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.
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:
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:
The deck yields 156 distinct ways to make four-of-a-kind and 624 total possible ways. A♠ A♥ A♦ A♣ 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:
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.