How accurate is the rule of 2 and 4?
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.
What the rules are
On the flop, multiply your outs by 4 for the chance of hitting by the river; on the turn, multiply by 2 for the chance on the river. They work because each out is about 1/47 ≈ 2.1% of the turn and 1/46 ≈ 2.2% of the river. The rule of 4 adds the two streets, which double-counts the runouts where both cards are outs; the more outs you have, the bigger that double count.
Exact against the shortcuts
Exact figures are 1 − C(47 − x, 2) / C(47, 2) with two cards to come and x / 46 on the river. Errors are in percentage points; red marks a miss of two points or more.
| Outs | Exact, two cards | Rule of 4 | Error | 4x − (x − 8) | Error | Exact, river | Rule of 2 | Error |
|---|---|---|---|---|---|---|---|---|
| 1 | 4.3% | 4% | −0.3 | 4% | −0.3 | 2.2% | 2% | −0.2 |
| 2 | 8.4% | 8% | −0.4 | 8% | −0.4 | 4.3% | 4% | −0.3 |
| 3 | 12.5% | 12% | −0.5 | 12% | −0.5 | 6.5% | 6% | −0.5 |
| 4 | 16.5% | 16% | −0.5 | 16% | −0.5 | 8.7% | 8% | −0.7 |
| 5 | 20.4% | 20% | −0.4 | 20% | −0.4 | 10.9% | 10% | −0.9 |
| 6 | 24.1% | 24% | −0.1 | 24% | −0.1 | 13.0% | 12% | −1.0 |
| 7 | 27.8% | 28% | +0.2 | 28% | +0.2 | 15.2% | 14% | −1.2 |
| 8 | 31.5% | 32% | +0.5 | 32% | +0.5 | 17.4% | 16% | −1.4 |
| 9 | 35.0% | 36% | +1.0 | 35% | 0.0 | 19.6% | 18% | −1.6 |
| 10 | 38.4% | 40% | +1.6 | 38% | −0.4 | 21.7% | 20% | −1.7 |
| 11 | 41.7% | 44% | +2.3 | 41% | −0.7 | 23.9% | 22% | −1.9 |
| 12 | 45.0% | 48% | +3.0 | 44% | −1.0 | 26.1% | 24% | −2.1 |
| 13 | 48.1% | 52% | +3.9 | 47% | −1.1 | 28.3% | 26% | −2.3 |
| 14 | 51.2% | 56% | +4.8 | 50% | −1.2 | 30.4% | 28% | −2.4 |
| 15 | 54.1% | 60% | +5.9 | 53% | −1.1 | 32.6% | 30% | −2.6 |
| 16 | 57.0% | 64% | +7.0 | 56% | −1.0 | 34.8% | 32% | −2.8 |
| 17 | 59.8% | 68% | +8.2 | 59% | −0.8 | 37.0% | 34% | −3.0 |
| 18 | 62.4% | 72% | +9.6 | 62% | −0.4 | 39.1% | 36% | −3.1 |
| 19 | 65.0% | 76% | +11.0 | 65% | 0.0 | 41.3% | 38% | −3.3 |
| 20 | 67.5% | 80% | +12.5 | 68% | +0.5 | 43.5% | 40% | −3.5 |
Which one to use
- The rule of 2 is always slightly low (each out is worth 2.2%, not 2%), by at most 3.5 points at 20 outs. Safe to use as is.
- The rule of 4 is fine for the common draws (gutshot, open-ender, flush draw) and wrong for big ones: it calls 15 outs a 60% shot when it is 54.1%. Above 8 outs, take off one point per out over 8, and the worst error across 1–20 outs drops to 1.2 points.
- Only use the rule of 4 when you will see both cards for one price, which in practice means all-in on the flop. Facing a bet with more betting to come, you are buying one card at a time: use the rule of 2, and see the price tables on the flush draw page.
- Both rules assume the outs are fixed and clean. Some draws gain outs after the turn (a set filling up goes from 7 to 10) and some outs give the opponent a better hand. When it matters, the spot calculator counts real equity against a range.
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%
- Two overcards (6 outs)24.1%
- Set to full house33.4%
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.