Why is one pair so common? Counting versus intuition
· 5 min read
A lopsided deck of possibilities
Roughly 42 out of every 100 five-card hands contain exactly one pair, and about 50 contain nothing at all. A flush shows up in about 1 hand in 500. That is a huge gap, and it comes entirely from counting.
Patterns are groups of hands
Each individual hand has the same probability: 1 in 2,598,960. A pattern like “one pair” is a group of hands, and the group is large because there are many ways to build it: 13 ranks, 6 suit pairs, then many choices for three other ranks and their suits.
A flush needs all five cards from the same suit. Once you fix a suit there are only C(13, 5) = 1,287 ways to choose the ranks, so even across four suits the group is small.
The key idea
Rare does not mean any single hand is unusual. A flush is rare because few hands qualify, while a particular nondescript hand is exactly as unlikely as a royal-looking one. Keeping “outcome” and “event” apart is the first step in clear probability thinking.
Test it yourself
Run the hand simulator at 100 and then 100,000 samples. At 100 hands the rarest types usually don't appear at all; at 100,000 the observed frequencies sit very close to the exact ones.