Why does a flush beat a straight? Because it is rarer. The entire poker hierarchy is just combinatorics — counting how many of the 2.6 million hands make each rank.
Poker Hand OddsLive
the mathematics of a five-card hand
Poker hands by probability (log scale)
Royal flush
1 in 649,740
Straight flush
1 in 72,193
Four of a kind
1 in 4,165
Full house
1 in 694
Flush
1 in 509
Straight
1 in 255
Three of a kind
1 in 47
Two pair
1 in 21
One pair
1 in 2
High card
1 in 2
Controls
Highlight a hand
There are exactly 2,598,960 distinct five-card poker hands. Ranking them by rarity gives the hand hierarchy: a royal flush occurs once in about 650,000 deals, while nearly half of all hands are just a pair or high card. The odds are pure combinatorics — counting how many of the C(52,5) hands match each pattern.
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How it works
There are exactly C(52,5) = 2,598,960 five-card poker hands. Ranking them by how few hands achieve each pattern produces the familiar hierarchy: a royal flush appears once in about 650,000 deals, while nearly half of all hands are a single pair or worse. The rankings are pure counting, not convention.
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