Something that seems pretty fundamental has always illuded by understanding.

When calculating outs it is never ok to make assumptions about unknown holdings, I know this from memory, but I don't fully understand why. Meaning, you cannot say I have 8 outs to the straight, but let's assume 2 of those outs have already been dispersed amongst the players.

For example: You hold A 2

Flop is:

J 8 3

You know you have 9 outs to the flush and 3 outs to the ace, or 12 outs.

What is baffling to me is that the math works out so that you get your flush at exactly the same percentage as the theoretical yield (~35%). As in, in theory and application you hit your flush exactly 35.4% every time you draw to one.

9/47 * 38/46
+ 38/47 * 9/46
+ 9/47 * 8/46
= 15.82 + 15.82 + 3.75
= 35.4%

Now, something else we know is that if you drew to 100,000 flushes in your lifetime you could easily say that some percentage of that time you did not have your full 9 outs.

This means from the omniscent view, sometimes our chance to make our flush was less than 35.4% because the ominscent view knows that some of our suit is already dealt out. So why is it that we make a flush exactly 35.4% of the time in theory and in practice?

Why can't we use an expectation based on the number of players, to figure out how many cards of our suit are likely in our opponents' holdings or in the muck?

Here is why:

For every time that your opponent has one of your outs and you are less likely to hit your flush, it is even more likely that he does NOT have one of your outs.

If he does NOT have any of your outs, you are more likely than 35.4% to hit your flush.

Each of these scenarios balances out your odds in the long run.

LOL, it all makes sense now. I knew there was something, but my head hurt.

If your opponents, at a 10 person table don't have your hearts, then of the 29 unseen cards, we have 9 that win for us, aka 9/29 > 9/47.

The real question is:
Why do you think we make a flush exactly 35.4% of the time in theory and in practice?

Are you 354 for 1000?

I was surprised to see that mine hit 357/1000, but then i realize that it's just because some of the hands were at pokerstars where flush draws hit 371/1000.

WHOA!

...sorry but you just totally blew my mind

The key here is that we're talking about the unseen cards only. We don't know what the other players have folded. So, like Head_Trauma said, in the omniscient view...if no one has folded a heart you're much more likely than 35% to make your flush. If several hearts have been folded you're in worse shape than 35%, but in the long run it averages out.

The long run is irrelevant. The fact is that it averages out.

unless you have reason to believe some of our outs are gone, in which case you can discount for them. Otherwise, if you sum the conditional probabilities of x outs being available, you'll get back to the exact same figure as if you assumed all outs were available and all unseen cards were unknown.

remember you aren't assuming 9 outs left in the deck. you are assuming 9 outs somewhere in the deck or other hands.


Head_Trauma should post more

God, I had to read Head Trauma's post 10 times before I got it.

I get it now!! Thanks all!

I agree with Actuary btw.

You are right. I have been lurking these forums for way too long. I plan on doing a lot of posting in the Limit Hold 'Em thread as that is my game of choice and I have a lot of hands that deserve analysis.

But I also like talking these theoretical/mathematical type issues as well

we still have a limit forum?

hope you revive it