I recently read a 2+2 "6max vs full ring" thread where one of the replies said that if the first 3 players fold, a 9 seat table should play the same as 6 seat table for that hand.

Obviously this is not perfectly true, because the more folds before you, the greater chance that a remaining opponent has a decent hand (because players tend to fold poor cards leaving better cards behind)

I wanted to find out if folds before you made significant changes in the hole cards of players who would act after you, so I wrote a simulation to find out.

Basically the simulation deals hole cards to all players at the table and then the players decide to fold or not based on the cards they are dealt. They consider only the rank of their holdings and ignore suites. They also ignore position and all act exactly the same way.

Obviously it's a very simplified version of the way people play, but I think it's good enough to find out whether a larger number of players folding before you should make any kind of change to your play.

I ran the simulation from the perspective of the button. The big blind and small blind are considered to have a hand if it falls within the same range as the other players non-fold range.

I ran each simulation 10 million times which seems to give an error range of about .01%.

For my first run I gave all players a non-fold range of AJ+, KQ, 55+

At a 6 or 9 handed table, the random chance of either the SB and BB having a hand within that range was ~17.07%.

At a 9 handed table if all the players before the button folded that chance went up to 17.94%. A difference of ~.9% or 1 extra hand in every 115.
If the button has a hand within that range and everyone before has folded the chance of the BB or SB having a hand within the range drops to ~16.40% or about 1.6% less often than when the button does not have a hand within the range.

At a 6 handed table all players folding before the button has less of an effect, improving the chance of the SB or BB hitting the non-folding range by only about .5% or 1 extra hit per 200 hands. However if the button had a hand within the range the chances of the BB or SB falling within the range fell by just over 1.5% to 16.03.

From the differences between the table size we can see that given a random button hand and confining all players to this static range a steal from the button will encounter a hand within the above range about 1 more time for every 270 hands at a 9 handed table than at a 6 handed table.


So far not very impressive but the hand range chosen is very tight when averaged across positions. By loosening the players up, the results become more significant since there is more information in a fold.

For the next simulation the range was A9+, K10+, QJ, 55+
The standard rate at which at least one of the blinds landed within that range was 23.57%

At 9 max, when everyone has folded to the button the chance of the SB or BB having a hand within the new range is 25.29% or about 1.7% greater than normal. If the button has a hand within that range the chance of the SB or BB falling within the range drops by 2.4% compared with when the button doesn't.

This shows that if all your opponents stuck to this static range you should expect 1 more call in every 40 when stealing with air than when raising with a hand that falls within the range.

Once again at a 6 handed table the effect of all players folding is less pronounced raising the chance of the SB or BB falling within the range by about .8%. The difference between when the Button has a hand within the range and when the button doesn't is 2.2% again slightly smaller than at a 9max table.

I ran the simulation a third time, this time tightening the range of pocket pairs that the players play to 77+. This is unrealistic given how wide the range of non-pairs are, but I wanted to confirm that calls with pairs that are lower than the non-pair calling range, lower the ammount of information given by a fold. This proved to be true, increasing the difference in SB or BB hit rate by about 2% at a 9 handed table when all have folded, compared with when they have called.

In conclusion, if all players play by the first range, an unopened raise from the button should be about .45% more likely to take down the pot at a 6 max table than at a 9 max and for the second, looser range that difference rises to .9%. This is probably not enough to make it reasonable to change your play at a 6 max table compared to a 9.

It is interesting to note that at a looser table the blinds are more likely to have a good hand when you attempt a steal. Also if you're at a table where people like to play small pocket pairs the information gained by whether they call or fold, is smaller than when they don't.

If anyone wants the source code for my sim, they can PM me. I must warn that it's pretty rubbish since it was my first python program and my aim wasn't to learn programming so a lot of it is a kludge, but if you would like to check the math that would be fine.

isn't AJo/AQo type hands a standard fold UTG in a 9 handed full ring game? Doesn't make them trash hands though, in fact they're perfectly playable for a raise a few positions down, as well as UTG in 6 max.

Yes of course. It would have been a bit too complex to give the players a non-static range.

As it was, with that static range players would see about 8.5% of flops, so while it's loose UTG, it's super tight on average.

