Haven't posted in this great subforum for a while so I found a question. Playing NL 9 player table tourney, preflop utg opens, cutoff calls, button calls, and Hero calls with 4h4d in the BB. Flop is monotone spades with the 4s giving Hero a set. What is the the probability that one of the three foes flopped a flush? You can assume a VPIP of about 20% using the Sklansky-Chubokov (Sklansky-Karlson) rankings for each the other players. (This is different from asking what are the chances that someone flops a flush given a suited starting handed (.08) btw)

The way I run it's 50/50

I coded up a simulation of this situation. Since you did not specify the other cards on the flop I assumed that they were not unduly significant so I assumed they were the nine and deuce of spades.

Over 10 million trials, I find that at least one villain flops a flush 14.44% of the time.

My simulator program shows it at 14.63% for 10 million trials. The 20% range I used is
A7o+,KJo+,A2s+,K9s+,QTs+,JTs, 22+. The slight difference in results could be mostly due to the opponent hands included in a 20% range.

According to the Sklansky-Chubukov hand rankings I found on the internet, I used the following for the top 20% range:

A4o+, KQo, A2s+, KTs+, 33+.

Thanks. Close enough. 14-15% That's about what I thought. Well I shoved my stack (~3xpot) in with the 44 and one guy called with a flush. The flop cards were rags something like 2s4s8s. I guess should be concerned about set over set too.

The probability here is considerably higher. If you shove 3 times the pot and someone calls you, the probability that they have a flush goes up much closer to 1. Why would you ever shove 3 times the pot in that situation is beyond me. Anyone with a flush is calling, and anyone without a flush (or higher set) is folding.

The only thing that you accomplish is getting drawing hands to fold - but you certainly don't get any value (unless you boat up - but if they have a flush and you boat up you will probably get paid anyway). Contrary to a lot of peoples' opinion, you want hands that are behind to call. You just don't want to give them good odds to call. But even the people with the worst math skills ever can figure out it isn't right to call that large of a bet on a draw.

What action do you suggest ? If Someone has a flush they're going to win most of my stack anyway. It was a rebuy tourney btw

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How about a normal sized bet - maybe consistent with other c-bets you have made. The point isn't that you are going to lose to someone who has you beat - it is that you want to get value from someone who doesn't have you beat.

If someone has a flush and you don't boat up, you are probably going to lose a lot. How much will depend on what comes on the turn and river, or on how they play the hand. A lot of times the person with the nut flush is not going to raise, because they want to keep you in. You may get away cheaply, especially if another spade comes. You may win it all if the board pairs and your opponent can't fold a flush.

It is not uncommon to see people shove on the flop when they fear losing to someone who is drawing. It makes sense - you feel that what is in the pot is yours, and you don't want someone outdrawing you. But that is where the money comes from - people putting in money when they are behind. If you are behind you want to lose as little as possible, and if you are ahead you want to win as much as possible. The overbet shove accomplishes the opposite.

Not sure if clear to OP, but you really have a poker question here, not a pure probability question. Yes, one of the pieces to understanding your best play is estimating how likely you face a flush in total, but that's just one small piece. What's my equity? Do I charge the draws? What's their range? What's my equity against their continuing range? etc

It's possible for you to improve.

This is different than a cash game hand. In this re-buy tourney players were calling and shoving loosely since chipping up was more important than losing your stack. Betting something like 2/3 pot allows multiple callers with a draw to get priced in. The flopped flush player would likely raise. My overbet allowed a drawing hand to make the maximum calling mistake.

If all 3 villains call with draws (unlikely but hypothetical), the drawers have payed 2x Pot combined. Yes, the one with the highest draw benefits, but so do you, moreso than by taking the pot right now (which is what will happen if you open-shove, unless one of them is already beating you).
And then you can save money instead of handing them your stack gift-wrapped before they even act.
But are they bad enough to actually make that mistake? You're talking about an online game, right?

Checking is an option too. But I don't visit the Probability forum to analyze hands. I gave you some things to chew on.

To answer your OP:

10 spades remain out of 47 unknown cards. There are 3 villains. By inclusion-exclusion:

3*45/C(47,2) - 3*210/C(47,4) + 210/C(47,6) =~ 12.14%

Oh, I didn't take into account their ranges. In that case it's a little more, probably what the others said based on their sims.

Thanks for the input everybody. I was mainly interested in probability that someone flopped flush. It seems to happen more often than it should. I didn't really mean to discuss the hand in detail since my memory of the all aspects of it is a bit fuzzy.

Interesting -- the chance someone is suited with that range is exactly the same (4/17) as the chance they're suited with a range of ATC.

It’s not surprising that a probability expert would notice something like that.

Oops no, I forgot to include the KTs+ part of the range. So then it's 16/51 or 31.37% compared to 23.53%

When you mistakenly called me an expert, the chance of my stat being wrong increased to 100%