Which is correct regarding a flush draw?

[Dave H.]

Two different sites and I quote:

[ QUOTE ]
N.B. If you take two sutied cards to the river, you have a 15/1 (6.4%) chance of making a flush in your suit by then.

[/ QUOTE ]

[ QUOTE ]
Finally for all the possibilities if you start suited and stay to see all seven cards (your two and the five board cards) the probability that you will make a flush is 5.77%. The odds against you are 16.3:1

[/ QUOTE ]

Which of the above is correct, please?

Also, I thought I read (and I searched for this a long time) that there was something like a 60% probability that the board could be exactly 2 suited. Can that be??

Thank you!

[Lost Wages]

Question 1)
If you start with 2 suited cards then you will make a flush (or straight flush):

C(11,3)*C(47,2)/C(50,5) = 8.42% = 10.9:1 against

However, that includes the times that the board contains 4 or 5 of your suit which you generally would not prefer.

The probability of making exactly a 5 card flush (or straight flush) is:

C(11,3)*C(39,2)/C(50,5) = 5.77% = 16.3:1 against

Question 2)
Not sure what you mean by "exactly 2 suited". Can you give an example?

Lost Wages

[Stork]

I'm pretty sure the second one is correct. I just remember hearing that you make the flush slightly less than 6% of the time.

[gaming_mouse]

5.77% is the chance of having a board with EXACTLY 3 of your suit.

6.4% is the chance of EXACTLY 3 OR EXACTLY 4.

Lost wages gives the correct calculation for having a flush made any way at all (including a str8 flush).

gm

EDIT: Actually, LostWages calculation is not quite right for making a flush any way at all. It counts certain hands multiple times. The correct answer is 6.4%, because having a board of 5 suited cards does not even affect the calculation by as much as 1 digit.

[gaming_mouse]

Dave,

In case you want to see the math, here it is:

ncr(11,3)*ncr(39,2)/ncr(50,5)=.0577 (EXACTLY 3)
ncr(11,4)*ncr(39,1)/ncr(50,5)=.0060 (EXACTLY 4)
ncr(11,5)/ncr(50,5)=.0002 (EXACTLY 5)

Add them all together for the final answer, which is about 6.4%

gm

[Lost Wages]

Good catch, I fudged it.

Lost Wages

[Dave H.]

Thanx for your help!

By 2 suited, I meant two and ONLY two suits on the board, i.e. 2 diamonds and 3 spades, 4 hearts and 1 club, etc.

[Dave H.]

Yes, I absolutely wanted to see the math. Thank you very much!

[gaming_mouse]

[ QUOTE ]
By 2 suited, I meant two and ONLY two suits on the board, i.e. 2 diamonds and 3 spades, 4 hearts and 1 club, etc.

[/ QUOTE ]

ncr(4,2)* (ncr(13,1)*ncr(13,4)*2 + ncr(13,2)*ncr(13,3)*2)/ncr(52,5)

This works out to 14.5%. So yes, the 60% is way off.

gm

[Lost Wages]

GM has it right. Link to the solution for every possible suit distribution.

Lost Wages