Accounting for hidden hole flush cards.

I've been reading Slansky's Theory of Poker, and I've come across a problem with expressed odds...

Every poker site and book I've come across always counts possible outs as if none of those outs can be in the other player's hole cards; folded or active.

I figure that this is neither zero nor trivial, so I was wondering if anyone has done a probability analysis of the number of outs to subtract by number of players on the table, based on the average probability of someone holding a blocking card to a flush or straight.

Except for face up cards, its irrelevant.

Like...lets say theres 10 red marbles in a jar of 50 marbles. Your chance of drawing a red marble is 10/50.

If you deal out 10 of these marbles though...on average 2 of those dealt marbles will be red. The marbles left in the jar are then on average 8/40 red, which is the same as 10/50.

Now, you can do some outs estimation based on player ranges though. Like if UTG raises, and you call and flop a gutshot that requires an Ace to complete...you might be able to consider that UTG's range is ace heavy...and so you're less likely to draw one. That all depends on his range distribution though and how accurate you are at guessing that distribution.

EDIT: I removed a couple of paragraphs, because JKDS did a much better job of explaining.


If you really want some info (that you will almost definitely mis-use), here's some basic stats on Aces. Remember that these stats hold true for any given card rank... so the %'s are the same for Kings or 2's as they are for aces.

Stats for everyone before you look at your cards:
In a 9-handed FR game, there is a 17.1% chance that no one will be dealt an ace pre-flop. In a 6-handed game, this goes up to 33.4%. If you're playing 4-handed, there is about 50% chance that no one has an ace pre-flop.

Stats for everyone else when you look and DO NOT have any aces:
In a 9-handed FR game, there is a 79.9% chance that someone will have at least one ace. In a 6-handed game, it goes down to 61.3%. In a 4-handed game, it is 41.0% likely that at least one of your opponents holds at least one ace pre-flop when you do not.

Stats for everyone else when you look and DO have exactly one ace:
In a 9-handed FR game, there is a 69.5% chance that someone else will have an ace. In a 6-handed game, it goes down to 49.6%. In a 4-handed game, it is 32.4% likely that you are not the only one with at least one ace.

TLDR: What MojoMoran is talking about isn't the same thing. The odds are the same either way in the type of scenario you're talking about. I've worked it out before with some complicated examples in previous posts to prove this in action.

Yeah, OK. I did it wrong. I misread the OP.

Jkds nailed it. Because the cards dealt to opponents are random your overall probability doesn't change.

In an individual hand the dealt cards may help your odds with a draw hand or they may hinder them, on average though your odds are the same. That's why unknown cards are ignored.

Originally Posted by davisrei

I've been reading Slansky's Theory of Poker, and I've come across a problem with expressed odds...

Every poker site and book I've come across always counts possible outs as if none of those outs can be in the other player's hole cards; folded or active.

Hey, now you mention it, the out counting theory doesn't take into account the cards in the middle of the deck or the bottom of the deck either, and they can never be drawn. Something ain't right.

What MoranMonkey was talking about does create differences between short-handed games and full-ring games, however. If its folds to the small blind in a six-handed game, it's less likely that the big blind has AA than if it folds to the small blind in a nine-handed game.

It's correct that with no information your opponents' hole cards are irrelevant. However consider the following:

You hold


The flop is


For the sake of this scenario, we know our opponent is playing a heart flush draw. We want to count our outs for improving our hand (don't ask why, just do it). We would typically say 5, 2 jacks and 3 tens. However, since we know our opponent is on a heart draw, we only really have 3, because both the Jh and Th give our opponent the flush.

We don't often know for sure that our opponent has a draw. We can get a little deeper. If we give our opponent a 50% chance of being on the draw, then only half of the hearts count in his favor. Therefore in the above example, we have 4 outs.

Originally Posted by Lucothefish

In an individual hand the dealt cards may help your odds with a draw hand or they may hinder them, on average though your odds are the same. That's why unknown cards are ignored.

Okay, that makes sense. I was only thinking in terms of outs lost to opponent's hole cards, but if you turn all of your opponent's cards over, then the unknown card count OTF goes down to 31 on a 9 handed table, which means that you can see some of your flush cards without hurting your drawing chances.

Originally Posted by davisrei

Every poker site and book I've come across always counts possible outs as if none of those outs can be in the other player's hole cards; folded or active.

This wording isn't accurate, possibly because it's not exactly what you meant to say. Of course, some outs can be unavailable because players folded them or opponents are holding them. But inferences you make from the action aside, your outs can be anywhere among the unknown cards - folded, held by opponents and undealt.

I tried to find the other spoon thread. There was a long lengthy discussion about this. It essentially said that we can't discount our outs in other players hole cards effecting our odds without discounting the non out cards. In a 6 handed game if you are drawing to a flush (9outs) there are 10 hole cards you can't see, so just trying to discount your outs held in those 10 cards would be stupid and have nothing to do with true odds.

The only way you can "discount" outs is with card removal but we would be discounting a very small fraction of an out so it's pretty much never useful in hold em games. I think it would be a little more useful of stud games since we see their door cards but don't take my word for it.

Originally Posted by Icanhastreebet

The only way you can "discount" outs is with card removal but we would be discounting a very small fraction of an out so it's pretty much never useful in hold em games. I think it would be a little more useful of stud games since we see their door cards but don't take my word for it.

Yeah in razz this is why A23 can be an underdog on third street to something like 543 depending on the other door cards.