// poker odds · drawing: turn and river

Odds of a set improving to a full house

How often does a set fill up by the river? The short answer is below, and under it the working: the formula with the numbers put in, and the count from dealing every possible card.

full house or quads by the river33.40%361 / 1,081
as a frequency1 in 2.99on average
odds against1.99 to 1misses to hits
on the turn alone14.89%7 / 47
Example: 7♣7♦ on 7♥K♣2♠7 outs on the turn · 10 on the river after a blank

Why the outs change

A set of sevens on K-7-2 fills up with the last seven (quads) or any of the three kings or three deuces (full house): seven outs on the turn. If the turn is a blank, say a nine, it pairs too: now the three remaining nines also fill you up, and the river has ten outs. The rule-of-four shortcut and the standard two-card formula both assume the outs stay fixed, so they undercount:

Fixed 7 outs:  1 − C(40, 2) / C(47, 2) = 27.84%   (wrong)
Real:          1 − (40/47) × (36/46)   = 1 − 85.11% × 78.26% = 33.40%
Engine, every runout: 361 / 1,081 = 33.40%

The engine dealt all 1,081 turn-and-river pairs to the example and agrees with the second line exactly. A flopped set is a 33% favourite to be a full house or better by the river, and against a made flush or straight that is usually the whole of its equity. To flop the set in the first place: 11.76%.

The bottom two pair spot uses the same idea from the other side: two pair has four outs to a full house, and a draw that is “ahead” still loses to those.

Hitting is half of it; the other half is the price. The pot odds pages show which draws can call a third-pot, half-pot or pot-sized bet, and what each one must win later when it cannot.

Pot odds by bet size →

More drawing: turn and river

Other odds

// how these numbers were computed
  • Counted, not quoted. tools/build-odds.mjs deals every possible flop, turn, river or starting hand and counts. Where a closed-form formula exists, the build refuses to publish unless the count matches it exactly.
  • Made hands come from the engine. Whether cards make a straight, a flush or a full house is decided by the same Rust/WebAssembly evaluator the trainer runs, not by hand-written rules.
  • Odds are not equity. Hitting your hand is not winning the pot. For that, the matchup pages deal every board against a specific hand and the spot calculator works against a range.
  • Raw data: odds.json.