So Ive been toying a bit around today with this concept of card removal and how it is effecting opening ranges

My startion question basically was: How much more frequent are you gonna get the opportunity to open up AK on the button (suits don't matter for now) opposed to 32 for example so that's assuming everyone folded to you

Obv this is a very complex thing to calculate, and i'm not really interested in exact numbers, so i've been simplifying things quite heavily, and was wondering if this was (again overly simplified) a somewhat decent/valid first estimation

The first thing I did, was predefine some standard opening ranges for every position in front of the button, so for example UTG & UTG1 15%, MP 18% and so on

Next thing I did was calculating the chance of a player folding a certain card, so let's start with the Ace in a 15% UTG range (for example 55+, A8s+, A5s-A2s, K9s+, QTs+, JTs, T9s, 98s, ATo+, KJo+)

So that means the UTG player is folding 104 combos that consist an ace, namely A7s-A6s, A9o-A2o

So the chance that a UTG player is folding an ace is then 104/1326 combos = 0.078431 aka roughly 7.8%

So far I don't think I've made any mistakes, but now things get more complicated obviously. In reality once the UTG player folds, it should effect the probability of UTG1 folding a certain card as well (since we know what the UTG player is opening)

Anyway for the sake of simplicity, so far I haven't try to work this out, since it would take me too far for what I'm only trying to accomplish. So basically I did this for every position up till the button which gives this table



The next thing I did was to take the sum of each probability which gives us plus minus the chance that either UTG, UTG1, MP, LJ, HIJ or CO folds a certain card (I believe statistically you should also take into account the intersection but again simplifying things here)

So now as you can see, there is a +- 43% chance that an ace is folded in front of you and about a 57% chance that a king is folded in front of you

So now I've applied another very simplified logic to try to measure relative differences

Usually you have a 16/1326 = 1.2% chance of getting dealt AK

As I'm sure you are aware, if you fully remove one card, that chance is being reduced to 12/1326 or 0.9%

So given that information I computed the following calculation: 16 - (4*0.43) - (4*0.57) = +- 12 combos of AK left

If you want to calculate the chance of A2 you get 16 - (4*0.43) - (4*0.87) = +- 10.75 combos of A2 left

So relatively speaking, if you were to attach weights you could (very simplified!) say that the chance of getting to raise first in with A2 on the button compared to AK is about 10.75/12 = 89.5%

For pairs you can apply the same logic, only replace the 4 by 3 now, since the full removal of one card takes away 3 out of 6 combos so then you get AA = 6 - (0.43*3) = 4.71 combos and for deuces = 6 - (0.87*3) = 3.39 combos hence a relative difference of 3.39/4.71 = +- 72%

Any feedback or comments would be really appreciated, since it's a long time since I took my statistics class, and can def have made some thinking errors on the way

Edit: So let's say if you were to construct a BTN RFI range you could assign the plus minus following weights, making AA the benchmark 100% weight

AK = 95%
A2 = 85%
22 = 70%

I have only skimmed your post but it appears to be very interesting (I reserve comment on your method). Several people have looked into card removal effects in various threads in the Probability Forum over the years using simulations and/or analytical methods.

I am not sure I fully understand the purpose of your "weights" and how they would be used. I guess you are saying that the blinds can use these factors to assess the button's actual weighted range after button makes a standard (perfunctory) raise after being folded to.

I ran a simulation tonight to shed some more light on the distribution of ranks that would be expected to be folded by the first six players at a 9-player table using the opening frequencies shown in OP's spreadsheet. I used the Poker Stove hand rankings that, I think, are based upon how starting hands fare against three other random starting hands.

Rank	Avg # Folded
A	0.551
K	0.681
Q	0.739
J	0.749
T	0.791
9	0.927
8	1.012
7	1.040
6	1.072
5	1.091
4	1.114
3	1.114
2	1.119
.
SUM	12.000

Of course, the rank frequencies "available" for Button's starting hand are the complement of the above folding frequencies. Accordingly, in the scenario that all six players fold to the Button, the Button's cards essentially are dealt from a deck stub that is rich with Aces and Kings and depleted of low cards.

As an example of the implications, if I did the calculation correctly, based upon the figures in the above table I find that in this scenario Button will have AK around 22% more frequently than in the "totally random" scenario (assuming we know nothing about anyone's holdings).

To disgress a little, how could this information be used to increase winrate?

