Odds of flopping a pair or two pair
How often does an unpaired hand flop a pair? 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.
The working
With two unpaired cards, say AK, three of each rank are left: six cards that pair you, among 50 unseen. The flop misses only if all three cards come from the other 44:
P(miss) = C(44, 3) / C(50, 3) = 13,244 / 19,600 = 67.57% P(pair+) = 1 − 67.57% = 32.43% P(two pair) = 3 × 3 × 44 / C(50, 3) = 396 / 19,600 = 2.02% By the river = 1 − C(44, 5) / C(50, 5) = 48.74%
| Flop | Probability | 1 in | Odds against | Count |
|---|---|---|---|---|
| Pairs exactly one of your cards | 28.96% | 3.45 | 2.45 to 1 | 5,676 / 19,600 |
| Pairs both: two pair | 2.02% | 49.5 | 48.5 to 1 | 396 / 19,600 |
| Two of one rank: trips | 1.35% | 74.2 | 73.2 to 1 | 264 / 19,600 |
| Two of one, one of the other: full house | 0.09% | 1,089 | 1,088 to 1 | 18 / 19,600 |
| All three of one rank: quads | 0.01% | 9,800 | 9,799 to 1 | 2 / 19,600 |
| Any of these | 32.43% | 3.08 | 2.08 to 1 | 6,356 / 19,600 |
| Neither card pairs | 67.57% | 1.48 | 0.48 to 1 | 13,244 / 19,600 |
Roughly two flops in three miss an unpaired hand completely, which is why continuation bets work and why AK is a drawing hand after the flop more often than not. The build dealt all 19,600 flops to A♠K♦ and sorted them by how many cards of each rank came; every count matches its formula. The engine's view of the same flops, by best five-card hand:
| Best hand (AK offsuit) | Flops | Share |
|---|---|---|
| High card | 10,496 | 53.55% |
| Pair | 7,920 | 40.41% |
| Two pair | 792 | 4.04% |
| Three of a kind | 308 | 1.57% |
| Straight | 64 | 0.33% |
| Full house | 18 | 0.09% |
| Four of a kind | 2 | 0.01% |
The engine finds more pairs (7,920) than the 5,676 flops that pair one of your cards: a paired board like 7-7-3 is a pair of sevens for everyone. And 396 of the flops that pair you also pair the board, which it scores as two pair. Its straight row is Q-J-T, the only flop that makes AK a straight.
Two overcards that miss still have 6 outs, 24.1% by the river.
More on the flop
- Flopping a set11.8%
- Flopping a flush or flush draw0.8%
- Flopping a straight or straight draw1.3%
- An overcard to your pair41.4%
Other odds
- Counted, not quoted.
tools/build-odds.mjsdeals 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.