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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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  • 1
    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
    Commented Feb 6, 2012 at 17:06
  • Probability is a measure of information. All "unknown" cards are equivalent, regardless of where they might be physically.
    – Lee Daniel Crocker
    Commented May 2, 2016 at 16:14

5 Answers 5

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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...
    – Emre K.
    Commented Feb 6, 2012 at 14:26
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    Don't feel bad. Every poker player has thought about this before.
    – k to the z
    Commented Feb 6, 2012 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
    Commented Feb 6, 2012 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
    Commented Feb 6, 2012 at 17:04
  • I've heard people talk about this and find it quite funny. It doesn't matter where you place the cards you do not see, the odds are still the same. I have heard people calculating burn cards etc. It doesn't make sense but @ktothez is right, we've all have thought about it. HOWEVER, if you happen to suspect that an opponent is on the same flush draw as you, for example, and you have top pair, then it is reasonable to consider that you have two 'blockers' to their flush (and he to your flush) and the flush is therefore less likely to hit.
    – Danny Mahoney
    Commented Aug 17, 2016 at 7:49
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(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?" 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
    Commented Feb 7, 2012 at 15:10
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    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
    Commented Feb 7, 2012 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
    Commented Feb 7, 2012 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.
    – paparazzo
    Commented Apr 30, 2016 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 .
    – user1934
    Commented May 2, 2016 at 2:54
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Statistically any down card is an unknown card. In a hole card or muck is a down card.

Certain cards are played more than others. For example Ace is the most played cards. Yes there are some ace rag hands you don't play with an Ace but there are are a lot you do. 2-7 you pretty never play or only play as a pair.

Not everyone plays top 10% and it will vary by position but consider that case. Not everyone has the same top 10% but this is one. You also need to consider how many ways to make the hand. There are 6 ways to make a pair, 4 ways to make suited, and 16 ways to make unsuited not paired.

            combinations
A   A       6
K   K       6
Q   Q       6
J   J       6
T   T       6
9   9       6
8   8       6
7   7       6
A   K   s   4
A   Q   s   4
A   J   s   4
A   T   s   4
A   9   s   4
K   Q   s   4
K   J   s   4
K   T   s   4
Q   T   s   4
A   K       16
A   Q       16
A   J       16
K   Q       16

If you add those up what is the percentage of each of those cards 

card count 
A     148   50.0%
K      68   23.0%
Q      20   6.8%
J      12   4.1%
T      12   4.1%
9      12   4.1%
8      12   4.1%
7      12   4.1%

Pretty interesting that if someone is playing 10% and they play the hand based on that alone then 50% of the time they are holding one or two aces.

Card removal is real but it does does NOT really matter

Card removal has the same effect on every player. AK going from slight dog to a slight favorite does not change the way the game should be played. You should open from mid to late with AK not because you have a better chance of pairing the board - play it because open in mid to late position it is likely the best hand. Don't open 46o because it has a worse chance of pairing the board behind multiple folds - fold it because it is a bad hand. Card removal does not change the relative strength of hands enough to be a factor. People new to poker math seem to fixate on this and my advice is acknowledge and then immediately ignore.

Take JJ as the sample case. It is a very good hand with very few card removal blockers. Behind a raise and re-raise it is likely beat. Because you are beat actually improves the chance of spiking spiking a third J on the flop by maybe a whole 1%. Does not make JJ any more playable in that spot.

If you play 87s multi-way then yes your chance of pairing goes up base on card removal. And if you hit top pair on the flop your chance of being top pair goes up as K8- and a lot of other combos don't play. Newsflash 99+ is playing the hand and still has you beat. Card removal is real but does not really even rise to statistical noise.

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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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  • The card removal theory means that if 6 people fold, they are LESS LIKELY to hold As or Ks. Its not confirmation by any means. Just because you know a case of folded Aces doesnt mean they are as easily folded as any other 2 cards. Theres nothing that 2 pair beats that AK doesnt beat, but AK flips against almost any pair, while 2 pair is crushed. I dont think this an opinion based issue.
    – sakon
    Commented Feb 23, 2017 at 9:20
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When I considered this question (from the point of view of a live full-ring game with 10 players where 25 cards are missing after the flop is dealt: 20 player cards, 3 flop cards, 2 burn cards) I came to the conclusion that with 50% of the cards missing but never knowing which ones we're talking about some kind of standard deviation calculation. For a player having 8 full-deck outs if we choose to calculate the effect of card removal to 2 standard deviations (95% probability) then my back of envelope guesstimation was that we should subtract 1-2 outs from the full-deck number. I'm not enough of a mathematician to reproduce that, but I'm sure others can once we agree we're talking about standard deviation.

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  • Nope unfortunately thats not true. As long as a card is unknown, it has the same probability of being any unknown card. The formula is always (number of outs)/(number of leftover cards). It is true that your outs may be in the burnt cards. But if you calculate the odds of that happening, and add it to the chance that they are not, the formula will be the same.
    – sakon
    Commented Feb 23, 2017 at 9:12

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