Odds Of Someone Having A Flush

I recently played at a rather loose table where it seemed to be acceptable practice to chase a flush no matter what amount of bets were made. Let's say you are at a 9 player table. There are 2 suited cards on the flop. By the river a third card of that suit has come down. Assuming no one at the table with two cards of that suit in their hand folded, what are the chances someone else has a flush when you don't? There are two different odds here. One for if you hold no card of that suit, and one for if you hold a single card of that suit.

If possible, what about the same question for all different number of players at the table?

Moderator-note: Migrated from answer below by same author:

Yes, the heads up case is pretty trivial, which is why I was interested in the 9 player case. I'm not sure why you would think the multiple player case would be not far off 1/22 though. If anything I would think it wouldn't be far off 1/22 x 8 = 36% because you now have 8 players with a chance to make a flush instead of just 1. Though it's probably going to be slightly lower than that because, as you say, the players are "competing" for the flush cards, which decreases the probability that any one of them will get 2.

For the purposes of this exercise you can just assume any two suited cards are going to see a flop, and hang on to the river if there is a flush draw, no matter what the bet, even all in against 7-2. And you can't bluff them off it. I know this is unrealistic even at the loosest of tables, but it gives you the worst case scenario for the odds you are facing with a potential flush on the board.

1 Answer

I will just cover a piece. If you don't have a flush card and there is a 3 flush on the board. There are 45 unknown cards (to you). They need 2 of 10 to make the flush. combin(10; 2) / combin(45; 2) = 4.54545455% = 1 / 22.

With multiple hands it gets complex as they are not independent events as they are competing for the same cards. And some hands can have just 1. Even 10 handed it will not move far off of 1 / 22. Maybe 1 / 10. Is 27 suited even going to see a flop?

I personally would just run a brute force analysis like this answer of mine

If you hold one flush card then they are combin(9; 2) and does not move the numbers much. If you hold the ace then you can bluff bigger flush and the number of suited hands that would see a flop goes down.

If you are on a table that chases draws then bet the flop hard when you have a made hand and there are draws.