The fact that players loosen up in later positions means that less information is given by their calls/raises and more is given by their folds.

Since the least information is given up by the folds of the first three players, my results might overstate the difference between 6 max and 9max.

You can't really put it like this. It's true that people fold worse hands, but it doesnt really help you at all, since 2 bad hands might be two good hands in a different combo. evaluate A7 and K7, both crap, but AK and 77 are good hands :-)

Unless you see the cards before you face up, you can't deduce anything based on folds except for the fact that people had less than premium hands.

Sure you can deduce that the deck remaining (to effectively deal the blinds) is rich in aces. Thats massive really.

I have worked on this problem before in other posts and the effect is most important for AX type hands. Thats because most of people are losing players, therefore you can expect them to have bad habits, ie to play most if not all AX if others before them have folded and also the good players are less tight than presented originally.

Additionally noway in hell is AQo a fold at UTG . I can go over the math if requested that show that even AJo is a hand you can play carefully as it has 73% chance to not be dominated (AQ is 80%) . AQ is fact has over 50% chance to be the strictly best (including low pairs) hand in the table. I consider table of 10 here. So imagine how better it is for 9 and less. Seriously play in some program like computer academy pro with the cards face up to get an appreciation through observation (unavailable in the daily poker experience which is very selective and results oriented and biased) of how frequent it is for nobody at the table to have anything good really.

A reasonable approach in this problem, focusing only on aces and maybe Kings is like this;
The first few play say only AA,AK,AQ,AJ lets say the first 2 do that . Then the other 3 play any AX with 50% chance below A9 say A8-A2 50% played and all above (ie some people play them and some dont) . The other 2 before the button play basically any AX .

That is a rather reasonable approach in my opinion of most games . People simply like to play AX suited small too from early positions regardless of whether its a good idea or not , its an experimental fact people play them when nobody has opened.

In fact i can show with extensive math that raising with A2 from button-1 is plus EV (raise to 3bb i mean) . Clearly its even better with A3,A4 etc. Obviously at least a limp is a given (some will debate that limping as the first to enter is a bad idea but we debate here how people play not how they should play )

So studying the chance of an ace having being dealt before out of 4 total in the past 7*2=14 cards you need to take into account these things and not be tight in selection. It makes sense to be tight in full ring but not when nobody has entered yet and many have folded and especially if you are not a solid player (even more likely to play any AX as limp or raise)

I say then first 2 almost certainly didnt have AA,AK,AQ,AJ. The next 3 didnt have AA,AK,AQ,AJ,AT,A9 and 50% for A8-A2 . The next 2 didnt have any ace.


I may go over the math in a detailed post but since this is a simulation on the original thread you can simply avoid my math and do the simulation with wider ranges to be more realistic and introduce also position adjusted ranges at least for AX hands.


The result of the detailed math i did shows (unless i made an error ) that you get a spectacular effect for the best of hands as a result of the fact others didnt play.

AA for example goes from 0.49% chance per remaining player (say you at the button have XY no A in them) to a massive 0.81% and KK too goes from 0.49% to 0.66%.

Hands like AK will also have a rise about 1.93% from 1.31%.

Overall other AX type hands will have a rise from the typical 1.31% each to 1.62%

I sure hope simulations with wider ranges ,that are more realistic to reflect the loose nature of most games these days and position adjusted, can prove those approxiations i did valid or at least reasonably close to the real thing.
Other top hands will have less serious bias though . Overall the top range may not experience a massive bias, it may still be as presented in the original thread opening post a rather small yet respectable bias (but more than originally presented) but i think the ultra premium hands like AA,KK,QQ,AK,AQ and all smaller AX will have a serious bias. KX even less noticeable though.

