Lets say we don't hold a club and there are 4 on board by the river. What are the odds of opponent having a flush?
Seems like odds= (9/45) + (9/44) = 40.45%

can somebody confirm this math? any more accurate way of devising the percentage?

You are double counting those times when both of his cards are a club. The best approach is to calculate the probability that each card is not a club, and subtract from 1.

So you have (36/45 * 35/44) = .636, so 1-.636 = .364

yeah that looks better. thanks!

In a real game the chance is higher than that calculation. DUCY?

not really. it seems accurate in calculating the odds of opponent having at least one club.

Why do you think so? If something, it seems lower to me - people tend to play suited hands a lot more!

It's accurate if your opponent is playing all their hands the same way, or randomly, or not looking at their cards.

If people play suited hands more, then on a flushy board, they're more likely to have a flush (and "more likely" means a higher chance).

That's one reason. Another is that prior to having 4 clubs on the board, there was a street with just 3 clubs on the board, and often a bet/fold opportunity at that point. Players with a club are more likely to continue in the hand.

For both of those reasons, the chance a live opponent on the river has the flush is certainly higher than the calculated random chance of 36%. Probably significantly higher. Players don't play randomly, and don't stay to the river randomly.

Are you sure? How likely are you to have at least one club when your hand is suited? How likely are you to have at least one club when your hand is off suit?

How would that change if OP made no assumption of how the betting went down?

No assumptions are needed. It is a fact that at the river there is a higher likelihood that players holding a club made it that far in this situation.

To heewaw's point, he is referring to players who already had the flush before the river. Certainly they didn't fold on the turn, so they are more likely to be in at the river than players not already holding a flush.

And to clarify based on the above points, there is a difference in the chance that someone dealt in to the hand was dealt a club, versus the chance that someone who made it to the river on a board full of clubs, was dealt a club.

How can you say no assumptions are needed? To say that it is a fact that they are more likely to have made it to the river with a club assumes that there has been betting, and presumably on both streets.

Here is a scenario. There is a raise preflop and one caller.The flop comes out KJ9 all clubs. It goes check/check. The turn is 6h, and it goes check/check. The river is 4c.

Is your opponent more likely to have a club because he made it to the river?

I think it is pretty clear that out of all poker deals that reached the river with four clubs on board in the history of poker, the chance that an opponent still in the hand has at least one club is higher than the random chance that an opponent was dealt at least one club.

This really struck me as very counter-intuitive, so I have filtered my database for all the hands I have been in position (meaning SB in heads up) and it has been checked to me on the flop and the turn.

1) There are 1240 hands in the database where the opponent checks - from those, there are 990 hands in the database when the river is checked down - 175 of opponents' hands were flushes. That's ~17.8 %.

2) There are 480 hands in the database where the opponent bets - from those, there are 143 hands that I call (seems like a sick leak ) - 68 of opponents' hands were flushes. That's ~47.6 %.

(Trying to avoid Simpson's paradox here.)

Without removal effects, if I count correctly, opponent's hand on the river (when the flop and turn is checked through) is flush in ~26.1 %.

You absolutely need assumptions.

This might and also not might be true - this heavily depends on how "suited" is the opponent's range.

This is silly, we're talking about probability which applies to averages over time. Of course there are exceptions, lots of them. Sometimes your 80% favorite gets sucked out on too.

It isn't silly at all. You were stating some facts, saying that they made no assumptions at all about the betting. I'm saying that you were making assumptions that there had been betting. When you are in a hand you know exactly what the betting has been, and you use this information to help you decide.

You are making an assumption that a monochrome flop will almost certainly be bet on the flop, the turn, or both, and therefore someone who calls down is more likely than average to have the suit. The second part is likely true. But very often it won't be bet, which actually makes it less likely your opponent has one of the suit.

Fair enough. But I think OP was asking a general question, not a specific scenario. And the general answer is, lacking other facts, there is a higher likelihood that the remaining opponent has a club, than random chance.

I think the OP was asking what is the probability that the two random cards will contain a flush card. He was answered in the first reply (and confirmed that his question had been anwered). Then you came and without any explanation, you said that the chance is higher. I understand the probabilities based on random hands don't really tell much, but so doesn't basically anything in poker where no assumptions have been made (see my evidence). To quote you, DUCY?

Touche'.

Maybe the OP will chime in on which answer is more useful to him.

You don't know their cards, therefore any range you put them on is an assumption. To assume they hold a random hand is a much bigger assumption than to assume they don't.

Exactly.

It's just that it is absolutely crucial to be aware of assumptions one is making, however reasonable they are.

If I assumed - preflop the opponent played 100 % of hands and then called on monotone flop and the turn came 4th to flush, I think it is safe to assume that the chance is much higher than that 36.4 %

If I assumed that for some reason (very basic example is 3bet pot in heads up where most hands of most 3bet ranges are suited) opponent has many suited hands and the flop went check-check, I think it is very safe to assume that the chance is much much lower.

But to somehow assume all situations at once doesn't really make sense. That's all I am saying.

Wouldn't the value of the clubs on the board also be needed? Faces on the board would decrease the flush odds, while low clubs would increase the odds that a club is held by a remaining player.

My initial question was just to know from a pure probability perspective what the odds are of opponent having a flush. first answer was most useful but of course points made regarding a player calling a bet on a 3 flush board being more likely to hold a card of that suit make some sense. ( but what if it goes runner runner club? and no betting on turn with bet on flop) Players also can be bluffing and floating with the intention to bluff on a 4th club so i think its good to know the theoretical odds of vilain holding club when deciding on a river call.


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This.

I played a hand yesterday that came to mind while skimming this thread:

no limit holdem tourney; four spots pay; 8 players left.

I raise KQ 3x, lag calls, lag calls, rest fold including blinds.

QJT

I check, they check.

9

I check, MP checks, LP bets 2/3 pot, I call, MP folds.

7

I check, LP covers and shoves 1.2 pots effective, I tank.

My thoughts:

a) a naked king shouldn't shove there.
b) most of his preflop calling hands that have a flush are those with little cards that shouldn't shove the river with the exception of straight flushes like 86.
c) he loves 3 betting vs me preflop; AQx is never in his range here and AsJx should check the turn.
d) AsKx is not in his range because he would 3 bet preflop.
e) He rarely has Asx, if ever. Perhaps never.

I call. He shows 86. I scoop.

edit: I messed up the board but I do remember that there was four to a straight flush out there, which he never holds unless it was 88.