Introduction
In Poker, hand rankings determine the value of different card combinations. New players often wonder why a Full House is ranked higher than a Flush. Although both hands are strong, the Full House consistently ranks above the Flush in Poker’s hierarchy.
This article will explain why a full house beats a flush.
What is a Full House?
A Full House comprises three cards with one rank and two cards of another. For instance, a hand like 8-8-8-K-K qualifies as a full house. The value of the hand is primarily determined by the rank of the triplet, followed by that for the pair's rank. Thus, a full house composed of three Aces and two Kings (A-A-A-K-K) will outrank one consisting of three Kings and two Queens (K-K-K-Q-Q).
A Full House is one of the strongest hands in the game, ranking above a Flush but below Four of a Kind.
What is a Flush?
A Flush comprises five cards from the same suit, like 2-6-9-J-K, all in spades. These cards don't need to be in consecutive order. The value of a flush depends on its highest card; if two players have flushes, the one with the higher top card wins. For example, a flush led by an Ace will outperform a flush whose strongest card is a King.
A Flush ranks lower than a Full House but higher than a Straight. It is a strong hand but can still be beaten by higher-ranked hands.
Full House versus Flush
In Poker hand rankings, a Full House always beats a Flush. A Full House combines two powerful hands—Three of a Kind and a Pair—giving it a higher value than a Flush, requiring only five cards of the same suit.
Example:
Player 1: Full House
- Cards: 10♣, 10♦, 10♠, 4♠, 4♥
- Description: Player 1 has three 10s and two 4s, constituting a Full House (10s full of 4s).
Player 2: Flush
- Cards: A♠, K♠, 5♠, 7♠, 3♠
- Description: Player 2 has five spades, forming a Flush (Ace high).
In this scenario, Player 1 wins with a Full House, which beats Player 2's Flush.
Does Full House Beat a Flush?
A Full House beats a Flush due to its relative rarity and the strength of its hand combination. Here’s the reasoning behind it:
- Combinations for Each Hand
The strength of a Poker hand is based on how difficult it is to achieve that hand from a deck of 52 cards.
Total Number of Possible Poker Hands:
The total number of 5-card hands from a standard deck of cards is calculated as follows:
Total combinations: C(52,5) = 52×51×50×49×48/5×4×3×2×1 = 2,598,960
Flush Combinations:
A Flush consists of 5 cards in the same suit, not in sequence.
- There are 4 suits (hearts, diamonds, clubs, spades).
- The number of ways to choose 5 cards from any one suit is calculated as C(13,5) = 13×12×11×10×9/5×4×3×2×1 = 1,287 for each suit.
Thus, the total number of flushes is:
- Total flushes = 4×1,287 = 5,1484
Since we need to exclude straight flushes (which are a different hand):
- There are 10 possible straight flushes per suit, giving us 4×10=40 straight flushes.
So, the final number of flushes is:
Total flushes (excluding straight flushes) = 5,148−40=5,108
Full House Combinations:
A Full House consists of three cards of one rank and two cards of another rank.
To calculate the combinations:
- There are 13 possible ranks for the three-of-a-kind, and we can choose 3
suits from 4 for that rank: C(4, 3) = 4.
- For the pair, we choose from the remaining 12 ranks and select 2 suits from 4: C(4,2) = 6.
Thus, the total number of full houses is:
- Total full houses = 13×4×12×6=3,744
- Probability of Each Hand
Let's calculate the probability of getting each hand:
Full House Probability:
3,744/2,598,960 ≈ 0.00144 or 0.144%
Flush Probability:
5,1082/598,960 ≈ 0.00196 or 0.196%
- Relative Frequency
Full House combinations: 3,744
Flush combinations: 5,108
Because a Full House occurs less frequently than a Flush, it is a stronger hand. The ranking system in poker rewards hands that are harder to achieve. Since full houses are rarer and involve a specific combination of cards, they have a higher rank than a flush.
Frequently Asked Questions
Does a Flush beat a Full House?
No. A Flush cannot beat a Full House. The likelihood of drawing a Full House is lower than drawing a Flush. In poker hand rankings, rarer hands are valued more. Hence, the Full House takes precedence over a Flush.
How often do Full Houses and Flushes occur?
In a standard deck of cards, the probability of getting a Full House is roughly 0.144%, whereas the likelihood of drawing a Flush is approximately 0.196%. This variance in occurrence rates plays a part in ranking a Full House as stronger than a Flush.
What happens if both players have a Full House?
If two players have a Full House, the player with the higher three-of-a-kind wins. If their three-of-a-kinds are identical, the hand featuring the higher pair is victorious. For instance, a Full House of tens over fours beats nines over aces.
Conclusion
In card games, the ranking of hands is a carefully designed system based on probability and the strength of each hand. Consequently, in response to the question "Does a flush beat a full house?"—the answer is clearly no. A Full House, consisting of three cards of one rank and two cards of another rank, is more challenging to achieve than just five cards, all being from the same suit (a Flush), making it inherently stronger.
Two Pair in Poker: Hand Ranking
Does 3 of a Kind Beat a Straight
