Flush draw probabilities.

Simple question, just cant reach a reasonable conclusion though.

How come you're 19.1% to turn an out with 4 to a flush at flop, and you're 19.6% to river an out if you still hasnt turned one, but if one is allin with a flush draw at the flop, one only has a 35% to hit?

How come it isnt 19.6%+19.1%=38.7% to hit? Please enlighten me.

[BruceZ]

[ QUOTE ]
Simple question, just cant reach a reasonable conclusion though.

How come you're 19.1% to turn an out with 4 to a flush at flop, and you're 19.6% to river an out if you still hasnt turned one, but if one is allin with a flush draw at the flop, one only has a 35% to hit?

How come it isnt 19.6%+19.1%=38.7% to hit? Please enlighten me.

[/ QUOTE ]

19.6% + (1 - 19.6%)*19.1% =~ 35%.

You have to miss on the turn before you hit on the river.

The problem comes from the fact that you're counting twice. The odds for the turn are correct, but the river odds only come into play when you don't get a flush card on the turn. When you get a flush card on the turn AND river, you're double counting. To write it out:

- The probability of hitting on the turn is 19.1%, and the river card is irrelevant. So, 19.1%
- The probability of hitting on the river is 19.6%, but this only counts if the flush hasn't already hit. We've already counted the case for when the flush hits on the turn AND the river. So, the probability of getting a flush ONLY on the river is:

(Probability of a non-flush turn) x (Probability of a flush card river) =
(80.9%) x (19.6%) = 15.85%

The total is therefore: 19.1% + 15.85% ~= 35%

To make this a bit clearer, consider the case when you're 50% to hit on the turn and 60% to hit on the river. Your odds aren't 110%.