The rule of four and two, and where it breaks

Your outs are the unseen cards that give you the best hand. After the flop you can see five cards, your two and the board's three, so 47 are unseen. After the turn, 46 are.

With one card to come, the chance of hitting is your outs divided by 46. Each out is worth a little over 2%, so doubling your outs gives a slight underestimate. With two cards to come, you get two tries at a card worth about 2% each, so four times your outs is the first guess.

The exact chance of hitting by the river counts the ways both cards miss and takes that away from one:

For a flush draw, with 9 outs, that is 1 − (38 × 37) ÷ (47 × 46), or 35.0%. The rule says 36%.

Up to 8 outs the rule of four is within half a point. From 9 outs on it overestimates the exact chance, and by 15 outs it says 60% against an exact 54.1%. The rule of two goes the other way: it always comes in a little low, by 2.6 points at 15 outs.

Multiplying by four adds the chance of hitting on the turn to the chance of hitting on the river. That counts the deals where you hit on both cards twice. With a few outs, hitting twice is rare and the double count barely matters. With 15 outs, both cards are outs often enough to push the estimate 6 points too high.

There is a simple fix for draws above 8 outs: take four times your outs, then subtract the number of outs above 8.

For those larger draws, this adjustment stays within about a point of the exact figures in the table.

The two-card column assumes you see the turn and the river for the price you pay now. That is true when you are all-in on the flop. It is not true when you call a flop bet and face another bet on the turn: then you pay again, and the one-card figure is the one to compare with each price.

Calling a flop bet with a flush draw because you hit 35% of the time by the river, then calling a big turn bet because you have come this far, pays twice for one draw. For a nine-out draw, use 9/47 = 19.1% from flop to turn and 9/46 = 19.6% from turn to river, and let implied odds cover the gap if the stacks are deep enough.

The table assumes every out wins. Some do not. Hold A♥ 5♥ on K♥ 8♥ 3♠ against a set of eights, and you count 9 hearts. One of them, the 3♥, pairs the board and gives the set a full house, so on the turn only 8 hearts win. If the river pairs the board too, your flush loses again.

The same thing happens when an out gives your opponent a bigger hand, as when your straight card also completes their flush. Count only the cards that win, and keep the shortcut's rounding error separate from any outs that do not win.