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Another stack exchange answer odds I'll hit my flush says that if you start with a flush draw, the odds of hitting on either the turn OR the river are about 35%:
Hitting on turn: 9/47 = 19.15%
Hitting on river (if you didn't hit on turn): 9/46 = 19.57%
Hitting on either turn or river: 19.15% + (19.57% * (1-0.1915)) = 35%
But what about if you started with just 2 hearts in your hand and one heart on the flop? In this case you have to get a heart on BOTH the turn AND the river.
What you've done for your hitting your flush is basically the same. What is important here is we must hit a flush card on both the river and turn.
Given we must hit a flush card on the turn, we know there are 10 cards left in the deck that will give us that event, 10/47.
Next we must hit our flush card on the river, you already did the numbers on this, 9/46.
So we need to multiple these two events together to get the probability of a runner runner event.
(10/47) X (9/46) = 90/2162 which is 0.04163~. Or about 4.163%. If you prefer it in odds it works out close to 23 to 1. You can get the odds by dividing 2162 by the the 90, which will give you about 24.0222. In this case the .0222 isn't massively important, so we can turn the 24 into 23 to 1 as out of the 24 times 90 goes in 1 of those times is good for us, hence the 23 to 1.
