability.

Instead, odds are also expressed as a loss:win ratio, or also known as odds against (see table above). To make it totally clear, we have used win:loss odds in remainder of this explanation.

The correct poker playing strategy is influenced by pot odds for every possible table action (check, fold, call or raise) at every single moment during a poker game.

Most likely, calling is the correct play when facing a call with higher pot odds, just like folding is the correct play when facing a call with lower pot odds.

For instance, if you can win a $1000 pot by making a call for just $1, essentially no hand should be folded, since you only have to win this call one time every thousand hands for it to be profitable.

The probability of winning is equal to the chance that a player will win with either the best hand at showdown or simply because the opponents will fold out of the hand before opening up.

Be sure to also read:

Pot odds in Texas Hold’em Poker

The approximate percentage probability that a player hits one of his outs on the first coming card in Texas Hold’em is calculated as follows:

(number of outs) x 2 + 1

In case you have the outs for a potential flush, i.e. 9 cards could improve your hand, then you’ll have about 19% chance to complete your flush with the next card (9 x 2 + 1 = 19).

With two more cards to come (the turn and the river), the approximate percentage probability is calculated as follows:

(number of outs) x 4 – 1

This means that you’ll have 35% roughly to hit a flush with two more cards to come (9 x 4 – 1 = 35).

For a correctly classified table action, the odds for winning a hand should never be lower than the applicable pot odds.