Well, it came up on another forum, so let's see if we can figure it out.

If there are 3 flush cards on the board and you have the flush, what are the odds that somebody else also has the flush?

Assume nobody folds two suited cards.

Are the odds different if the 3rd board flush card comes on the flop vs turn vs river?

Whatever the odds are, they're not long enough. It happened to me against last night. I had [8d]. Flop came with two of the higher diamonds than mine, turn came with another higher diamond. So there are only two higher diamonds out there. Other guy goes all-in on the river, about 150% pot bet. I could feel that he had me beat but I called anyway like a doofus. (Note to self: trust your reads, not the odds!) Did he have one of the higher flush cards in his hand? Nope, he had both of them. They call that a royal flush. Sheesh.

Anyway. I'm going to leave this as a question for now while I recover from losing my stack on this one. I'll take a shot at answering it pretty soon. Anybody who wants to go first, please do so!

If he had the royal, and then overbet the pot on the river, he's lucky he got a call. The only person to call him would have to have a flush and since all the high flush cards are out (him having 2 of them) its unlikely to get a call. Of course this happens once in a while but the odds of it still have to be far off, especially since many people don't play ALL suited cards. I'm not that good with the statistics, but it does have to be rare.

OK.

Obviously the situation can only happen if the board has 3 of a suit. Let's say it's 3 diamonds on the board. What are the odds that both me and my opponent have 2 diamonds?

Let's start with the situation as heads-up. There are 3 diamonds on the board, leaving 10 diamonds in the deck, with 49 cards to choose from. The odds that I get two diamonds in my hand is therefore (10/49) * (9/48).

For my opponent, there are now only 8 diamonds left and 47 to choose from. His odds of getting two diamonds given that I have two diamonds is now (8/47) * (7/46).

Multiply all those together and you get (10*9*8*7)/(49*48*47*46) = 5,040 in 5,085,024 or 0.099% -- say 0.10%

So heads-up, given a single-suited board, you will see both of us flopping the flush only 1 time in 1,000.

If the 3rd flush card comes on the turn or the river, all that does is slightly lower the # of cards to choose from. On the turn it's the same numerator (10*9*8*7) but the denominator is (48*47*46*45) -- or 0.11%. On the river: (10*9*8*7) / (47*46*45*44) -- 0.12%

To review. When heads up and a three-flush board, the odds that you will see
Flop flush over flush: 0.10%
Turn flush over flush: 0.11%
River flush over flush: 0.12%

These are reassuringly low numbers. However, this really isn't the question. The question is, GIVEN that I have a flush, what are the odds that my opponent also has the flush.

This one is easier. Put 3 diamonds on the board & 2 in my hand. Leaves 8 diamonds for my opponent and 47/46/45 cards (flop/turn/river) left in the deck.

On the flop: 8/47 * 7/46 --> 2.59%
On the turn: 8/46 * 7/45 --> 2.71%
On the river: 8/45 * 7/44 --> 2.83%

Those are pretty high numbers!! And remember, this is heads-up. If several people saw the flop, the odds will be much higher!

How much higher? I'm not positive how to do that calculation. But my guess is that you will be pretty close to the answer if you simply multiply those %s by how many opponents saw the flop with you. So if you have 3 opponents and the 3rd flush card comes on the river, and you have the flush, the odds that one of the 3 also has a flush are about 2.83 * 3 = 8.49%. (Wow!!)

Am I doing this right? Anybody have comments based on their poker experience?

One thing to remember when making the calculation is that you are actually looking for the odds of someone flopping (or making) a higher flush than you. So the odds will start to become significantly smaller as your top suited card goes up.

Arx