Does Four Of A Kind Beat a Full House in Poker & Why?
To give you the answer right away, yes four of a kind beats a full house in poker.
However, if you are truly interested in becoming a better poker player, you should also learn the logic behind these hand rankings in poker as they will help you understand the game better and become a more successful player.
So, to help you with this, we have prepared a detailed article to explain why does four of a kind beat a full house.
The Four of a Kind Combination in Poker Explained
A 5-card combination that consists of four cards of the same rank and one card of a different rank is called four of a kind in poker.
A much more popular name for this combination, and the one that most players use, is quads.
Two examples of quads in poker:
- K♠K♣K♦K♥6♠ – four-of-a-kind kings or quad kings with a 6 kicker
- 3♣3♠3♦3♥A♦ – four-of-a-kind threes or quad threes with an A kicker
Rules for Ranking Four of a Kind Combinations in Poker
There are two rules that are used to rank quads in poker:
- The rank of the four cards that make the central part of the combination
- The rank of the fifth card (the kicker)
Here are a couple of examples of how these rules are applied in practice.
- Hand 1) 9♠9♦9♣9♥5♣ (four-of-a-kind nines or quad nines with a 5 kicker) vs.
- Hand 2) 8♠8♦8♣8♥4♠ (four-of-a-kind eights or quad eights with a 4 kicker)
In this example we can use the first rule since the rank of the cards in the central part of the combination in both hands is different.
Hand 1 outranks Hand 2 because the rank of the four cards in Hand 1 (9) outranks the rank of the 4 cards in Hand 2 (8).
The second rule is used in situations where the rank of the central part of the combination is the same in multiple hands. This scenario can only happen in hands where multiple players are playing the board, at least in Texas Hold’em.
- Player 1 is holding Q♠J♦ as his hole cards.
- Player 2 is holding 10♠9♦ as his hole cards.
- The board is 7♠7♣7♦7♥2♠
In this hand, the best 5-card combination that Player 1 can put together is 7♠7♣7♦7♥Q♠ (quad 7s with a Q kicker), and the best 5-card combination that Player 2 can put together is 7♠7♣7♦7♥10♠ (quad 7s with a T kicker).
Since the rank of the central part of the combination is the same in both hands (7), the kicker comes into play.
In this case, Player 1 wins because the kicker in his combination (Q) outranks the kicker in Player 2’s combination (T).
The Total Number of Four-of-a-Kind Combinations in Poker
Poker is played with the standard deck that consists of 52 cards divided into 4 suits (hearts, diamonds, spades, and clubs) with each suit containing 13 card ranks (A, K, Q, J, T, 9, 8, 7, 6, 5, 4, 3, 2).
From this we can calculate that there are:
- 624 possible four-of-a-kind combinations
- 48 different kickers
You can calculate the number of possible four-of-a-kind combinations by multiplying the number of different card ranks by the number of kickers and to get the total number of kickers you need to deduct the four cards from the central part of the combinations from the total number of cards in the deck.
The number of kickers:
- 52 – 4 = 48
The number of four-of-a-kind combinations:
- 48 x 13 = 624
The Full House Combination in Poker Explained
In poker, a 5-card combination that consists of three cards of the same rank and another two cards of the same rank is called a full house.
The full house combination is also referred to as a boat by players.
Two examples of a full house combination in poker:
- 9♠9♦9♣5♠5♦ – nines full of fives
- 2♠2♦2♥10♠10♦ – deuces full of tens
Rules for Ranking Full House Combinations in Poker
There are two main rules based on which full house combinations are ranked in poker:
- The rank of the three cards of the same rank
- The rank of the two cards of the same rank
Here is how these rules are applied in practice:
- Hand 1) Q♠Q♦Q♥K♣K♦ (queens full of kings) vs.
- Hand 2) J♠J♦J♥K♣K♦ (jacks full of kings)
In this example, we can use the first rule since the trips’ part of the full house combination has a different rank in both hands:
Based on this rule Hand 1 outranks Hand 2 because the rank of the trips’ part of the combinations in Hand 1 (Q) outranks the rank of the trips part of the combination in Hand 2 (J).
Only in situations in which the rank of the three-of-a-kind part of the full house combination is the same in both hands, do we use the second rule to determine which full house combination is stronger.
- Hand 1) K♠K♦K♥8♣8♦ (kings full of eights) vs.
- Hand 2) K♣K♦K♥5♣5♦ (kings full of fives)
As we mentioned, because the rank of the trips (K) is the same in both of these full house combinations we apply the second rule and use the rank of the pair to determine the winning hand.
Because the rank of the pair in Hand 1 (8) outranks the rank of the pair in Hand 2 (5), Hand 1 is the winner in this example.
The Total Number of Full House Combinations in Poker
To put together a full house combination in poker, a player needs to collect three cards of one rank and two cards of another rank.
Since there are 13 possible card ranks in the standard 52-card deck, there are 13 possible ranks for the three-of-a-kind combination and 12 possible ranks for the two-of-a-kind combination (discounting the rank that the three-of-a-kind combination is made).
If we multiply the number of ranks for the three-of-a-kind combination by the number of ranks for the two-of-a-kind combination we will get the total number of different full house ranks.
- 13 x 12 = 156
Now, to get the total number of possible full house combinations in poker, we also need to include all the different suits.
Because there are four suits this means that the three-of-a-kind part of the full house combinations can be made in four different ways using the same card ranks and the two-of-a-kind part of the combination can be made in six different ways using the same card ranks.
To find out the total number of ways in which both parts of the combination can be made, we need to multiply the number of possible card ranks by the number of ways in which each part of the combination can be made.
- Three of a kind: 13 x 4 = 52
- Two of a kind: 12 x 6 = 72
To get the total number of combinations, we now only need to multiply these two numbers.
- 53 x 72 = 3,744
Based on this calculation we can see that there are 3,744 possible full-house combinations in poker.
Does Four of a Kind Beat a Full House in Poker?
Okay, so we’ve already established that four-of-a-kind beats a full house in poker. Now let’s find out why is that.
In the game of poker, all of the card rankings are determined based on frequencies. The lower the frequency of putting together a hand, the stronger the hand.
In the table below are all of the made hands in poker ranked as well as some of the math that goes with them.
|Four of a Kind||624||0.02401%||4,164-to-1|
|Three of a Kind||54,912||2.1128%||46-to-1|
As you can see, both the four of a kind and the full house are in the upper part of the rankings,. However, there is one big difference that separates them, and that is the total number of possible combinations.
There are 624 possible combinations of four-of-a-kind in poker, which means that the odds of putting together a four-of-a-kind combination in any given hand are 4,164-to-1 or 0.02401%.
On the other hand, there are 3,744 possible full house combinations and the odds of putting together this combination in any given hand are 693-to-1 or 0.1441%.
This means that a poker player is roughly six times more likely to put together a full house combination than a four-of-a-kind combination in any given hand and that is why four-of-a-kind beats a full house in poker.