Flush Odds?

[thegreatbuddha]

Out of curiostiy and frustration, could someone tell me the odds of a possible flush on the board in Hold'Em/Omaha? By which i mean three cards of the same suit on the board after the river, without regard to pocket cards.

I'm asking because I've been playing at UB lately, and from what i have seen, roughly 2 out of 3 hands qualify for a flush. That seems way to high to me.

[rob6597]

Well, IF I did the math right, here's what I came up with:

For 3 and only 3 cards of 1 suit on a 5 card board:
(13 choose 3) * (39 choose 2) / (52 choose 5) = 8.15%

For 4 and only 4 cards of 1 suit on a 5 card board:
(13 choose 4) * (39 choose 1) / (52 choose 5) = 1.07%

For all 5 cards of 1 suit on a 5 card board:
(13 choose 5) * (39 choose 0) / (52 choose 5) = 0.05%

For either 3, 4, or 5 cards of 1 suit: 8.15% + 1.07% + 0.05% = 9.28%

This assumes that you don't hold any cards of the suit in question.

I think this is for ANY PARTICULAR suit, not for ALL suits. So, its 9.3% for hearts, 9.3% for spades, etc. for a total of 37.1%. Can someone confirm this?

[TLR]

Intersting question
It loks you are correct in the 37% figure

[RiverMonkey]

This assumes that you don't hold any cards of the suit in question.

True, your math does make that assumption. And the numbers are of some interest when you're thinking about your opponents' draws.

When you're thinking about your own flush draw, you'll want to run the math differently.

You want to know:

(1) What are the odds against completing my 4-flush on the next card; (a) turn, (b) river?

(2) What are the odds against hitting my 4-flush by the river assuming a 4-flush on the flop

(3) Odds of hitting back-door flush draw (i.e. you have 3 to a flush on the flop and want to know the odds against getting a runner-runner flush.) You should never be drawing to just a back-door flush draw and that is why you rarely see the math for this one.

Math
-----
(1a) 47 cards to come (turn) -> odds against are (47-9):9 = 4.22:1
(1b) 46 cards to come (river) -> odds against are (46-9):9 = 4.11:1

(2) Use probabilities and then convert back to odds.

Prob. By River
= [Ind. Prob. on Turn] + [Ind. Prob. on River]
= [1-outs/47]*[outs/46] + [outs/47]

Odds against by river: 1.86:1 (i.e. will come in ~35% of the time).

(see http://www.flopturnriver.com/phpBB2/...pic.php?t=7339 for more details on the math)

(3) Runner-Runner flush odds:

3 to the flush on the flop
-> odds against picking up a 4-flush on turn = (47-10)/10 = 3.7:1
-> odds against hitting 5-th flush card on river = (46-9)/9 = 4.11111:1

prob_turn = 1/4.7

prob_river = 1/5.1111111

Combined prob of both event happening = (1/4.7)*(1/5.11111) = 0.04162

or, approx. 4.16%

or, approx. 23:1 against

Or, more simply: 10/47*9/46 = about 4.2% = about 23:1 against.

This is why one usually gives a back-door flush draw a value of about 1.5 outs.

[thegreatbuddha]

So roughly 1 out of every 3 boards will have a potential flush.

Like I said, I was playing at UB, and 1/3 seemed a bit high...

Thanks for clearing that up everyone.

[ChezJ]

i remember reading somewhere that 40% of all boards have 3-flushes (or better). so 37% sounds right.

now what are the odds that somebody was dealt 2 cards of the particular suit on the board? pretty low. and what are the odds that they played them to the flop? even lower (except at microstakes!).

so if you have two pair or even top pair on a 3 flush board, go ahead and bet it. assume your hand is good until proved otherwise (by a raise). you can make people fold by betting, but you can't by check-calling.

ChezJ

[thegreatbuddha]

Well, I'm still waiting on my Neteller confirmation so I can play the RM tables, so I'm stuck in play chip land, or as I like to call it, "All-In All The Time."