The 2bet RFI ranges already take all of this into account. Card removal is best reserved for constructing 3! and 4! ranges. Arty is very good at explaining those constructions. Should be readily available in his past threads.

I think card removal is also interesting for the first raise, Robert. It's why I'd open A2s UTG in preference to 65s. The latter doesn't block someone waking up with a real hand. Indeed, if I have 65s UTG, it becomes slightly more likely that subsequent players have Axs and will 3-bet me.
This. I don't understand the last bit of the OP at all, but the first few paragraphs are very interesting, although pre-flop card removal has been studied to some extent (by Sklansky, among others) in the past.
I'll echo TB's question about how understanding the card removal effects is strategically useful. I mean, I already know that the BB is slightly more likely to have a "playable hand" (especially one containing an ace) if it's folded to me in the SB, but my opening range already takes that into account. Indeed, if four players have already folded, my own chance of being dealt a playable hand is slightly increased.
That's cool. Just out of interest - and if you don't mind running the numbers - can you calculate how much more likely than "at random" the BB is to have AA if all five opponents in a 6-max game fold to him? I intuitively understand that AA gets the most walks in the BB (because it blocks so much of the opening ranges), but I wouldn't know where to start in calculating the probability of getting a walk with aces in that spot. It just seems to happen 100% of the time. Feel free to ignore this request. It's probably been calculated in the probability forum before anyway.

I admit that some RFI ranges are different, I should not assume all are alike. However, none of my 6-max open raises account for card removal. I open the A2s and 45s UTG. And neither is a full ring UTG open for me, but both show up at UTG+2.

I take it you are asking for the probability BB has AA given 5 opponents fold in a 6-max game. Assuming no one folds an ace, the probability is 6/C(42,2) = 0.70%, not very likely at all, but 1.54 times the probability BB gets an ace pair assuming nothing about opponents.

Alternatively, and as a fairly silly exercise, the probability can be flipped the other way around. Given BB has AA, what is the probability that BB receives a walk in a 6-max game?

Using the "tight" 6-max opening hand ranges given on the Upswing website, I performed three different simulations (each of 1 million trials):

(1) If BB has a random hand, BB receives a walk 14.3% of the time

(2) If BB has 32o, BB receives a walk 12.6% of the time

(3) If BB has AA, BB receives a walk 18.6% of the time.

Thanks for your work whosnext, those numbers (at least relatively speaking) are very close to mine in most cases, so guess it sort of was a decent estimation then.

I think it also effects RFI, since the later the position of the RFI, the more likely he is to have at least a relatively strong hand (made up of higher cards instead of lower cards) because of the card removal. That basically was the premise of this whole thread and experiment.

The weights could be applied to a RFI range, instead of a 100% frequency range which we often put into solvers. Obv for utg1 it's not gonna make much if any difference, but for BTN & SB spots, the results could start to deviate quite a bit, when we know our opponent is more likely to have AK than for example 54s. This could have several implications on cbet strategy, x/r strategy, but also preflop defend and 3bet strategy as well to only name a few.

I hope none of this affects how you play poker or understand poker theory. Expand on this quote and explain it out loud and you might learn something.

I guess you don't understand, or maybe I don't, who knows. The fact is though that software like piosolver is designed for heads up play, so it does not take into account any card removal as far as i'm aware, hence why some of the output can be skewed. not sure why you felt the need to be so denigratory though, I was just toying around. It doesn't effect your OWN RFI range (bc yeah, you shouldnt adjust your opening (the frequency of them) ranges bc card removal does this by itself I get that), sure, but it does effect the RFI range of villain the later his position, so again that was the premise of my thread.

I mean it sort of does to a certain extent should effect your own RFI range, but that's just in constructing them, bc if you are one of the first players to get dealt 54s and decide to raise the chance is higher that someone wakes up with a hand opposed to when you have AT, for obvious reasons. Anyway, that's not significant and most players already take this into account obviously by constructing their ranges, and again it was not the premise of this thread. However you could make an argument that you could also tighten up your RFI range on BTN or sb even after taking into account the initial card removal effect already, bc it's more likely the blinds will have a playable hand, but that's really another debate and not what I was after.

Yes, that's what I was getting at. Thanks for crunching the numbers!
Agreed. The BvB ranges in particular are presumably impacted even more in full ring games. I've lost count of the number of times I've woken up with kings in the SB in a tournament, done a few fist pumps, and then run them into the BB's aces!

