Counting Outs

The denominator is 45

On the flop you can see seven cards: your 4 and the board's 3. So 52 minus 4 minus 3 leaves 45 unseen, and there are 990 different turn-and-river pairs that can come. By the turn it is 44 unseen.

Rule of the arithmetic: unseen cards on an Omaha flop is always 45, never 47 — that is the hold'em number, where you only hold two cards, and it gives 1,081 run-outs instead of ours.

Not a heuristic and not a preference: our engine asserts it. The hit-probability function refuses to run unless the unseen count comes out to 45 on the flop and 44 on the turn (the postflop rule spec), because the most common way to get every number on this page wrong is to import a habit.

With that denominator fixed, here is the chance that at least one of your outs arrives by the river. This is what we call the hit probability:

• 8 outs: 32.7%  ·  9 outs: 36.4%  ·  12 outs: 46.7%
• 13 outs: 49.9%  ·  16 outs: 59.0%  ·  17 outs: 61.8%  ·  20 outs: 69.7%

Memorise the two ends of that list and you can interpolate the middle. Do not memorise anybody else's table.

The two-and-four rule, and where it stops working

The shortcut everybody learns: multiply your outs by four to get your chance of hitting by the river. Against our table:

• 8 outs → shortcut 32, truth 32.7%. Off by 0.7.
• 9 outs → shortcut 36, truth 36.4%. Off by 0.4.
• 12 outs → shortcut 48, truth 46.7%. Off by 1.3, and now in the dangerous direction.

Use the shortcut only at 9 outs or fewer. In that range it runs low — by at most a point and a half, worst around five outs — and you can do it at the table.

Flop to river, when you intend to see both cards. Above 9 outs it overestimates every single time, and the error grows with the hand you were most excited about.

At 13 outs the shortcut says 52 and the truth is 49.9%, off by 2.1 — it just moved you from "behind" to "ahead" on a hand where you are not. At 20 outs it says 80 and the truth is 69.7%: an overestimate of 10.3 points. Omaha wraps normally land between 13 and 20 outs, so the entire interesting range of this game sits inside the broken part of the rule.

That the rule overestimates at high out counts is not our discovery — it is stated publicly, along with a correction formula. PokerCoaching, "Outs in Poker" <https://pokercoaching.com/blog/outs-in-poker/> But that article never mentions Omaha and its truths are computed against 47 unseen cards, so neither its numbers nor its correction formula can be carried over. The table above is ours.

Count the nut outs first

Two draws, same street, and the one with more cards is the worse hand.

• 9♦5♠4♥3♠ on 6♥7♦9♠: 13 outs, of which 0 are nut outs. Every single card that completes your straight also completes a better one for somebody.
• J♠T♥6♣5♣ on 9♥8♦2♣: 8 outs, of which 8 are nut outs.

Measured against a stack-off proxy range — hands that already have two pair or better on this flop, or a big draw of their own, 800 of them sampled — the thirteen-out hand is worth 26.3% and the eight-out hand 35.1%.

Report every draw as two numbers: total outs and nut outs. A single number is not a description of an Omaha draw.

On the flop, on an unpaired board with three different suits. On a paired board almost no straight is the nuts, so the nut count stops meaning what it means here; that board type gets its own treatment, and the tool for reading it, in C19.

If one number were enough, the 13-out hand would be the better of those two, and it is worse by nearly nine points. That is the whole reason this course counts nut outs before it counts outs.

We say "proxy range", never "the opponent's range": this project has no postflop range data at all, and naming it something it is not would be inventing data we do not have.

What to take away

One more warning before you trust any of these numbers too far. Hold 7♥7♠6♦T♣ on 7♣4♦8♥: 8 straight outs, 0 of them nut outs, so the formula says 32.7%. Enumerate all 990 run-outs and the real chance of finishing with a straight or better is 60.3% — higher by 27.6 points, because the pair in your hand hit the board and every full house and quad run-out is invisible to an outs count.

The formula only ever counts down. Treat it as a floor, not an estimate.

• The denominator is 45. Everything else follows from that.
• The two-and-four rule works to 9 outs. Past that it flatters you, and at 20 outs it is off by more than ten points.
• Two numbers, always. Total outs, then nut outs.
• The formula is a lower bound. When we grade you, we enumerate; the shortcut is for your head, not for the scoreboard.

These are rules of this course, not an equilibrium solution: each one is a line we drew because our engine showed the situations on either side behave differently. Next we take the big draws apart and find out what 69.7% is actually worth.