[JMAnon]

Hey Benwood,
I don't think there is anything wrong with calculating odds for a back door draw by treating it as a draw to a draw. It is quite a complicated way to do things (as opposed to throwing the hand into a pot equity calculator). If that is the approach you want to take, here is an example of how I would do it:

If 20 of 40 (to use even numbers) cards will give you a draw on the turn, you have a 50% chance of picking up the draw. To see if it is worth taking a card off, I next need to know how often the draw will come in.

Now say 16 of the remaining 39 cards will complete the draw on the river. Thus, I will have about a 40% chance of making the draw if I pick it up on the turn.

Thus, when all is said and done, I will make my hand 20% of the time (50% * 40%), and 30% of the time (50% * 60%)I will see the river and miss. Now to figure if I should take a card off in the posited scenario, I need to make some assumptions.

To simplify, I will assume the pot is heads up, and that there are 4 big bets in the pot before the flop action. If my opponent bets at me, and I assume he will bet on the turn, it will cost me 1.5 big bets to see the river. If I miss on the turn, it will cost me .5 big bets. If I make it to the river, the pot will contain 7 big bets. Lets assume my opponent always pays me off when I hit by check-calling, for a total final pot of 9 big bets.

So 50% of the time, I lose .5 big bets by taking a card off and missing the turn

30% of the time, I lose 1.5 big bets by taking a card and missing the river

20 % of the time, I win 6.5 big bets (9 - the 2.5 big bets I put in).

In 100 trials, I expect:

to lose .5 big bets 50 times for a total of -25 big bets;

to lose 1.5 big bets 30 times for a total of -45 big bets;

and

to win 6.5 big bets 20 times for a total of 130 big bets.

You would expect to win 60 big bets over 100 hands by taking a card off in the given example. Thus, taking a card off in that situation would be + EV versus folding if you were drawing for the whole pot. Unfortunately, not many draws to the whole pot are as good as the one posited here.

Now imagine you are drawing for half the pot. Everything stays the same, except that when you win, you win a mere 2 big bets when your draw comes in (1/2 of 9 - the 2.5 bets you put in). You now expect to lose 30 big bets over 100 hands.

I recognize that this is a highly stylized example, but it demonstrates how difficult your approach is. If you need to factor in scooping draws and 1/2 pot draws together (like in the poster's hand), it becomes quite complicated quite quickly.

The example also illustrates how bad it is to draw to a backdoor low, because in the example, the pot was big on the flop and it still would have been a big mistake to take a card off, without even factoring in the chance of getting 1/4ed.