The formula
probability by the river = 1 − (47−outs)/47 × (46−outs)/46
Where it comes from
After the flop there are 47 cards you have not seen. The chance of missing is the chance that neither the turn nor the river brings one of your outs, and what we want is exactly the opposite. At the table it is approximated by multiplying the outs by four after the flop and by two after the turn, which lands close without a calculator.
How to work it out by hand
- Count your outs: the cards that give you the winning hand
- Work out the chance of missing both cards and subtract it from one
- Compare against the equity you need: bet ÷ (pot + twice the bet)
What is worth knowing
Two cautions the arithmetic does not carry. The first is that not every out is clean: if completing your flush hands someone else a straight, that card does not help you and has to come out of the count. The second is that comparing against pot odds is only complete if you will not have to pay more on later streets, which is why players add implied odds, what they expect to win afterwards if they hit. On nine flush outs the rule of 4 gives 36 % against a real 34.97 %: it overshoots slightly, and that is why close decisions are worth doing properly.