There is no debate. You are constructing ranges incorrectly.

Ok m9.

A couple of years ago I did a simulation of how often I would have certain hands in the BB position if 9 handed and one player 'open shoves', the earlier players having open folded with my idea of current ranges.

Here is the output for a btn open allin (range 27%), the folded earlier players open ranges were 21%, 17%, 13%, 12%, 12%, 12%, using Skalansky-Chubukov ranges.



If I got the simulation right the BB only has AA 83% as often compared with dealing 2 cards off a complete fresh deck.

I picked 27% RFI at the time as this was about the average 10bb Btn shove frequency that I found from my hand histories, but although the Btn appears to open 27% this 27% will be biased toward hands like Ax due to 'bunching'. If I used a tool like Icmizer or Holdem resources to try to get the perfect defending range I think I should tighten the 27% to perhaps 25% as these tools don't take into account bunching from the open folds (as far as I know) so I should make some adjustment for this.

In real life I don't think it is possible to get very precise info. from a batch of hand histories and using a simple average frequency would not be all that useful. To complicate things further the range seen would be composed of a mix of players playing somewhere between too tight through to too loose.

One tool I know that does take into account card removal and bunching is Crev but it gets depressingly hard to fill in enough ranges for more than a few active players.

I am now not so sure my simulation is giving the correct results here, the code is a complete mess . it would be easier to start again rather than try to understand it.

I might have another go sometime for a bit of fun.

For some interest I did find an old program written by Barry Greenstein called aces.cpp and this calculates how often you get AA when the other 8 players fold to you in the BB. It gives an answer of 132 to 1, about 66% more often than normal.

I looked for a link to the program but it has disappeared.

This shows how much the 'bunching' is by the time you get to the BB with all the others folding when having reasonable opening ranges.

As previously mentioned in the thread above, I have programmed relevant opening ranges for both 6-Max and Full-Ring (9-Man) tables using ranges widely available on the internet.

Since I have the relevant opening ranges already programmed, I decided to look further into the "Pocket Aces in the Big Blind when Folded To" issue.

I ran two separate simulations each of 1 million trials last night. The first simulation is for the first five players at a 6-max table all folding to the big blind. The second simulation is for the first eight players at a 9-man table all folding to the big blind.

The table below summarizes the key results. The table gives the average number of each rank folded (10 cards folded by the first 5 on a 6-max table and 16 cards folded by the first 8 on a 9-man table).

In addition, the table also shows the average probability of the big blind being dealt pocket Aces in the two cases respectively. Note that the probabilities were derived using information on how many Aces were folded prior to the big blind and not simulated per se, so the resulting average probabilities should be very accurate.

The last row of the table converts the probability into the odds of the big blind being dealt pocket Aces when folded to in the two respective scenarios.

Rank	Avg # Folded by first 5 Players in 6-Max	Avg # Folded by first 8 Players in 9-Man
A	0.320	0.599
K	0.621	0.938
Q	0.651	1.012
J	0.680	1.040
T	0.709	1.078
9	0.718	1.179
8	0.837	1.357
7	0.875	1.405
6	0.869	1.427
5	0.897	1.456
4	0.914	1.486
3	0.942	1.501
2	0.967	1.522
.
SUM	10.000	16.000
.
Prob of Pocket Aces in BB	.5883%	.6853%
Odds of Pocket Aces in BB	169 to 1	145 to 1

Of course, the probability of being dealt pocket Aces in the big blind is 6/1326 = 0.4525% or 220 to 1 odds in general. Having a bunch of players fold prior to the big blind essentially means that the big blind is more likely to get pocket Aces since players' opening ranges are weighted toward Aces (so folding implies that each player likely did not have an Ace, leaving more of them to appear in the big blind's hand).

I see in the previous post that Barry Greenstein found that the odds of the big blind at a 9-man table has pocket Aces when folded to is 132 to 1 which is approximately 10% lower (meaning Barry found the probability is around 10% higher) than what I found above. Presumably we are using different opening ranges for the players.

Pretty cool, thanks for all your work bud

What program did you use to run these simulations?

I write my own programs from scratch.

Boss! Do you know if there are any readily available programs that give you poker odds if some cards are known?

I was trying to figure out a 6max scenario. Where the SB has 99 and the BB has AKo. It folds to the blinds, and I was trying to figure out the odds of someone folding a 9 because I thought it would be higher than one of the 4 players folding an Ace or King.