Poker Equity & OddsOdds of filling up from a set or two pair
Short answer
A set on the flop becomes a full house or quads by the river 33.4% of the time, and 14.9% on the turn alone. From the turn, the river fills it 21.7%. Two pair on the flop fills up 16.7% by the river, 8.7% on the river once the turn has missed. The rule of 4 says 28% for a set because it counts only the 7 outs on the flop and misses the boards where the turn and the river pair each other. Every turn and river counted in the Poker Equity & Odds engine.
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Filling up, spot by spot
| Spot | Hand | Board | Outs | Next card | By the river |
|---|---|---|---|---|---|
| Set on the flop | 8♠ 8♥ | K♣ 8♦ 3♠ | 7 | 14.9% | 33.4% |
| Trips on the flop (pair on the board) | 9♠ 8♥ | 8♦ 8♣ 3♠ | 7 | 14.9% | 33.4% |
| Two pair on the flop | K♠ 8♥ | K♣ 8♦ 3♠ | 4 | 8.5% | 16.7% |
| Set on the turn | 8♠ 8♥ | K♣ 8♦ 3♠ 2♥ | 10 | 21.7% | |
| Two pair on the turn | K♠ 8♥ | K♣ 8♦ 3♠ 2♥ | 4 | 8.7% |
Trips with a pair on the board (9♠ 8♥ on 8♦ 8♣ 3♠) comes out exactly like a set: the same 7 outs and the same 33.4%. On the turn a set has more outs than on the flop, because the turn card is one more card it can pair.
Why a set is 33.4%, not 28%
one of the 7 outs
A third eight, or a king or a three to pair the board.
turn and river pair
Two of a new rank, like a seven and a seven, also fill you up.
With 8♠ 8♥ on K♣ 8♦ 3♠, the rule of 4 gives 7 x 4 = 28%, and the exact chance of one of those 7 outs arriving is 27.8%. Both miss the runouts where the turn and the river are the same new rank. That pairs the board and fills the set too, and adds 5.6%, for 33.4% in all. Two pair doesn't get that help: a paired turn and river only puts a pair on the board, a third pair it can't use, not a full house, so it stays close to its outs-only number. Its only extra runouts are a turn and river that are both threes, which puts trips on the board: 0.3%.
A set against a made flush
The flush came on the turn and you hold the set. You win on exactly the 10 river cards that fill you up, 10 of the 44 rivers, which is 22.7%. Whether calling a turn bet is right then depends on the price: the should you call with a draw guide and the pot odds cheat sheet do that part.
Count it at the table
On the flop, count 7 outs for a set and 4 for two pair, and remember that the set's real number is a little better than the rule says. The outs chart has every count of outs, and the flop equity cheat sheet shows what a set is worth against a draw.
Count the outs, then add the board pairing itself.
Let the app count the outs
Tap in your hand and the board, and Outs & Odds lists every card that fills you up and the exact chance one comes, free and offline.
Questions
What are the odds of a set filling up by the river?
33.4% from the flop, counting quads. That is every turn and river: 361 of the 1,081 pairs make a full house or better.
What are the odds of a set filling up on the turn?
14.9%: 7 of the 47 cards you can't see. If the turn is a blank of a new rank, the river is 21.7%, because the turn card adds three more outs.
What are the odds of two pair making a full house?
16.7% by the river from the flop, 8.5% on the turn alone. Two pair has 4 outs, and unlike a set it gets almost nothing from a turn and river that pair each other.
Does the 33.4% for a set include quads?
Yes. Quads by the river are 4.3% of the 33.4%. The rest are full houses.
Where the numbers come from. Every figure on this page comes from the app's own engine, the code that runs on your phone. Outs come from the Outs & Odds tool, hit chances are exact counts, and equities count every runout with no sampling. Engine build 482550e8. More worked numbers: flush draw odds and the pot odds cheat sheet.
