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Why Does Four of a Kind Beat a Full House.

Why Does Four of a Kind Beat a Full House?

Most poker games use the standard poker hand rankings to determine each winning hand. The rarer a hand occurs mathematically, the higher it stands in the hand rankings.

Full house and four-of-a-kind both represent extremely strong poker hands. Picking up one of these hands in a game of Texas Hold’em gives you a strong chance of winning the pot.

Which of these premium hands wins in a head-to-head battle? Does four-of-a-kind beat a full house?

In this case, the answer is yes: four-of-a-kind beats a full house. Let’s take a look at the math behind both:

Why Does Four of a Kind Beat a Full House?

To calculate the odds of making any particular five-card poker hand, we need to figure out the probability of making that hand if we randomly drew five cards from a 52-card deck. 

Texas Hold’em probabilities are calculated a bit differently, with the object of trying to make the best five-card hand out of seven possible cards. Here’s a look at the probabilities behind four-of-a-kind and full house hands:

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The Math Behind Four of a Kind

When you randomly draw five cards out of a standard 52-card deck, you have an 0.024% chance of making four of a kind. That equates to 4,165-to-1 odds against drawing four-of-a-kind.

Four-of-a-kind consists of four of the same-ranking card in a five-card hand. An example of four-of-a-kind looks like this:

Why Does Four of a Kind Beat a Full House.

The deck yields 156 distinct ways to make four-of-a-kind and 624 total possible ways. A♠ AAA♣ 4♠, for instance, represents a distinct four-of-a-kind hand. The kicker can be any of the four suits, giving you four ways to draw that distinct four-of-a-kind.

In Texas Hold’em, with all five community cards on the board, you have an 0.168% chance of making four-of-a-kind (594-to-1 odds against).

The Math Behind a Full House

Drawing a full house marks a pretty rare occurrence, but the hand known as the “full boat” comes up significantly more often in a game of poker.

You need to draw three-of-a-kind and a pair in the same five-card hand to make a full house. An example of a full house could be a hand like this:

Why Does Four of a Kind Beat a Full House.

If you randomly draw five cards out of a 52-card deck, you have an 0.1441% probability of making a full house. That translates to 693.17-to-1 odds against drawing a full house.

The deck gives us 156 distinct ways to make a full house. Multiplied by all possible suit combinations, there are 3,744 total ways to make a full house.

The probability and the total number of ways to make a full house are higher for a full house than four-of-a-kind, making four-of-a-kind a rarer and stronger hand in a game of poker.

In Texas Hold’em, you have a 2.6% chance of making a full house with all five community cards on the board. That equates to 35.7-to-1 odds against making a full house.

Note: Are you here just to learn how to play poker...or do you want to know how to win too? Get this free guide with 10 quick poker strategy tips if you want to come out on top.

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Note: Are you here just to learn how to play poker...or do you want to know how to win too? Get this free guide with 10 quick poker strategy tips if you want to come out on top.

free poker guide

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About the Author
Geoffrey Fisk

Geoffrey Fisk

Freelance writer and poker player based in Las Vegas, Nevada.

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