Does 4 of a Kind Beat a Full House in Poker

Introduction

Understanding hand rankings is crucial for making the right decisions in Poker, especially in high-stakes situations. One of the most common confusions for newer players is whether a Four of a Kind (quads) beats a Full House (boat).

The answer is yes, Four of a Kind outranks a Full House. But beyond just knowing this fact, it’s important to understand the reasons why quads rank higher than a boat.

Here, we will explore the reasons behind this hierarchy.

What is Four of a Kind?

A Four of a Kind in Poker is a hand composed of four cards with the same rank, along with an extra unrelated card known as the kicker. This strong combination ranks just below hands such as the straight flush and royal flush in terms of strength.

Example of Four of a Kind

10♠ 10♣ 10♥ 10♦ 3♠ (Four Tens with a Three kicker)

In this case, the player holds four tens and a kicker. Unless the opponent has an even higher-ranked quads (rare) or a straight flush, this is nearly unbeatable.

What is a Full House?

A Full House consists of three cards of one rank and two cards of another rank. It is a strong hand and ranks just below Four of a Kind.

Example of a Full House

A♠ A♣ A♦ 4♠ 4♣ (Aces full of Fours)

This hand is very strong and beats hands like flushes, straights, and trips, but it falls short when facing quads.

Why Four of a Kind Beats a Full House

Here’s why Four of a Kind beats a Full House:

Rarity

Poker hands are ranked according to how difficult they are to make. The fewer combinations of a particular hand, the more valuable that hand becomes. This is the foundation of poker hand rankings.

Full House Combinations

• There are 3,744 possible combinations of a Full House in a standard 52-card deck.

Calculation:

Choose the rank for the three cards (13 ranks) → C(13, 1)

Choose 3 cards from that rank’s 4 suits → C(4, 3)

Choose a different rank for the pair → C(12, 1)

Choose 2 cards from that second rank’s 4 suits → C(4, 2)

Total: C(13, 1) × C(4, 3) × C(12, 1) × C(4, 2) = 3,744

Four of a Kind Combinations

• Four of a Kind is much rarer, with only 624 possible combinations.

Calculation:

Choose the rank for the four cards (13 ranks) → C(13, 1)

Choose all 4 suits from that rank → C(4, 4)

Choose 1 kicker from the remaining 48 cards → C(48, 1)

Total: C(13, 1) × C(4, 4) × C(48, 1) = 624  

Four of a Kind is six times rarer than a Full House, making it much more valuable.

Poker Hand Strength

• Four of a Kind represents a more structurally dominant hand than a Full House. It is extremely difficult to beat quads unless the opponent has an even rarer hand (straight flush or higher quads). The ability to have four cards of the same rank creates an almost impenetrable wall against most hands in poker.

• A Full House, while strong, is vulnerable to hands like quads, straight flushes, and royal flushes. It is constructed by combining two different ranks, which means it doesn’t have the same level of dominance over the deck as Four of a Kind.

Probability

Full House Probability

• The probability of being dealt a Full House in a 5-card poker hand is about 0.1441%, or roughly 1 in every 693 hands.
• Full House Combinations: 3,744
• Total Possible Poker Hands: 2,598,960
• Probability Formula: P(Full House) = 3,744/2,598,960 ≈ 0.1441%

Four of a Kind Probability

• The probability of being dealt Four of a Kind is much lower, around 0.0240%, or roughly 1 in every 4,165 hands.
• Four of a Kind Combinations: 624
• Total Possible Poker Hands: 2,598,960
• Probability Formula: P(Four of a Kind) = 624/2,598,960 ≈ 0.0240%

The stark difference in probability shows how much rarer and, therefore, stronger Four of a Kind is compared to a Full House.

Four of a Kind Vs. Full House

Here are the strategic differences between Four of a Kind and Full House.

Game Theory

• Due to its rarity, Four of a Kind is typically played differently than a Full House. Players holding Four of a Kind tend to adopt a more aggressive strategy, recognizing that there are only limited hands capable of defeating them.
• Although a Full House is strong, it can be defeated by quads or higher hands. Consequently, players with a Full House might exercise more caution in the game if there is a chance their opponent has Four of a Kind or even a straight flush.

Psychological Edge

• Holding quads provides a psychological advantage. Players with Four of a Kind often have the confidence to dominate the pot, knowing that they are ahead most of the time. This confidence can influence opponents to fold even strong hands like Full Houses, as they fear being beaten by quads.

Bluff and Deception

• With a Full House, players may be more prone to bluffing or semi-bluffing to maximize the pot or scare off opponents who might have lower hands. However, a Four of a Kind is rarely used in bluffs since it is often a sure winner.

Scenarios Where Four of a Kind Beats a Full House

Let’s examine some practical examples where quads triumph over a Full House:

Example 1

• Player A’s Hand: Q♣ Q♦ Q♥ Q♠ 5♣ (Four Queens)
• Player B’s Hand: 8♠ 8♣ 8♥ K♦ K♣ (Eights full of Kings)

In this case, Player A’s Four of a Kind (Four Queens) beats Player B’s Full House (Eights full of Kings). Despite Player B having a strong hand, quads are simply too powerful.

Example 2:

• Player A’s Hand: 9♦ 9♠ 9♣ 9♥ 2♠ (Four Nines)
• Player B’s Hand: J♣ J♦ J♠ 7♥ 7♠ (Jacks full of Sevens)

Again, Four Nines beats Jacks full of Sevens, no matter how well Player B plays. The dominance of quads is undeniable.

Additional Reasons: Why Four of a Kind Beats a Full House

Dominance Over Other Hands

• Quads can beat many more hands than a Full House can. It is almost unbeatable unless facing a straight flush, making it significantly stronger.

Eliminating Opponents

• Having Four of a Kind puts you in a prime position to knock out other players from the pot, especially in tournaments where each hand matters. Full Houses are strong, but can sometimes cause you to lose big if you face quads.

Rarity Multiplied by Certainty

• The combination of rarity and certainty makes Four of a Kind superior. While a Full House is strong, players always have a lingering fear of being beaten. With quads, that fear mostly vanishes.

Frequently Asked Questions

Why does Four of a Kind beat a Full House?

A Four of a Kind beats a Full House due to its rarity and difficulty in forming. With just 624 possible combinations compared to the 3,744 ways to achieve a Full House, it occurs less frequently. The infrequency and strength associated with having four cards of identical rank place quads higher in poker hand rankings.

Can a Full House ever beat Four of a Kind?

No, a Full House cannot beat Four of a Kind. Regardless of the ranks involved, Four of a Kind will always rank higher on the poker hand hierarchy. Even the best Full House (Aces full of Kings) loses to the lowest possible Four of a Kind (twos). The hand rankings in Poker are set so that rarer, stronger hands, like quads, beat more common hands, like Full Houses.

What beats Four of a Kind in poker?

A: Four of a Kind is a very strong hand, but it can be beaten by only two hands: a Straight Flush and a Royal Flush. A Straight Flush consists of five consecutive cards of the same suit (e.g., 5♠ 6♠ 7♠ 8♠ 9♠), while a Royal Flush is the highest possible straight flush (A♠ K♠ Q♠ J♠ 10♠).

Conclusion

Four of a Kind is not only a rarer hand but also a structurally superior one compared to a Full House. Whether you are calculating odds or making critical decisions at the Poker table, knowing what beats Four of a Kind beats a Full House is key to mastering poker. Understanding the ‘why’ behind the ranking gives you a deeper appreciation of how Poker hands are valued and can help you improve your play.