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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

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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
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Don't feel bad. Every poker player has thought about this before. – k to the z Feb 6 '12 at 14:34
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@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
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@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
up vote 8 down vote

(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).

http://forumserver.twoplustwo.com/36/stt-strategy/lt-card-removal-bunching-effect-simulation-321075/

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:

http://forumserver.twoplustwo.com/28/internet-poker/official-poker-site-data-analysis-discussion-thread-264915/

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?" then 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

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