Holding a strong hand in a game of poker often leads to winning money. The strongest hands in the poker hand rankings don’t often occur in the game, and seeing two monster hands go head-to-head marks an even rarer event.
Is a straight or flush strong enough to win against four-of-a-kind? Does four-of-a-kind beat a straight?
In this case, four-of-a-kind wins against both straights and flushes. Let’s dig into the math behind these three hands and figure out the probabilities of making each in a poker game.
Why Does Four of a Kind Beat a Straight or a Flush?
The lower the probability of making a poker hand, the higher that hand stands in the poker hand rankings. To figure out poker hand probabilities, we need to calculate the chances of drawing a particular hand if we randomly chose five cards from a standard 52-card deck.
Texas Hold’em, the world’s most popular poker game, involves making the best possible five-card hand from seven total cards. This yields different probabilities versus drawing just five cards.
Four-of-a-kind happens less often than straights and flushes in a poker game, and here’s a breakdown of the math behind the probabilities of each:
The Math Behind Four of a Kind
If you draw five random cards from a 52-card deck, you have an 0.024% chance of making four-of-a-kind. This translates to 4,165-to-1 odds against making four-of-a-kind from a random five-card draw.
The deck gives us 156 distinct ways to make four-of-a-kind. An example of a distinct four-of-a-kind hand could look like this:
Any combination of four queens with a four kicker makes the same distinct four-of-a-kind hand. Multiplying the 156 distinct four-of-a-kind hands times the four possible suits for the kicker, the deck yields 624 possible ways to make 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 Straight
A straight consists of five sequential cards, including any mix of suits. The following hand is an example of a straight:
A random draw of five cards from a 52-card deck yields an 0.3925% chance of making a straight. This equates to 253.8-to-1 odds against drawing a straight.
A standard poker deck gives us ten distinct ways to draw a straight. Multiplying by all possible suit combinations, the deck yields 10,200 different possible ways to make a straight.
In Texas Hold’em, with all five community cards on the board, you have a 4.62% chance of making a straight. This probability translates to 20.6-to-1 odds against making a straight in Texas Hold’em.
The Math Behind a Flush
To make a flush, you need to put together five cards of the same suit. An example of a flush looks like this:
The 52-card deck yields 1,277 distinct flush hands. Multiplied by four different suit possibilities for each distinct flush, a regulation poker deck gives us 5,108 total ways to make a flush.
When drawing five random cards from a deck, you have an 0.1965% probability of making a flush (508.8-to-1 odds against). In Texas Hold’em, if all five community cards are on the board, you have a 5.82% chance of making a flush.
Four-of-a-kind, flushes, and straights are all strong hands in most variants of poker. Four-of-a-kind occurs the least out of the three hands, however, making it the winner against a straight or a flush.