Does Four of a Kind Beat a Full House in Poker & Why?
Thanks to the nature of the game, you can play poker for months or years at a time before you hit rare combinations such as a straight flush or four of a kind. Nevertheless, you still need to learn how these hands stand in the hand rankings.
Knowing does four of a kind beat a full house or whether is it better to have a full house or four of a kind is a must if you want to be a winning poker player.
With that said, you also need to know some basic math behind these combinations as this knowledge will help you improve your game even further.
For those looking for a simple answer: yes – four of a kind beat full house in poker!
If you’d like to know more about this topic, including why one combination is stronger than the other and what’s the math behind this, keep on reading.
Does Four of a Kind Beat a Full House In Poker?
Yes, as explained, four of a kind beats a full house in poker.
When it comes to hand rankings in poker, the main thing you need to know is that all hands in Texas Hold’em are ranked based on the probability of getting them.
This is the first aspect of the game in which math plays a major role, as you need to do some calculations to find out possible combinations, probabilities, and odds.
To spare you some time, we have done these calculations for each hand rank and presented the results in the table below.
| Hand | Combinations | Probability | Odds |
| Royal Flush | 4 | 0.000154% | 649,739-to-1 |
| Straight Flush | 36 | 0.00139% | 72,192-to-1 |
| Four of a Kind | 624 | 0.02401% | 4,164-to-1 |
| Full House | 3,744 | 0.1441% | 693-to-1 |
| Flush | 5,108 | 0.1965% | 509-to-1 |
| Straight | 10,200 | 0.3925% | 254-to-1 |
| Three of a Kind | 54,912 | 2.1128% | 46-to-1 |
| Two Pair | 123,552 | 4.7539% | 20-to-1 |
| One Pair | 1,098,240 | 42.2569% | 1.37-to-1 |
The odds of getting a royal flush in any given poker hand are 649,739-to-1 which makes it the rarest hand combination in poker. Therefore, based on the rule of frequency, it is also the strongest hand in the game.
On the total opposite part of the poker hand ranking specter is the one pair combination. With odds of 1.37-to-1, it is the most common paired hand in poker, which means that it is also the weakest paired hand in the game.
The two hand combinations that we are interested in the most, the four of a kind and the flush are among the stronger hands in poker.
The odds of getting a four of a kind combo are 4,164-to-1 which is enough for the third spot among all poker hands.
The only two hands that beat four of a kind are the royal flush and the straight flush.
With odds of 693-to-1, the full house is one spot lower in the hand rankings and it is outranked by the royal flush, the straight flush, and the four of a kind.
As you can see, all hands in Texas Hold’em are ranked by their frequency without any exceptions, and you can use math to determine the strength of each and every one of them.
What Is Four of a Kind in Poker?
In poker, a four of a kind hand combination consists of four cards of the same rank and a fifth card of a different rank known also as the kicker. Another term that is commonly used for this hand combination in Texas Hold’em is quads.
For example:
- A♠A♣A♦A♥7♠ – four of a kind, aces, or quad aces
- 2♣2♠2♦2♥J♦– four of a kind, deuces, or quad deuces
How Four of a Kind Combinations Are Ranked
There are two steps that will help you determine a strength of a four of a kind hand:
- First, the rank of the four cards with the same rank
- Second, the rank of the kicker
For example,
- J♠J♦J♣J♥5♣ vs. 10♠10♦10♣10♥9♠ (quad jacks with a 5 kicker vs. quad tens with a 9 kicker)
In this example, quad jacks with a 5 kicker outrank quad tens with a 9 kicker because the rank of the four cards with the same rank in the first hand (J) outranks the rank of the four cards with the same rank in the second hand (T).
The rank of the kicker only comes into play in situations where the rank of the four cards with the same rank is the same in both hands. This can only happen when both players are playing the board.
For example:
- You are holding A♠K♦ and your opponent is holding K♥Q♣, and the board is 8♠8♣8♦8♥J♠.
In this scenario, the best four of a kind hand that you can put together is 8♠8♣8♦8♥A♠ (quad eights with an ace kicker) while the best four of a kind hand that you opponent can put together is 8♠8♣8♦8♥K♣ (quad eights with a king kicker).
