Odds of flopping 2nd nut flush while someone else flops nut flush

[EricW]

The odds of flopping a flush while holding KQs are about 118:1.

In a 10 handed game, what is the probability of you holding KQs and flopping a flush while anothber player holds Axs and flops a flush at the same time? Would it be approx. (1/118)(1/118)? Or does it get adjusted because you already have a flush and the chances of someone holding 2 of the same suite becomes a bit less?

[RonMexico]

I think the wording of the question is important here. Are you asking "What is the probability that villain holds Axs given that you hold KQs and have already flopped the flush?" Or is it "After I see my KQs in the hole, what is the probability that another player is dealt Axs and then we both flop a flush?" I suppose another variation could be that you already know that both you and villain hold said hands, and then the flop is to come.

In the first case, I get about 1.49%. In the second case, I'm getting .018%. And for the last scenario, I get .49%. Does anyone concur?

[EricW]

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I think the wording of the question is important here. Are you asking "What is the probability that villain holds Axs given that you hold KQs and have already flopped the flush?" Or is it "After I see my KQs in the hole, what is the probability that another player is dealt Axs and then we both flop a flush?" I suppose another variation could be that you already know that both you and villain hold said hands, and then the flop is to come.

In the first case, I get about 1.49%. In the second case, I'm getting .018%. And for the last scenario, I get .49%. Does anyone concur?

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Hmmm I don't really understand the differences in your questions. I don't think that the percentage you came up with in the first wording of the question could be right because the chances of me flopping a flush with 2 suited cards is .84%. There should be no way that the probability of me flopping a flush and someone else flopping a nut flush at the same time should occur more often me flopping a flush alone.

I guess the best I can word it is what is the probability of me flopping a flush with any two suited cards, in this case KQs while someone else flops a flush as well AND holding Axs therefore flopping the nut flush.

[AaronBrown]

There are 4 ways to hold KQs out of 52*51/2 = 1,326 possible pocket holdings, so the chance is 4/1,326 = 0.3% or 1 chance in 332.

Given you have KQs, there are 10 ways for another player to hold Ax of your same suit, out of 50*49/2 = 1,225, so the chance is 10/1,225 = 0.8% or 1 chance in 123. Given that there are 9 other players, the chance is 7.3% or 1 in 14 that one other player has Ax of your suit.

Given that both of those things happen, there are 9 remaining suited cards to flop a flush with. These can be selected in 9*8*7/(3*2*1) = 84 ways out of the 48*47*46/(3*2*1) = 17,296 flops. The chance of you both flopping flushes is 84/17,296 = 0.5% or 1 chance in 206.

If you multiply all of these together, you get the chance of you being dealt KQs, someone else being dealt Ax of the same suit, and the flop giving both of you flushes.

However, none of these numbers are relevant for poker. If you flop a King high flush, with neither the Ace nor the King on the board, there are 8 unseen suited cards, one of which is the Ace. There are 18 cards dealt to the other players at the table out of 47 total unseen cards. The chance that the suited Ace is in someone else's hand is 18/47. Given that, the chance that the player's other card is suited is 7/46. The product of those chances is 5.8% or 1 chance in 17. Given that most people see the flop with suited Ace hands, there's a small but not negligible chance that someone has you beat.

[RonMexico]

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The product of those chances is 5.8% or 1 chance in 17. Given that most people see the flop with suited Ace hands, there's a small but not negligible chance that someone has you beat.

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I agree with this number (made a miscalculation the first time in the first question I posed). And yes, this is the important number. If you take the Bayesian approach when flopping a non-nut flush, you'll find that the probability that villain flopped the nut flush is higher than most would expect. This actually happened to a friend of mine in a live tourney, and he was convinced by intuition that the probability of his opponent holding the nut flush was miniscule.