Does four of a kind beat two pair?

Yes. Four of a kind beats two pair in poker — every time. In the ranking of 10 hand categories, four of a kind is #3 and two pair is #8. Categories are compared first: even quad deuces beats aces and kings — the category always decides first.

#3
Four of a Kind — wins

Four cards of the same rank ("quads").

Q♠Q♥Q♦Q♣7♦
#8
Two Pair — loses

Two different pairs.

A♠A♦9♣9♥5♠

How rare is each hand?

Rarity is exactly why the order is what it is. Out of 2,598,960 possible five-card hands, four of a kind shows up in 0.024% of them (624 combinations), while two pair appears in 4.75% (123,552 combinations). The harder a hand is to make, the higher it ranks.

Does two pair ever beat four of a kind?

No — never. The category comparison is absolute: the highest possible two pair still loses to the lowest possible four of a kind. Ranks inside a category only matter when both players hold the same category.

Prove it to yourself

Which hand wins? Tap it.

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