Because both hands have the same rank of the four of a kind combination (8), the kicker is used to break the tie which means that your hand outranks your opponents because your kicker (A♠) outranks your opponent’s kicker (K♥).
The Number of Four of a Kind Combinations in Texas Hold’em
The standard deck used for Texas Hold’em consist of 13 different card ranks (A, K, Q, J, T, 9, 8, 7, 6, 5, 4, 3, 2), which are sorted in 4 different suits (hearts, diamonds, spades, and clubs). In total, the number of cards in the deck adds up to 52.
Based on this information, we can calculate the number of possible four of a kind combinations by multiplying the number of different ranks by the number of possible kickers:
- 13 x (52 – 4) = 13 x 48 = 624
We deducted the number of cards already in a four of a kind combination (4) from the total number of cards (52) to get the total number of possible kickers in each combination (48).
Based on this calculation, we can see how people come up with the number 624 when talking about the number of possible four of a kind combinations in poker.
What Is a Full House in Poker?
In poker, a full house combination consists of three cards of the same rank + two cards of another rank. It is a combination of a three of a kind (trips) and two of a kind (one pair).
Another term that is often used for this combination is a boat or a full boat. The name of each individual full house is a combination of the names of the three of a kind and the pair it consists of.
For example:
- 8♠8♣8♦K♣K♦ – eights full of kings
- 2♣2♠2♥A♥A♦ – deuces full of aces
How Full House Combinations Are Ranked
When it comes to ranking full house combinations, there are two steps, each individual full house combination is ranked based on the:
- Rank of the three of a kind part of the combination
- Rank of the pair part of the combination
For example:
- J♠J♦J♥K♣K♦ vs. 10♠10♦10♥K♣K♦ (jacks full of kings vs. tens full of kings)
In this example, the full house combination jacks full of kings outranks the full house combination tens full of kings, because the rank of the three of a kind in the first hand (J) outranks the three of a kind combo in the second hand (T).
The second part of the full house combination (a pair) is used to decide the winning full house combination only in situations where the first part of the full house combination (three of a kind) has the same rank in both hands.
For example:
- A♠A♦A♣7♠7♣ vs. A♥A♦A♣5♠5♦ (aces full of sevens vs. aces full of fives).
If we try to apply the first rule in this situation, we won’t get far because the rank of the three of a kind part of the full house combination is the same in both hands.
Because of this, we need to apply the second rule and compare the second parts, i.e. the pairs.
In the first hand, the pair part of the full house combination consist of two sevens (7♠7♣) while the pair in the second hand consists of two fives (5♠5♦).
This means that the first full house outranks the second one because a pair of sevens outranks a pair of fives in Texas Hold’em.
The Number of Full House Combinations in Texas Hold’em
To form a full house in poker, you need three cards of one denomination + two cards of a different denomination.
There are 13 possible ranks for the three of a kind and 12 possible ranks for the two of a kind (discounting the one rank that is already in the three of a kind part of the full house).
- 13 x 12 = 156
So, without taking into account the suits, there are 156 different ranks of full houses.
There are four different suits for each rank which means that you can choose three cards of the same rank in four different ways. Furthermore, you can choose two cards of the same rank in six different ways.
- 13 x 4 = 52 (the total number of ways in which you can pick the three of a kind part of the full house)
- 12 x 6 = 72 (the total number of ways in which you can pick the two of a kind part of the full house)
When we multiply the total number of ways in which we can pick the three of a kind part of the full house with the total number of ways in which we can pick the pair segment, we get the total number of possible full house combinations in poker.
- 53 x 72 = 3,744
There are 3,744 unique full house combinations in Texas Hold’em.
If we divide this number by the number of total different poker hands we can calculate the probability of having a full house.
- 3744 / 2598960 = 0.1441
So, there is a 0.1441% chance of getting a full house on any given hand.
We hope that this article not only answered your question does full house beat 4 of a kind, but also helped you understand the math behind these two poker combinations!
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