Hey all, I've read my share of poker books, and I understand the basic odds of hitting draws etc., but I've had some questions brewing lately considering the idea that at a full ring table of 9 players, that you could consider that on average your live outs when drawing should be lower than what we generally consider to be our true number of outs. i.e.. 9 outs remaining for a flush draw, 8 outs for OESD. Take the following example for instance:

Let's say you have a flush draw on a full table of 9 players. Two spades in your hand, 2 on the flop plus one other non-spade card, lets say it’s a heart for this example.

1 burn card, plus 16 unknown cards that are now not possible for you to be dealt. 17 dead cards total.

9 spades, 13 diamonds, 13 clubs, 12 hearts remain in 47 total unknown cards, the 17 that have been dealt and the 30 remaining in the deck.

Meaning approximately:
19% are spades
27% are clubs
27% are diamonds
25% are hearts

So of the 17 cards that have been dealt or burned, would it not be safe to say that since the cards are dealt randomly, that on average, approximately 19%(or 3.23 cards) of those 17 cards would be spades and therefore cannot come out on the turn. Making your actual average number of outs on the turn, 9 total outs minus 3.23 average number of outs that have been dealt. Making the average number of outs that will be available to be dealt on the turn to be 5.77 outs? I appreciate your thoughts.

If you don't know anything about your opponents' hands, then, while you are technically correct, the probability of hitting your flush is properly calculated by ignoring your opponents' unknown cards.

There have been several threads on this issue over the years. Here is a link to one from the Probability Forum.

https://forumserver.twoplustwo.com/2...-draw-1619362/

Thanks for your response! I get that, but doesn't it seem entirely unrealistic to think that every single one of your outs is live when 36% of the cards you are considering to be live are not. To think that those dealt cards are only comprised of every other suit besides the one you need?

I wonder why everyone always takes the pessimistic view of outs when they first try to figure out this scenario. It's actually just as likely that your outs are more live than the actual odds indicate. The effect cancels out and the real odds are:

outs/unseen cards = probability of hitting on the next card

The calculated probability doesn't assume they are all live. It is the average, and includes the times all are available, and times all are dead, and everything in between. It is the actual frequency you will hit them.

Yes, NewOldGuy worded it better than I did above in post #2.

When I said you should "ignore" your opponents' unknown cards, I meant to treat them as run-of-the-mill unknown cards like the other cards in the deck stub not distributed to any player.

Sorry for any confusion.


Edit to add: Bob148 gave the proper explanation and expression above as well.

i think it's like after you drink half a cup of coffee, the concentration of coffee is still the same for the remaining half.

I'm not sure I understand what you want to say tough.

But I think when opponents fold on the flop we can often assume they don't have a flush draw and then at least one of the folded card is not the spade. I think the odds are affected there but I never really looked at it.

Burned card or dealt unknown cards do not change the probability.
You could look at it this way: unseen card brings no information to you. You have absolutely no reason to change your original decision when you did not get new information.

Anyway you can get soft info later. Imagine the flop with FD and several players call a bet. Now it is kind of more likely somebody else holds FD too, blocks your outs and what is worse could have higher FD than you, makkng your outs realy dead even with reverse implied odds.

Since it is the theory subforum, I would like to theorize about what is “bugging” the op.

OP, you are on the verge of quite an increased understanding of poker negotiation. Keep thinking about it, all day every day, and dream about it at night!

By placing bets, poker players relay information about their range, in a hand. Also, blockers matter.

Without any betting, you have no information and no reason to discount any of your outs.

Start thinking about COMBOS of cards your opponents might start the hand with.

Say you hold just the ace of the flush draw on the board. Your opponent bets. It is significantly less likely that your opponent holds a flush draw. Not just because of a single suited card in your hand, but that the Ace is a key card in the formulation of a range of suited starting hands.

Or, you are in position and your opponent checks to you. You can bet the Ace, since you block a chunk of his/her calling range.

So cards can influence your decision to bet or decision to call, but should be part of a RANGE VERSUS RANGE analysis.

You were very close to answering your own question here. If you would go one step further you would see that it also means that the remaining fraction of dead cards could also not come out on the turn, meaning that on average the deck still holds the same distribution of spades vs other suits.

@Mcfly - if you were by yourself (no opponents) drawing from a 47-card deck with 9 spades, you would agree the probability is 9/47 for the top card being a spade. Now what about the bottom card? Also 9/47, agree? Same with, say, the 7th card down in the deck, or any card I ask about, it's 9/47 to be a spade.

The deck is currently in one big pile. Now remove the top 6 cards and put them into piles of two, keeping them face-down. Now the top card is what was formerly the 7th card. Why should the probability suddenly not be 9/47 just because you rearranged part of the deck into piles of two?

It seems like this is a milestone every analytical poker player goes through and comes to the realization that yes, players actually have been calculating the probability correctly for decades.

My post questioning this idea was over 10 years ago (under another ID). I thought I was the one to finally discover this mistake that gamblers were making, just like many others have.

There are many ways to 100% convince oneself of the truth of the matter if you are so inclined.

With one opponent with unknown cards, you can enumerate all the possibilities of how many spades the opponent holds (and therefore how many spades are left in the deck with attendant probability of hitting your flush).

If you correctly work through all the probability math, it will turn out that the overall probability of hitting your flush via all of this hard work equals the easily-derived probability of hitting your flush assuming that your opponents unknown cards are (essentially) part of the deck of unknown cards.

Others have made the point above, but anyone can work through the math if they still have doubts.

#1 in the FAQ to this forum:

https://forumserver.twoplustwo.com/1...stions-676280/




And 5.77/30 = 9/47 (give or take due to rounding),

so you're going to get the same answer.

Hey all, yes I see now. I was bouncing it around in my head for awhile, but now I think I get it. Even if 3.23 of my outs are in the dealt 17, there are still 5.77 outs in the remaining 30 cards in the deck, giving me the same probability of hitting my flush draw as if we were drawing to 9 cards from an untouched 47 card deck which is 19.23%. Why was that so hard?! Pretty sure every poker book needed to be rewritten there for a moment. Hah!

@robert_utk I have been working to understand preflop ranges and blockers and it's been helping my game a lot. The idea of having the Ace of the flush draw on the board is a great tool I'll be sure to add to my collection. Thanks for all of your responses!