Poker odds, counted
The probabilities every player ends up looking up: hitting a draw, flopping a set, running into aces. Each one here was counted by dealing every possible card and checked against its formula, and each page shows that formula with the numbers put in, so you can check it with a pencil.
The outs chart
On the flop you have seen five cards and 47 are unseen. With x outs, the chance of hitting by the river is 1 − C(47 − x, 2) / C(47, 2); on the turn alone it is x / 47 and on the river x / 46. The rule of 4 is shown for comparison, in red where it is two or more points out. From 14 clean outs a draw is a favourite with two cards to come.
| Outs | Turn | River (one card) | Turn or river | Odds against, two cards | Rule of 4 | Rule of 2 |
|---|---|---|---|---|---|---|
| 1 | 2.1% | 2.2% | 4.3% | 22.5 to 1 | 4% | 2% |
| 2 | 4.3% | 4.3% | 8.4% | 10.9 to 1 | 8% | 4% |
| 3 | 6.4% | 6.5% | 12.5% | 7.01 to 1 | 12% | 6% |
| 4 | 8.5% | 8.7% | 16.5% | 5.07 to 1 | 16% | 8% |
| 5 | 10.6% | 10.9% | 20.4% | 3.91 to 1 | 20% | 10% |
| 6 | 12.8% | 13.0% | 24.1% | 3.14 to 1 | 24% | 12% |
| 7 | 14.9% | 15.2% | 27.8% | 2.59 to 1 | 28% | 14% |
| 8 | 17.0% | 17.4% | 31.5% | 2.18 to 1 | 32% | 16% |
| 9 | 19.1% | 19.6% | 35.0% | 1.86 to 1 | 36% | 18% |
| 10 | 21.3% | 21.7% | 38.4% | 1.60 to 1 | 40% | 20% |
| 11 | 23.4% | 23.9% | 41.7% | 1.40 to 1 | 44% | 22% |
| 12 | 25.5% | 26.1% | 45.0% | 1.22 to 1 | 48% | 24% |
| 13 | 27.7% | 28.3% | 48.1% | 1.08 to 1 | 52% | 26% |
| 14 | 29.8% | 30.4% | 51.2% | 0.95 to 1 | 56% | 28% |
| 15 | 31.9% | 32.6% | 54.1% | 0.85 to 1 | 60% | 30% |
| 16 | 34.0% | 34.8% | 57.0% | 0.75 to 1 | 64% | 32% |
| 17 | 36.2% | 37.0% | 59.8% | 0.67 to 1 | 68% | 34% |
| 18 | 38.3% | 39.1% | 62.4% | 0.60 to 1 | 72% | 36% |
| 19 | 40.4% | 41.3% | 65.0% | 0.54 to 1 | 76% | 38% |
| 20 | 42.6% | 43.5% | 67.5% | 0.48 to 1 | 80% | 40% |
Two cards to come only matters if you see both for one price, which usually means all-in on the flop. Facing a bet with more to come, use the one-card columns. How good is the rule of 2 and 4? has the error for every row and a better shortcut. Whether a draw can call a given bet, and how much it must win later if not, is on the pot odds calculator.
Before the flop
On the flop
Drawing: turn and river
tools/build-odds.mjsenumerates: all 1,326 starting hands, all 19,600 flops, all 2,118,760 five-card boards, all 1,081 turn-and-river runouts, depending on the question. Where a textbook formula exists, the build fails unless the count is identical.- Straights, flushes and full houses are recognised by the site's Rust/WebAssembly engine, the same one the trainer runs. The straight-draw table on flopping a straight comes from asking the engine about every turn card on every flop.
- “Someone has a bigger pair” is exact inclusion–exclusion, cross-checked against a seeded simulation of whole deals.
- Odds are not equity. Making your hand is not the same as winning: for that, see the preflop matchups, the starting hands and the spot calculator. How often you end up with each made hand, royal flush to high card, is on the hand rankings. Raw data: odds.json.