flush odds

[irishiain]

Alright, basic odds/percentage question i know but maybe you guys can help me out.
Im four to the flush after the flop and i know its like 19% to make it on the turn and then 19.6% to make on the river if the turn doesnt help you.
So my question is how is the percentage chance of making my draw on either the turn or the river combined?
I know its meant to be like 35% but am having trouble calculating it......
any help would be appreciated
thanks

[chillin411]

There are basically two ways of calculating the probability the event of completing a flopped flush draw.

If we know the probability of two separate events, then we can calculate the probability of either of those events occuring with the formula

U representing union, commonly known as "or"
∩ representing intersection, commonly known as "and"

P(A U B) = P(A) + P(B) - P(A ∩ B)

We subtract the probability of A and B occuring because otherwise, we're double counting the times when both events occur.

Using 9/47 for P(A) and 9/46 for P(B) (the probabilities of hitting the flush on the turn or river, respectively, in decimal form 19.57% and 19.15%)

P(A U B) = (9/47) + (9/46) - (9/47)*(9/46) = 34.97%

Alternatively, we can calculate the probability of not catching the flush, and subtracting from 1, or 100%.

P(A U B) = 1 - P((A U B)')

We know that 38 cards don't make the flush for us on the turn, and 37 cards don't make the flush for us on the river, so the probability of not making the flush is equal to:

(38/47) * (37/46) = 65.03%
P(A U B) = 100% - 65.03% = 34.97%

Most people find the second way the easier way of calculating it, but either one will give you the same answer.

[orion7781]

I don't understand why you subratct the intersection. Technically it's right if you are strictly calculating whether you'll make your flush on the turn OR river. But for our purposes we should also count the time we make it on the turn AND river, right?

[cybrarian]

If you are calculating it by looking first at the odds of not making your flush at all, then any other event includes you hitting your flush at some point, on the turn, on the river, or even that both cards are of the flush suit. The two events are exclusive and exhaustive - you will either see zero cards of the flush suit, or one or more cards of that suit.

[orion7781]

Hm, that makes sense. thanks.

[chillin411]

Basically, the reason why you're subtracting the intersection is because you're double counting it. I find the easiest way to explain this is using a picture.

A U B (what we're trying to solve for) is represented by the red, orange, and yellow venn diagram in the first part.

If we just add A and B together, we end up getting an extra orange piece, or in messy probability terms, an extra (A ∩ B).

To solve this, we just subtract the extra intersection we get, and that gives us the correct answer.

[nsidestrate]

Venn diagrams make me feel funny.

[orion7781]

[KOJAK2U]

Thanks for the help...I get it now