What's a backdoor flush draw worth?

Back to a bit of math to recover from all the excitement of Al's visit.

Just how much is a backdoor flush draw worth? Let's assume that you're reasonably certain that if it comes in you're good (say you have the ace of the flush suit).

Postflop waiting for a backdoor flush, there are 10 cards of the relevant suit out. So the number of turn/river combinations that give you the flush is 10 × 9 = 90.

Let's compare that to a single out. If you have 1 out card, then you can get it on the turn or river in 2 × 46 = 92 ways (choose whether you get it first or second, then choose the other card). So that suggests that a backdoor flush draw is pretty much the same as a one outer. Let's try to confirm this with PokerStove.

For the flush computation, we'll give one hand A♦J♦ and the other A♣J♥ on a flop of 2♦8♥10♠. The first hand is obviously freerolling for the backdoor flush against a chopped pot here, so the difference in equity must equal the chance of a backdoor flush.

And ... drumroll please ... PokerSove says: 52.27% to 47.73%, so a 4.54% edge.

For the one outer, let's look at A♣A♥ against 6♣6♥ on a board of A♦6♦6♠. The aces here have 4.44% equity.

Hang on, our calculations suggested that the backdoor draw should be slightly less likely than a one outer. What's up?

Ahh, sneaky, in the one outer computation, we didn't assume that our opponent could not hold the out and nor in the backdoor computation did we assume that our opponent had no cards in the key suit. Making these assumptions still leaves us 90 turn/river combinations for the backdoor flush, but reduces the one outer to 88 combinations.

But the main moral is:

If you intend to see both the turn and the river (e.g. calling an all in bet, or calling a bet all in), count one extra out for a backdoor flush draw that you believe will be good.