Quote Originally Posted by rob6597
This assumes that you don't hold any cards of the suit in question.
True, your math does make that assumption. And the numbers are of some interest when you're thinking about your opponents' draws.

When you're thinking about your own flush draw, you'll want to run the math differently.

You want to know:

(1) What are the odds against completing my 4-flush on the next card; (a) turn, (b) river?

(2) What are the odds against hitting my 4-flush by the river assuming a 4-flush on the flop

(3) Odds of hitting back-door flush draw (i.e. you have 3 to a flush on the flop and want to know the odds against getting a runner-runner flush.) You should never be drawing to just a back-door flush draw and that is why you rarely see the math for this one.

Math
-----
(1a) 47 cards to come (turn) -> odds against are (47-9):9 = 4.22:1
(1b) 46 cards to come (river) -> odds against are (46-9):9 = 4.11:1

(2) Use probabilities and then convert back to odds.

Prob. By River
= [Ind. Prob. on Turn] + [Ind. Prob. on River]
= [1-outs/47]*[outs/46] + [outs/47]

Odds against by river: 1.86:1 (i.e. will come in ~35% of the time).

(see http://www.flopturnriver.com/phpBB2/...pic.php?t=7339 for more details on the math)

(3) Runner-Runner flush odds:

3 to the flush on the flop
-> odds against picking up a 4-flush on turn = (47-10)/10 = 3.7:1
-> odds against hitting 5-th flush card on river = (46-9)/9 = 4.11111:1

prob_turn = 1/4.7

prob_river = 1/5.1111111

Combined prob of both event happening = (1/4.7)*(1/5.11111) = 0.04162

or, approx. 4.16%

or, approx. 23:1 against

Or, more simply: 10/47*9/46 = about 4.2% = about 23:1 against.

This is why one usually gives a back-door flush draw a value of about 1.5 outs.