Ok, this one is going to be huge if it’s correct. If it’s not, I’m just making an ass of myself, but I have yet to find the flaw in this. Please, feel free to search for it. In fact, I WANT you to find a flaw in this reasoning. I want to be able to play normal poker again as I was taught, because if this is true, it’s going to change the way you and I play.
Yesterday night, I was lying in bed and I was thinking about some hand where I got it all in with a straight draw, a flush draw and 2 overcards on the flop. So, I assumed I had 21 outs, because I put my opponent on top pair. He indeed had, but he had a also had a better (nut) flush draw, together with one of my straight outs. So my REAL number of outs was actually 12. This got me thinking: we always assume that, if we count our outs, they’re still in the deck, so they’re live. Maybe here and there, we discount one given jow the hand was played. But that’s pretty rare. If we have a flush draw, we always assume to have 9 outs. For this to be true, all 9 outs have to be still in the deck. Is this the case? Let’s find out.
When you’re playing full ring, the number of opponents at the table is 9. That means, if you’re playing a hand that saw the flop, there are 5 known cards: 2 hole cards + 3 on the flop, and there are 47 unknown cards. If you’re drawing to a flush, the previous way to calculate your outs was 9/47=19.1% to complete on the turn, and 9/46=19.6% on the river. But this is assuming that all 9 flushcards are still in the deck. If we have 9 opponents, that means that 18 hole cards are out of the deck. What’s the chance that none of those 18 hole cards was a flush card? It’s pretty small, that’s for sure. The way to calculate it is: 38 of the 47 unknown cards are non-outs. So the chance of 1 card being a non-out is 38/47. If one non-out is being dealt, the chance of the next one being a non-out is 37/46. The chance of the next one 36/45. And so forth. The chance of none of the 18 unknown cards being an out, is therefore the result if you repeat this process 18 times and multiply the chances of each possibility with each other. Calculations are found below.
# non-outs # unknown cards p
38 47 0.808511
37 46 0.804348
36 45 0.8
35 44 0.795455
34 43 0.790698
33 42 0.785714
32 41 0.780488
31 40 0.775
30 39 0.769231
29 38 0.763158
28 37 0.756757
27 36 0.75
26 35 0.742857
25 34 0.735294
24 33 0.727273
23 32 0.71875
22 31 0.709677
21 30 0.7
If you multiply these chances with each other, you’ll find that there’s only a 0.736% chance of all your outs being live, being that all of your 9 flushdraw cards are still in the deck. So next time you’re assuming that you have 9 outs full ring, better think about this. Shocking? It gets worse.
Now, you might ask, “if my number of outs is not 9, then what is it?” The answer is: we don’t really know. It’s still 9 a good 0.7% of the time, but that’s just the probability for that specific case. If we can calculate this probability though, then we’re also able to calculate the expected value of your number of outs as well, based on the number of opponents (and therefore, dead cards) at your table. Let’s do that for 9 opponents.
The probability of a card being dead is: #dead cards / #unknown cards = 18/47 = 38.298%
So, if you have 9 presumable outs, there are probably 9*(18/47)=3.45 outs dead. That means, on average there will be 5.55 outs still left in the deck. That’s a lot less than 9! This is getting worrisome. We can extrapolate this to any number of outs for any number of players if we want to. Basically, the higher number of opponents, the more dead cards and the less live outs we have. I can’t believe no one ever noticed this before!
All this time PokerStrategy, TV-shows, Twoplustwo, etcetera, have been spewing fog in terms of how many outs we have with a flush draw. Why? Do they want to us to keep drawing with worse odds to make more money? Are they conspiring with the online pokersites to produce more rake?
Luckily, no. The answer is simple. If you have 9 outs out of 47 unknown cards, that’s’ basically the same as 5.55 outs out of 29 live cards. So our equity doesn't change one tiny bit. Phew, scared you for a minute now, didn’t I 
But next time you're saying: “I have 9 outs” when you're drawing to that flush, realize that you’re actually making an assumption that most likely isn’t true. A better way to say it is. “I can expect to have 5.55 outs”.
Yes, I was bored last night and didn’t feel like sleeping
