Flush Draw Theory

[Fish Bot]

Every book I have ever read says to count 9 outs for a flush draw because you are never sure how many if any are out.
But sitting at a full table I know over 30% of the cards have been dealt out. Chances are that if you are on a flush draw some of your outs have already been discarded. I took a deck of cards and dealt out 9 hands 20 times, 19 out of the 20 times had at least 1 of every suit dealt out.

If I am on a spade flush draw-when counting my outs, should I really be counting 9. Or would it make since to take the average # of spades left wich would be closer to 7 spades. My thinking is if 30% of the cards were dealt out that means 4 spades were dealt out on average, I recieved 2 of them. That means that on average 2 other spades were mucked or in other hands and my acutal outs would be closer to 7 and not 9.
Obviously if my thinking is correct, this changes the odds of completing a flush draw dramatically.

Please, anyone who has more thoughts or insight into this reply and let us discuss furthur.

[arahant]

Yeah, you can 'count' fewer than 9 outs, but then you have to divide by fewer than 47 remaining cards...you'll get the same fraction for your odds calculation.

[bsheck]

It's 9 outs out of 47 on the flop, so about 19% of the unseen cards are the suit you need. If you're playing with 8 opponents, then 16 more cards have been dealt. That leaves 31 left in the deck. If you estimate that 3 spades have already been dealt, then that leaves 6 left out of 31 cards in the deck. That's about the same 19%. So your odds of making the flush are the same.

[AKQJ10]

Discussed here, more or less....

But your way of looking at it is far better than saying, "I can't calculate the pot odds unless I know EXACTLY how many flush cards are out." You're taking into account the EV for the number of flush cards in folded or live hands. It's not wrong, just adding needless complexity that doesn't affect your calculation at all.

You have two spades, two spades are on the board, one card is to come. Just for kicks, let's try accounting for the spades in the nine unseen hands (whether mucked or live). There are 18 cards in those hands, which is 18/46 of the remaining deck. So the EV for number of spades in those hands is (18/46) * 9, or 3.52.

So we expect that 9 - 3.52 => 5.48 spades are left in the remaining 28 cards (stub plus burn cards). Hence there are 22.52 non spades left on average. Our odds against catching a spade on the river are 22.52 : 5.48 => 4.11 : 1. This happens to be precisely the same as 37 : 9.

Do you see why this is? The proportion of spades in the pool of unseen cards isn't changing just because those cards were or weren't dealt out. None of those cards in someone else's hand is more or less likely to be a spade. (NOTE: That's slightly untrue. One or more of your live opponents might also have a spade draw. But this possibility is negligible.)

You could do the same math I just did for the three burn cards, too, taking them out as candidates for the river card. That wouldn't affect your pot odds either, since you don't know their identity. But if the dealer accidentally exposed the 5 [img]/images/graemlins/diamond.gif[/img] and burned it, then it would affect your calculation.

Since it's easier to consider all 46 cards to be one pool of cards, and doesn't change the result one iota, that's how most people choose to calculate it.

Cross posted to the wiki. Everyone's invited to help clean up that article.

[binions]

[ QUOTE ]
Every book I have ever read says to count 9 outs for a flush draw because you are never sure how many if any are out.
But sitting at a full table I know over 30% of the cards have been dealt out. Chances are that if you are on a flush draw some of your outs have already been discarded. I took a deck of cards and dealt out 9 hands 20 times, 19 out of the 20 times had at least 1 of every suit dealt out.

If I am on a spade flush draw-when counting my outs, should I really be counting 9. Or would it make since to take the average # of spades left wich would be closer to 7 spades. My thinking is if 30% of the cards were dealt out that means 4 spades were dealt out on average, I recieved 2 of them. That means that on average 2 other spades were mucked or in other hands and my acutal outs would be closer to 7 and not 9.
Obviously if my thinking is correct, this changes the odds of completing a flush draw dramatically.

Please, anyone who has more thoughts or insight into this reply and let us discuss furthur.

[/ QUOTE ]

I am reminded of the drunk in the 10 handed Omaha game who, when told his foe had 17 outs on the turn, said "Good thing there are only 6 cards left in the deck."

[Jeremy517]

[ QUOTE ]

If I am on a spade flush draw-when counting my outs, should I really be counting 9. Or would it make since to take the average # of spades left wich would be closer to 7 spades. My thinking is if 30% of the cards were dealt out that means 4 spades were dealt out on average, I recieved 2 of them. That means that on average 2 other spades were mucked or in other hands and my acutal outs would be closer to 7 and not 9.


[/ QUOTE ]

The problem with your thinking is that cards from other suits are already also gone. You can calculate the odds of hitting your flush based on every possible combination of cards remaining, but ultimately it still works out to be exactly the same.