Obviously in a very tight table the effect will be less interesting. Maybe the very fact we had this occur in this table places a bias towards this being tight overall and therefore the results should not be as impressively higher as i claimed, therefore feel free to accept what i said only if prior activity in this table has provided you with evidence that the people there are reasonably loose and not seriously nitty. After a while even nitty originally tight players realize its ridiculous to play poker by being very tight though. The difference between a by nature loose player and one that is flexible but still TAG at heart is that the TAG realizes even when playing loose the problems and responsibilities such behavior carries while the simplistically loose player is a brutally naive degenerate or a gambler at heart and pays little attention to such issues (not the good skill LAG though that are thoroughly self aware of their image and implications and methodically exploit many rigid conservatives)

Yes but although K7 is a bad hand for most positions and probably even for button-1 , A7 is not the same so you can conclude indeed that not even A7 appeared for most of these guys before. A7 i am willing to prove with math if requested is plus EV to raise with from button or button -1 and many other low AX too even A2 (A2 3bb raise from button -1 in this setup is +0.15bb if i am right) , certainly not bad to limp with from earlier positions even if someone then raises you, you can be confident due to removal effect from a7that your ace is still usually good (but of course dont lose your stack to a bad kicker) . My point is even if you limp (and didnt raise with A7) from say button-1 or -2 a late position player can still raise you with 66 or KQ, KJ etc and they are not at all any better. The chance they have AX though if combined with the fact that others before you folded and the fact you have one Ace is seriously reduced (having one ace is always a serious 50% reduction on AA for instance and 25% reduction on AK,AQ type raising hands ). Also if you hit an ace at flop for example its even less likely the other guy had Ace when he raised you. My point being that it not necessarily so totally bad to limp with small aces or raise at button or before 1 ,2 with them . Not bad at all if you remain aware of the bad kicker issue . Plus never forget its not about what some tight people or classic older books think is proper play , its about what the barbarians playing out there today think ,lol.

I have the responsibility to inform you that i made an error likely in my estimation of AA,AK,KK in the post before and that i need to revise my numbers lower (still higher than without the folding effect though) . I will have the corrected result after i post an example of how such estimations ought to be done in the absence of simulations given as function of the ranges of players considered.

I don't think you're taking into account play after the flop. A2 - A7 may be likely to be the best hand remaining if everyone has folded to the cut off but you'll need to be very skilled to play it after the flop which is why I think it may be ev- overall, at least for most players.

Even if you're making a profit with these hands, you may be losing over the long term. For example, lets say you raise 3xBB and get the blinds to fold >2/3 of the time. You're making a profit pre-flop but when you get called the only flops you can play with confidence are the ones that give you 2 pair or trips. You have to fold to almost any raise before you, or if your opponent has a strong hand you'll probably lose a continuation bet to a reraise.

It might still be profitable on an individual hand, but these hands will wreck your table image because they tend to make your pre-flop play look loose and your post flop play look timid.

Here is the general problem facing here;

Lets understand it with an example;

Consider a player that folded from earlier to mid position who has a range of say 22-AA ,AK,AQ,AJ,AT,A9 all suited AX, KQ,KJ,KT,QJ,QT,JT. (this is just an example a more realistic player could have included some suited connectors although clearly this is a tiny 1-1.5% change in the range no big deal)

This is the probability we are asking here;

What is the probability this guy had an ace given the fact he folded.

Using Bayes theorem it would have to be this;

P(had an ace| fold) = P(fold|had an ace)*P(had an ace)/P(fold)

P(fold)=Probability to receive a hand outside the playing range and fold.

P(had an ace ) is the chance to be dealt an ace which is 1-(48,2)/(52,2) (= 1 - chance to not get an ace)

P(fold|had an ace) is the probability to fold when you have an Ace.


For our example the player that plays say

22-AA ,AK,AQ,AJ,AT,A9 all suited AX, KQ,KJ,KT,QJ,QT,JT from that given position
or in pokerstove notation; 22+,A2s+,KTs+,QTs+,JTs,A9o+,KTo+,QTo+,JTo
(21.3% range in an unbiased spectrum)

has a rough fold probability of (i say rough because his range numbers are affected by the fact others before him may have already folded but its a secondary effect so it can be ignored for now) of 100-21.3%=0.787

So P(fold)=0.787

His probability to have an ace (also affected by the folds of others before but lets ignore the correction for now since they are unknown anyway but assumed small) equals the probability to have received an AX type hand which for an unbiased spectrum is as explained 1-C(48,2)/C(52,2) or 1- C(46,2)/C(50,2) if you assume we view it from the perspective of the button observer that happens to not have an ace either). Anyway we are after rough estimations here so feel free to use say

