The question you opened
Nine outs, two cards to come
You hold four hearts after the flop, so nine hearts are left to finish the flush.
About what percent of the time do you make it by the river?
about 35 percent
Multiply outs by four when two cards are still coming: nine times four is 36. The exact figure is 35 percent, so the shortcut is one point high.
There are 47 unseen cards after the flop from your point of view, and nine of them finish the flush. The chance of missing on the turn is 38 over 47. If you miss, the chance of missing again on the river is 37 over 46. Multiply those and you get about 0.65, so the chance of hitting is about 35 percent.
The shortcut is called the rule of four and two. With two cards to come, multiply outs by four. With one card to come, multiply by two. Nine outs gives 36 percent and 18 percent, against true values of 35 and 19. The error stays under two points for any realistic number of outs, and it shrinks further if you subtract one for every out above eight.
The number only matters next to a price. If the pot holds 100 and someone bets 50, you are risking 50 to win 150, so you need to be right one time in four, about 25 percent. With 35 percent equity the call is correct. If the bet were 150 into 100, you would need 37.5 percent, and the same hand becomes a fold. Equity is never good or bad on its own. Two adjustments belong in any honest version.
The four times figure assumes you get to see both cards, which is only true if you are already all in. Facing a bet on the turn as well, the two times figure is the safer one for the flop decision. And hitting a flush does not always win, since a higher flush or a paired board can beat it, so outs are sometimes worth less than they are counted. The transferable idea is that a probability is only actionable when a price sits next to it.
Technique: Rule of four and two