Counting Flush Draw Outs

Hey, I haven't really posted here at all, but I have been reading for a long time. I didn't really know where to post this, so I figured I would just throw it in the Beginners section

Anyway, I have always wondered about counting the outs of a flush draw (although it could apply to other draws I suppose)

This pretty much just makes things more complicated, but I've been thinking something...

So assume you have 2 diamonds, and the flop comes with 2 more diamonds. Generally you would just say you have the 9 other diamonds to catch the flush.

The thing I have been wondering though, is this:
There are 47 other cards that you have not seen, so usually you would just say the odds of catching the diamond by the river are (9/47)+(9/46) - ((9/47) * (9/46)) or something like that, my math isn't all that great.

But, if there are 9 other players, who each have 2 cards, for a total of 18 cards already gone, could you not assume that there are 18 * (9/47) = 3.45 diamonds already in other peoples hands?

Therefore you only have 9 - 3.45 = 5.55 outs?

I've heard before that you don't look anything into those cards, because you don't know what they are, but you also don't know the composition of the cards you are basing your calculations on in the first place. You are assuming that there are 9 diamonds in the cards remaining to come out, when really there is most likely less than that.

Maybe I have this all wrong, just thought I would post it. It got me out of doing homework for a while.

EDIT: I don't think my last calculation was even near correct. It wouldn't be 3.45, but it would be some sort of number anyway. Not sure how to calculate it exactly, maybe somebody with more probability math experience can do it.

You're right and wrong. Statistically speaking, there should be ~3.5 diamonds among the hands of the rest of the players. You can't assume that when making your calculations though.

Ya, thats what I figured. But I never quite understood why you can't assume that?

I mean, if it is supposed to be ~3.5 most of the time, would things not eventually work out that way?

I don't know, maybe it is one of those things you run into in math where you just have to go, it is that way because it is. I get those types of things sometimes.

You can't assume the holdings of anyone at all unless they've given you reason to believe (folding, betting, calling) what they held.

However, if you want to REALLY get into calculations, you can assume that of those 3.5 diamonds among other players, how likely are they to have a playable hand and how likely are they to fold? Then how many diamonds on average will be played and how many will be folded? etc. etc.

No point though, that type of analysis only proves a marginal edge. Unless you KNOW the cards they folded, just assume that it's only you and your opp. Calculate your outs based on cards not shown, and figure out what outs will beat your opponents LIKELY holdings.

On another note, wouldn't this game give a huge edge to the mathematically-inclined if the dead (burned) cards were dealt face up? You'd be able to discount outs for straights/flushes/full houses/three of a kinds. And you'd get a better idea of the odds of your opp's holding.

It works out the same.

9 outs with 47 cards is 19.15%

If you're just considering the undealt cards, there are 29 of those left. Your 5.55 outs with 29 cards is ............. (drumroll) .........19.15%

Haha, ya I just thought of that while I was eating. But you beat me to it.

just assume that every card you haven't seen is unknown (obv) and therefore every new card has the same chance to be any of those.

Originally Posted by Xioustic

On another note, wouldn't this game give a huge edge to the mathematically-inclined if the dead (burned) cards were dealt face up? You'd be able to discount outs for straights/flushes/full houses/three of a kinds. And you'd get a better idea of the odds of your opp's holding.

Like stud?

Originally Posted by swiggidy

Originally Posted by Xioustic

On another note, wouldn't this game give a huge edge to the mathematically-inclined if the dead (burned) cards were dealt face up? You'd be able to discount outs for straights/flushes/full houses/three of a kinds. And you'd get a better idea of the odds of your opp's holding.

Like stud?

My games are limited entirely to Hold'em, never played anything other than that except for 7 card draw in my life. So I wouldn't know. :P