1-C(46,2)/C(50,2)=15.51%

P(had an ace)=15.51% (pending not severe corrections)


All that is left is to find the probability he folded given the fact he had an ace)

A player has an ace when he has a hand AA,AK-A2. Our player folds when he has these hands not present in his range; A2o-A8o and doesnt fold with the other AX hands and AA. The basic probabilities of all those ace hands are;

AA: 0.49%
AK-A9: 1.31% each
A2-A8 off =2*4*4/50/49-2*4/50/49=2*3*4/50/49=0.98% each
A2s-A8s=2*4/50/49=0.33%

So the sum of the ace hands he folds is 0.98*7= 6.86%

So each time someone with that range who has an ace folds with probability
6.86/15.51=44.2%


Now we can get back to the original equation those numbers and obtain;

P(had an ace | the fact he folded) =44.2%*15.51%/0.787 =11.07%


We need to do that for all the players in the table that folded one by one using their range to obtain the probability that each one of them didnt have an ace.


Very roughly the answer will be equal for each one of these guys to about

probability of the AX hands they fold to appear in their hand/(probability of range they fold)



One might naively assume the chance they had an ace given the fact they folded is equal to the chance their ace was the ones they fold but that would ignore the essential fact that the given information that the guy folded somewhat enhances the chance he had one of these hands we know he folds. As a result the true probability is somewhat larger enhanced by that factor 1/folding range which is usually something like 1/0.8 or 1/0.65 for most players depending on position and preferences.


This correction was needed in my original estimates.


One needs to go now over all the players and after assuming their ranges find the probability each one had an ace given the fact they folded. Then combine all these probabilities to see what is the chance 0 aces were used so far or 1 ace or 2 etc. From that then and maybe doing something similar for Kings one can see the proper bias in the remaining to "deal" hands that the blinds say have now.

Without bothering you with the details that are boring (and will have to be redone each time one changes the ranges) past this point one arrives at roughly something like this;

(under the reasonable assumption -concering AX hands- that the first 2 play AA,AK-AJ, the next 3 play always AA,AK-A9 and 50% of the time A8-A2, the last 2 play any AX)

probabilities of Xi player that folded having an ace;

(first level approx ie not taking into account prior players did fold- this will get more interesting as we arrive at late players though and it is of course the main effect we are after)

X1: 12%
X2: 12%
X3: 6%
X4: 6%
X5: 6%
X6: 0%
X7: 0%

Similar logic for having a King (first level approx)

X1: 13%
X2: 13%
X3: 12%
X4: 12%
X5: 12%
X6: 11%
X7: 11%

Obviously the effect is more intense for Aces practically the last 2 play any ace and for this reason have 0% chance to have had an ace.

Just to illustrate that there is reason behind the massive difference between the stength of a hand like AX and that of same KX the equity of A7 say vs a random hand is 60-40 but k7 is 56-44 and if you had as opponent a 20% hand it becomes 44% for the A7 and 37% for K7. Trashy looking aces are seriously better than the same trashy looking kings vs different kinds of opponents . People may second guess folding weak aces but not weak kings, that is a lot easier. Still there are many people that play stupid kings too (you have seen them lol they usually pay you or give you a bad beat out of nowhere land) but not the majority of people. The division in AX though is less serious. Even generally tight people will play low AX when it makes sense and the key idea here is that when 3-4-5 others have folded yes its time to use that small AX because it is becoming a small table suddenly. Additionally one can show with some math that limping or raising with trashy looking KX in late positions is marginal to negative EV but not so much with trashy AX, the later the better.

Its simple approx. theory now to find from the above tables the chance that 0 aces were dealt or 1+ ace and same for Kings. Then from these numbers and the fact we now have 52-2*8=36 cards left you can find the chance of AA,AK,KK etc.

Using the above tables;
Chance of 0 aces dealt; 64.5% (products of 1 - probabilities for all 7 players), 1+ ace ~35.6% (multiple aces are possible of course but they decline dramatically in % especially due to the fact ignored in this first level model that if one ace is already dealt the chances of further aces on the others left are reduced seriously and not as predicted by multinomial expansion here so we can lump 1 or more aces under 1 ace category at ~35.6% , the contribution of more aces is also less important to the main effect.

