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Standard odds calculation assumes that all unseen cards are still in the deck. For instance for a flush draw with two suited cards showing on the board, we assume 9 outs.(13-2 hole cards-2 on the board). On the flop we assume about 36% chance to win.

This however is not always the case, if the hand is played heads up there is the possibility that one or more of the burnt cards are of the suit we are expecting to catch. If the hand is played in a 6-max or full ring game, there is also the possibility that some of our out cards have been folded by our opponents.

I feel like this must have some sort of mathematical significance when calculating odds for an underdog hand. I'm talking about a draw vs. an already made hand. A draw needs the cards to drop on the board in order to win. Cards that would improve the made hand might also be folded but the made hand does not need those to win so it does not cancel out the disadvantage to the draw caused by folded hands.

Am I on the right track here? If so, is there a mathematical model that would make the standard odds calculation more accurate?

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see my comment to Chris' answer. For folded cards, there's more to it then simply "it's not significant". There have been studies made using billions of online hands on the subject. It's hard and even harder to model due to what is know as the "card removal effect" ; ) – TacticalCoder Feb 6 '12 at 17:06
Probability is a measure of information. All "unknown" cards are equivalent, regardless of where they might be physically. – Lee Daniel Crocker May 2 at 16:14
up vote 10 down vote accepted

No, those cards have no significance to odds calculation.

If I shuffle a fresh deck, what is the chance the the top card is the Ace of Spades? 1 in 52. If I deal off the top 10 cards face down, what is the chance that the card on top now is the Ace of Spades? Still 1 in 52. The same probability will apply to the rest of the cards in the deck, including the King of spades, etc, down to the 2 of spades. Since there are 13 total spades, the chance that the card on top is a spade will be 13 in 52, or 1 in 4, no matter how many cards you deal (face down) first.

In fact, you could deal off 51 cards face down, and the chance that the final remaining card is a spade would be 1 in 4.

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And I studied cs... – AntonAnsgar Feb 6 '12 at 14:26
Don't feel bad. Every poker player has thought about this before. – k to the z Feb 6 '12 at 14:34
@Chris Marasti-Georg: I agree about burnt cards but OP also mentioned folded (...there's some chance some of our cards have been folded by our opponents...) / mucked cards. There have been extensive discussions in "poker theory" forums (and on Usenet) about the difficulty to correctly model the fact that there's a near 100% probability that opponents who are folding preflop without betting anything aren't holding premium hands. And this definitely does have an influence: it's been proved AFAIR that due to the card removal effect, community cards, for example, aren't evenly distributed. – TacticalCoder Feb 6 '12 at 16:59
@Chris Marasti-Georg: Barry Greenstein talk about this "card removal effect" and states this in one of his books: "as players fold (preflop), the probability of an Ace or King coming on the board increases". It can be seen this way too: hands that are folded preflop are not randomly distributed (because AA, KK, QQ, etc. should be part of a random distribution, but in this case they're not, because they're not folded preflop etc.). – TacticalCoder Feb 6 '12 at 17:04

(oh well instead of commenting I may as well post this as an answer)

You're asking two different questions and they have two different answers.

Are burnt cards significant in odds calculation?

No. They're not different than any other card still in the deck.

there is also the possibility that some of our out cards have been folded by our opponents.

Indeed. And that is a very complicated topic.

It's called the "card removal effect" and it has implications for computing odds and it explains "weird" facts:

  • community cards aren't randomly distributed
  • some players (depending on their playing style) are apparently consistently above or below AIEV when they go all-in
  • etc.

The "card removal effect" is described here as this: "The Card removal effect, or the card bunching effect, describes the changes to opponent card ranges considering that other players have folded preflop." (it changes opponents' card ranges, and it of course also affects the flop/turn/river cards).

Barry Greenstein himself wrote, (page 150 of "Ace on the River"), the following:

"...If several players fold first, Ace-King suited is a favorite over most pairs. ...(snip)... The reason for this is that players are more likely to play hands having an Ace or King than those containing smaller cards. Therefore, as players fold, the probability of an Ace or King coming on the board increases"

The card removal effect is also used here to try to explain weird results for AIEV:

So the card removal effect is very real, it's a topic of studies and there's definitely more to "Are folded cards significant in odds calcution?" than a simple "no".

It's very complicated to model : )

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You have to have a very strong handle on the ranges of all players at the table to be able to model this correctly. Players folding in EP may be folding as high as AJ or AT. Many tighter players in late position will fold hands like A8o or A6s, and players that do play hands like A8o will also be playing hands like 56o or 79s, which means that you can't put that much correlation on their folding or playing vs. the number of Aces or Kings in the deck. – Chris Marasti-Georg Feb 7 '12 at 15:10
Also, you need to define "significant." If the card removal effect can effect a 1% shift in odds, is that significant? I would say no - in nearly every case there is a greater than 1% margin of error in assigning a range to your opponent, which makes the card removal effect insignificant. – Chris Marasti-Georg Feb 7 '12 at 15:11
@Chris Marasti-Georg: I don't disagree with you that one would need to determine how "significant" it is but... It's apparently sufficiently significant to make the community cards not follow a true random distribution. And it's also apparently sufficiently significant to change the odds of AKo vs low pocket pairs from "slightly underdog" to "slightly favorite" in the case Barry Greenstein outlined. And these are just two examples, there are probably many more where people have been able to notice "weird" results/numbers. All I'm saying is that it's more complicated than it looks like : ) – TacticalCoder Feb 7 '12 at 19:22
I don't call 1% significant. There is no case a 1% statistical change would change my play at the table. I would open with AK late because it is a favorite against random hands, have position over the blinds, and have blinds to steal. Going from a slight dog to a slight favorite versus a pair does not enter that decision. AK versus a pair could go down 1% and I would still play it. – Paparazzi Apr 30 at 11:02
community cards aren't randomly distributed. Unless this is because you aren't including the community cards which don't show when someone takes the pot pre-flop (or before the river) I don't see how this can be true. If there is going to be a flop, the same cards are going to come regardless of what hands anybody has. This isn't the same as how the community cards are likely to fit based on pre-flop betting . – Michael May 2 at 2:54

So an 8-handed table and it is folded round to the SB who has 22 and raises, AK suited in the BB shoves and SB calls. Stats show AKs is 50.1% equity but this cannot be the reality because of the chance that amongst the 12 cards in the muck there is at least one other ace or king. Taking out the SB and BB cards there are 48 cards left of which 12 (25%) are in the muck. To win BB has to hit a KKKAAA, flush, straight or counterfeit. There is 0.63% chance of a chop. Taking just the pair outs if we remove 25% of them this leaves 4.5 cards left to hit. If I take into account just one 'out' card is removed the odds reduce to 46.2% for the BB. If I remove two they reduce to 41.9%. Greenstein argues that if the table has folded to the SB then the chance of an ace or king being in the muck is low because otherwise they would have raised. Surely it depends on so many things, their range, their position, their stack size, if they are on or approaching the bubble. So I don't think this is true to say. I have known a player that satellited into an EPT for a few Euros fold AA in LP because the min-cash was over 10,000 Euros. So yes, in my view, card removal does come into play in this situation and does affect the value of known hands. If people ask me if I would rather be holding AK suited or 22, I would always take the small pair.

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