Quick question reguarding three-flush on board

[bluffist]

Without going into the storage shed and trying to find my old college stats book, I need a little help. In holdem, can someone give me the odds that any three like-suited cards will land within the five cards on the board?

Assume ten players already dealt two cards each. Assume 5 of the suit are already in those 20 hole cards. This leaves 8 more within the remainig deck of 32. The flop, turn and river containing exactly three of a like suit.

I don't really care what's in any of the ten hands, just the odds for a three flush on the board.

There are ten different ways these three cards could fall, and the odds of each vary, it seems. The probability for a spade coming in on the first flop card is 25%, but the odds for the second spade coming third card is aboout 23.33%

It all gets a little confusing how to come up with a single probibility for a 3-flush hitting the board in any card position.

Can anyone show how to do this?

Thanks in advance. /images/graemlins/cool.gif

[BruceZ]

With exactly 5 of the suit in your opponent's hands, the odds of getting exactly 3 of that suit on the board is:

C(8,3)*C(24,2)/C(32,5) = 7.7%

That's only for a particular suit, not all suits.

If you don't know how many of the suit are out, then it's:

C(13,3)*C(39,2)/C(52,5) = 8.2%

for a particular suit. For 3 of any suit, it's 4 times this or 32.8%.

[bluffist]

Thanks, that's great. I wonder if you could clarify something.

Suppose I stood and watched a holdem table for hours, and recorded only the cards on the board for each hand. Suppose I recorded 1000 hands, and each hand had 10 players.

Approximately how much of the time would I expect to see ANY 3-flush on the board? Is it 38.2%?

Thanks again.

[BruceZ]

Approximately how much of the time would I expect to see ANY 3-flush on the board? Is it 38.2%?

Make that 32.8%, yes. That assumes you only record hands that go to the river, and that all 5 card boards are equally likely. In reality, there could be a higher percentage of 3-flushes than this because the rainbow flops and boards that had no chance of a 3-flush may be less likely to go to 5 cards.