Essentially then we have 64.5% of the time 4 aces left in a deck of 36 cards left to deal the other 2 players (of course this is metaphorically speaking in equivalent mathematical terms since the hands have already been dealt) and 35.6% 3 aces.
The AA probability then will be the sum of the 2 ways properley weighted by the probabilities of 1 or 0 aces before ie
P(AA)= 0.645*4*3/36/35+0.356*3*2/36/35=0.78%

This is to correct the previous number in the post before.


Similarly one can go ahead and do calculations for KK or AX using the above tables.

Simulations of course bypass the problems here that this is in fact a first order approximation.

I have taken into account post flop play when called. I have also taken into account the need to fold when reraised because the equity you have then vs a hand that would usually be worthy of a reraise is below 30% even . When you get flat called therefore we have the only situation that matters postflop. It does however for the case of button-1 occur only 34% of the time (+- whatever range objection in my math) .When it does our equity vs the range that flat called us and not reraised us is 44% vs 56%. We are not terribly bad there and its up to us to not get killed. In fact why should one see as terrible a 44%. It just fine with me. Of course its a naive all in equity which this isnt but in all reality post flop play is usually adjusted for position in line with all in equity and you can estimate its outcome by taking an average pot possibly double the preflop one (sometimes it ends at flop sometimes it ends later etc) . My point is if you raised with A2 and the caller was KJ you have 57%. Why will you be outplayed by KJ post flop here?
Plus i dont care if i am outplayed or not here i only care what people think they can do . I am not asking here to believe all the people doing these with AX will win overall or know what they are doing. But i am willing to claim that they should win on average if done with care. If you raised with A7 and the caller is small pair or broadway XY you have little to worry about, certainly you will win a fair share of flops and later streets . You will be careful not to give heavy action to people unless as you said you have something strong but nothing prevents you from connencting with an ace or from bluffing them because they didnt connect 65% of the time with their XY etc. Depending on who calls you you will be acting first or not post flop. So this effect is somewhat balanced 2 to 1.

Remember its only 34% that ends as post flop play and you have 44% equity then. The rest of the time the decisions are easy. How is suddenly a 34% where you have 44% equity going to make the idea so negative ? Basically 52% of the time they have nothing to give you action with and they reraise you only 13% of the time (3 opponents left) . Additionally often the one that calls is not the big blind making the post flop game even easier due to a bit of dead chips available (i mean its not 1-1 in chips used hence 44% is not so bad necessarily). The very fact you have the ace reduces seriously the AX hands that dominate you in the range of the callers plus there are only 3 players left. In any case even limping and not raising is an option and i am quite confident in all the games i have played that i see people play low aces in these situations where most have folded happily.

If you show me math that proves the choice to play A2o from button -1 is negative EV i will gladly study it and even give you my own math but recall this is not a thread about what AX is ok to play , only a thread of what usually people do.

Ax is a standard button open for any winning player. Also, it won't wreck your table image, because any player good enough to be paying attention to your image knows its standard to raise Ax(and a lot worse) from the button.

Nobody is forcing you to put more money in if you are called. Sometimes it is best to just check it down. As you said, you have already made a profit, only bad play post flop could squander that. And even if people only ever defend their blind with better hands, position more than makes up for that.

Yeah I realised my table image point was retarded shortly after posting. Who cares even if it does make you look loose/weak. You can make a profit by simply tightening up and being more aggressive.

I think I'm so disinclined towards Ax because I play STT's at the donkaments. At these games Ax is too high risk for too little reward early on. Also giving off a loose image early will lower the effectiveness of your steals later. Of course, once the blinds get big and couple of people are knocked out, even A2 can be one of your most profitable hands since it increases the chance of your opponents folding by so much.

Nice thread, I've wondered before how big the effect of this was but never tried to quantify it.

One comment -

I presume that the purpose of calculating this is because we are on the button, and want to know how likely it is that the blinds will play. If so, I suggest breaking it down with respect to our hole cards, rather than just stating that our hand is within a certain range - others are less likely to have a playable hand if we have KQo than if we have 77.