Two overcards: odds of pairing by the river
How often do two overcards pair 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.
Why it is 6 outs
A♣K♦ on 8♥5♠2♣ is behind any pair, but an ace or a king on the turn or river makes top pair. Three of each are unseen: six outs. They are clean only against a hand like 8-x; against a set they are worthless and against A-8 the kings still win but the aces do not.
The working
On the flop you have seen five cards, so 47 are unseen. They come off in pairs, turn and river, and there are C(47, 2) = 1,081 ways for that to happen. You miss only when both cards are among the 41 that do not help:
P(miss) = C(41, 2) / C(47, 2) = 820 / 1,081 = 75.86% P(hit) = 1 − 75.86% = 24.14% (3.14 to 1 against) Turn = 6 / 47 = 12.77% River = 6 / 46 = 13.04% (if the turn missed)
Then the engine dealt every one of the 1,081 runouts to the example hand and counted how many finish with a pair of aces or kings: 261 / 1,081 = 24.14%. The same count as the formula, runout for runout, so on this board nothing else gets you there.
Is it a call?
Calling a bet of B into a pot of 1 risks B to win 1 + B, so you need B ÷ (1 + 2B) of the final pot. The table compares that with the draw's chance of arriving. “All-in now” means you see both cards for one price; “one card only” means there will be another bet on the turn, and then the draw has one card at a time. Where one card is not enough, the right-hand column says how much extra you must win on the hands where it hits (implied odds) to make the call break even.
| Bet faced | Equity needed | All-in now (24.1%) | One card only (12.8%) |
|---|---|---|---|
| ⅓ pot | 20.0% | Call | Fold, or win 0.94× pot more when it hits |
| ½ pot | 25.0% | Fold | Fold, or win 1.92× pot more when it hits |
| ⅔ pot | 28.6% | Fold | Fold, or win 2.89× pot more when it hits |
| Pot | 33.3% | Fold | Fold, or win 4.83× pot more when it hits |
| 1.5× pot | 37.5% | Fold | Fold, or win 7.75× pot more when it hits |
This assumes every out wins and nothing else does: if your opponent can make a better hand on the same card, those outs are dirty and the real equity is lower. The spot calculator prices it against a whole range instead of a single hand.
AK misses the flop entirely about two times in three; see the flop odds for unpaired hands.
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
- Flush draw (9 outs)35.0%
- Open-ended straight draw (8 outs)31.5%
- Gutshot (4 outs)16.5%
- Combo draw (12 to 15 outs)54.1%
- Set to full house33.4%
- The rule of 2 and 4, checkedup to 12.5 pts off
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.