Probabilities of the Standard Poker Hands

A five-card poker hand is dealt from a standard 52-card deck, and the total number of distinct five-card hands, ignoring order, is $\binom{52}{5} = 2{,}598{,}960$. Every hand's probability is just its own count divided by that number, and hands are ranked precisely by rarity — the harder a hand type is to assemble, the higher it ranks.

Counting a specific hand type is a combinatorics exercise. A flush (five cards of one suit, ignoring straight flushes) is counted by choosing 1 of 4 suits and then 5 of that suit's 13 cards: $4 \times \binom{13}{5} = 5{,}148$ raw combinations, minus the straight flushes contained within, giving 5,108 pure flushes — a probability of $5{,}108 / 2{,}598{,}960 \approx 0.197\%$. A full house (three of one rank, two of another) is counted by choosing the triple's rank (13 ways) and 3 of its 4 suits ($\binom{4}{3}=4$), then the pair's rank from the remaining 12 ranks and 2 of its 4 suits ($\binom{4}{2}=6$): $13 \times 4 \times 12 \times 6 = 3{,}744$ hands, a probability of about 0.144% — rarer than a flush, hence ranked higher.

By contrast, one pair — the most common named hand — occurs in about 42% of all hands, which is exactly why it sits at the bottom of the ranked hand hierarchy: it's the easiest pattern to stumble into by chance.