[Bozeman]

Let me try to sort through the errors made here:

"P(getting it on the turn) + P(getting it on the river assuming I didn't get it on the turn) + P(next two cards being of my suit)

P(turn) = 1 / (9/38 + 9/37 + 9/37 + 8/38) = 1.07083:1 (right? that seems ridiculously good to me)"

First, as has been pointed out, odds are different than probabilities. Probs. can be added, but odds can not.

The number you have used for probs are the odds, they should be 9/47 instead of 9/38, etc..

While you can express it as a sum of probs., you need to make sure that all events are mutually exclusive to do this, so P(flush)=P(turn)+P(river given not turn) (P(turn) includes P(turn and river). P(flush)=9/47+9/46*38/47=35.0%

Or

P(flush)=P(turn not river)+P(river not turn)+P(river and turn)=9/47*38/46+38/47*9/46+9/47*8/46=35.0%

(or of course the easy way: P(flush)=1-P(not turn, not river))

In your equation, you found the prob. of turn and river hitting by adding, instead of multiplying.

(The probability of one of two exclusive events happening is the sum of their individual probabilities, while the probability of two independent events both happening is the product of their individual probabilities.)

Craig