[BruceZ]

[ QUOTE ]
Hopefully the good folks at the 2+2 forums can help settle a bet between me and an acquaintance.

Here's the bet. An acquaintance bet me that I don't know how to calculate how flop outs correspond to odds. For instance, most of us know that a flush draw in hold 'em has 9 outs, and is about 1.9: 1 on the flop. His bet was that I couldn't do the math that proves it, and he doesn't believe my answer. We are using 2+2 to officially settle our bet.

Here was my answer (for the uninitiated, I have included how to do this with a 9 out flush draw as it is IMO a little easier to follow it that way):

((Outs/unseen cards on turn)+(Outs/unseen cards on river))
-((Outs/unseen cards on turn)* (Outs/unseen cards on river))
=% hand will be made

100-% hand will be made= x

x/% hand will be made= odds: 1

So did I win the bet? Below is an example of how the math works out for a 9 out flush draw.

((9/47)+(9/46))-((9/47)* (9/46))=35%

100-35= 65
65/35= 1.857
The odds of making a flush are 1.857:1 when you have 4 of the suit on the flop.

[/ QUOTE ]

Your expression gives the right anwer (1.860-to-1) because of 2 errors which cancel. It should be

9/47 + 9/47 - (9/47 * 8/46) = 1.860:1.

The probability of the turn card being a flush card and the probability of the river card being a flush card are each 9/47 before either of these cards are dealt. If the flop card is a flush card, the probabilty is 8/46 that the turn card is also a flush card, so the probabilty of getting a flush card on both is (9/47)*(8/46). This happens to give the same answer as yours. Note these two other equivalent forms:

9/47 + (38/47)*(9/46) = 1.860:1

That is, 9/47 on the turn plus 9/46 on the river of the 38/47 of the time you miss on the turn.

1 - (38/47)*(37/46) = 1.860:1

That is, 1 minus the probabilty of missing on both the flop and the turn.