Probabilities of poker hands

Probability is the likelihood of one outcome compared to another. In poker, it is possible to calculate the probability of being dealt a certain hand, and how often this hand is likely to win, as long as the cards are dealt randomly.

In the standard deck of 52 cards, there are 13 ranks (from two through to ten, Jack, Queen, King, and Ace) and 4 suits (diamonds, hearts, clubs, and spades). When playing poker, there are 9 possible types of hands within these cards. A poker hand is a particular arrangement of card suits and ranks.

A royal flush is the strongest hand whilst no pair is the weakest. Stronger hands will always beat weaker hands.

Types of poker hands

Royal Flush - A Royal Flush is made out of 10, Jack, Queen, King, and Ace, all of the same suit.

Straight Flush - A Straight Flush is five cards in a row, all in the same suit.

Four of a Kind - A Four of a Kind is the same card in each of the four suits.

Full House - A Full House is a pair plus three of a kind in the same hand.

Flush - A Flush is five cards in the same suit, not in numerical order.

Straight - A Straight is five cards in numerical order, but not in the same suit.

Three of a Kind - Three of a Kind is three of one card and two non-paired cards.

Two Pair - A Two Pair is two different pairings or sets of the same card in one hand..

One Pair - A Pair is a pairing of the same card.

No Pair / High Card - If no player has any of the above card combinations, then the player with the highest card wins.

5 card poker hands

Stud poker is a classic, popular version of poker. Each player is dealt 5 cards, and the player with the best card hand wins.

There are 2,598,960 distinct poker hands in this variation of poker.

Rank	Hand	Distinct hands	Probability (%)	Odds
1	Royal flush	4	0.000154	1 in 649,739
2	Straight flush	36	0.00139	1 in 72,192.33
3	Four of a kind	624	0.02401	1 in 4,165
4	Full house	3,744	0.1441	1 in 693.1667
5	Flush	5,108	0.1965	1 in 508.8019
6	Straight	10,200	0.3925	1 in 253.8
7	Three of a kind	54,912	2.1128	1 in 46.32955
8	Two pair	123,552	4.7539	1 in 20.03535
9	One pair	1,098,240	42.2569	1 in 1.366477
10	No pair / high card	1,302,540	50.1177	1 in 0.9953015
Total		2,598,960

7 card poker hands

Texas Hold’Em, a popular variation of poker, involves a hand consisting of 7 cards. The player then picks their best 5 cards.

The total number of combinations in Texas Hold’Em is 133,784,560.

Hand	Distinct hands	Probability (%)	Odds against
Royal flush	4,324	0.0032	1 in 30,939
Straight flush	37,260	0.0279	1 in 3,589.6
Four of a kind	224,848	0.168	1 in 594
Full house	3,473,184	2.6	1 in 37.5
Flush	4,047,644	3.03	1 in 32.1
Straight	6,180,020	4.62	1 in 20.6
Three of a kind	6,461,620	4.83	1 in 19.7
Two pair	31,433,400	23.5	1 in 3.26
One pair	58,627,800	43.8	1 in 1.28
No pair	23,294,460	17.4	1 in 4.74
Total	133,000,000

Royal Flush

A royal flush is incredibly unlikely in Hold’Em, there being only a 0.0032% chance of hitting one during a poker hand. With over 133,000,000 distinct hand possibilities, only 4,324 can lead to a royal flush. On average, every 30,979 hands played will lead to this hand.

Straight Flush

Being the second-best hand in Hold’Em, a straight flush is also highly improbable. There is a 0.0279% chance of scoring a straight flush, with only 37,260 distinct hands available in the game. Normally, every 3,589.57 hands will lead to a straight flush.

Four of a kind

Playing a four of a kind is unlikely in a game of Hold’Em, with every 594 hands leading to this poker hand. This equates to a probability of 0.168% as there are 224,848 different hand possibilities.

Full house

A full house is another unlikely combination in Hold’Em, with only a 2.60% likelihood of scoring this hand. There are 3,473,184 distinct hand combinations, meaning that every 37.52 hands will lead to a full house.

Flush

There is a 3.03% chance of getting a flush in Hold’Em, as there are 4,047,644 distinct hand possibilities. On average, every 32.05 hands will lead to a flush.

Straight

In Hold’Em, there is a 4.83% chance of playing a straight. Normally, every 20.65 turns will lead to a straight as there are 6,180,020 different ways to form this poker hand.

Three of a kind

In Hold’Em, three of a kind is a poker hand that is seen regularly. There are 6,461,620 ways to create this hand, corresponding to a probability of 4.83%. On average, every 19.7 hands will lead to a three of a kind.

Two pair

There is a 23.5% chance of getting a two pair in Hold’Em, making it the second-most likely poker hand to get. On average, every 3.26 hands will lead to this hand because there are 31,433,400 different ways to form a two pair.

One pair

One pair is the most likely poker hand you will form in Hold’Em poker. There are 58,627,800 combinations available for one pair, meaning there is a 43.8% possibility of scoring this hand. Every 1.28 turns will lead, on average, to a pair.

No pair / high card

There is a 17.4% chance of getting no pair, or high card, when you play Hold’Em. On average, with 23,294,460 distinct hands, you will play this hand every 4.74 hands.