Ok, so I was thinking about hand combinations...

... and I got carried away.

This is long, possibly wrong, possibly not worth reading. But it might make you think.

There are 13 pocket pairs with 6 hand combinations of each and 12+11+10+9+8+7+6+5+4+3+2+1 = 78 unpaired hands with 16 combinations of each where 4 are suited and 12 are unsuited.

It is prudent already here to consider blockers. If I am trying to read my opponents hand I already have two blockers, which means the total amount of hand combinations available to my opponent is not 1326 but rather 1225. If I have a pocket pair the opponent has 5 less pocket pairs and 48*2 = 96 unpaired combinations less. Of the unpaired combinations 24 are suited.

Knowing that there are 1326 (or 1225) combinations gives an easy shortcut to translate VP$IP and PFR that are both typically listed as percentages into approximate number of hand combinations. If we use the 1326 number every percentage point is worth 13.26 hand combinations. So AA is 6 hand combinations and 6/13.26 is about 0.45%. Knowing what we know (our hand) allows us to modify that. If we have no ace in our hand 6/12.25 is about 0.49%, so not much difference there. If we happen to have an ace, even A3o or something like that there are suddenly only 3 combinations available and the chance is 3/12.25 or 0.24%. If we have two aces the chance drops to 1/12.25 or 0.08%.

This is important basic stuff, because it’s the building block of all hand range considerations. I think an interesting example is ace-rag against super tight opening ranges (not that I recommend this).

Let’s set a tight opening range as 118 hand combinations:

• All pocket pairs (78) – 66.1%
  AK (16) – 13.6%
  AQ (16) – 13.6%
  AJs (4) – 3.4%
  KQs (4) – 3.4%

Now give ourselves two cards and let’s make it A7o. Pairs lose 3 combos from each AA and 77. AK and AQ lose 4 combos each and AJs loses one combo.

This leaves our opponent with 103 hand combinations:

• All pocket pairs (72) – 69.9%
  AK (12) – 11.7%
  AQ (12) – 11.7%
  AJs (3) – 2.9%
  KQs (4) – 3.9%

Now let’s see a flop of A94r. Pairs lose 2 more combos of AA, 3 combos of 99 and 3 combos of 44. AK and AQ lose 4 combos each and AJs loses one combo.

This leaves our opponent with 86 hand combinations:

• All pocket pairs (64) – 74.4%
  AK (8) – 9.3%
  AQ (8) – 9.3%
  AJs (2) – 2.3%
  KQs (4) – 4.7%

While A7o is a rubbish hand, it’s interesting here exactly how much of a tight opening range it’s ahead of here. A7o is behind AA (1), 99 (3), 44 (3), AK (8), AQ (8) and AJs (2) – a total of 25 hand combinations out of 86 (29.1%).

If we were calling pre-flop and now OOP, checked to the villain who cbets – let’s assume he’s not a serial cbettor and put him on any hand that beats us along with any pair above 99 as well as KQ (as it is a drawing hand against our made low/medium pair range.) That’s 24 more pocket pair combinations and 4 KQs combinations putting his cbetting range at 53 hand combinations (61.6% of 86) of which we are ahead of 52.8%. 25 hand combinations are strong and will call/bet at least two streets. Of the 24 weaker made hands probably 12 combinations will be inclined to pay for one more street, as well as the 4 combos of KQs if they hit.

I think I’m eluding my point here. I think I need to investigate a K-high flop. What I wanted to get at is that in narrow ranges blockers can mean a lot.

Ok on to a K-high flop. Now the flop is K52r. Pairs lose 3 combos of K, 5 and 2. AK loses 4 more combos, KQs loses 1 combo.

This leaves our opponent with 89 hand combinations

• All pocket pairs (63) – 70.8%
  AK (8) – 9.0%
  AQ (12) – 13.5%
  AJs (3) – 3.4%
  KQs (3) – 3.4%

A7o is bad here and completely missed the flop, but it’s still interesting to judge the strength of the opponents range.

• AA - 3 combinations
  AK - 8 combinations
  KK - 3 combinations
  KQs - 3 combinations
  55 - 3 combinations
  22 - 3 combinations

That's 23 combinations of TPGK or better. If we were playing T9s there would be 3 more combinations of AA and 4 more combinations of AK. That means that by us holding a weak A the opponent when he sees a K-high flop has lost 7/30 = 23.3% of his strong made hands.

Of course if we held T9s he’d also have 4 more combos of AQ and one more combo of AJs so it’s a little bit different from the villain perspective.

With us holding A7o our opponent has hit TPGK or better 23/89 = 25.8% of the time.
With us holding T9s our opponent has hit TPGK or better 30/94 = 31.9% of the time.

If we assume the opponent here checks all lower pocket pairs and bets AQ and AJs thinking that his 20 hand combos of bluffs / semi-bluffs to 30 hand combos of value is reasonable we know that he’s actually betting 15 combos of bluffs to 23 hands of value. And because this is pretty damn close to the same proportion this leads me to believe that my second attempt at coming up with a useful example may also have failed.

My gut feeling of the factors involved tells me that there should be an argument for playing naked aces for blocker value against super-tight ranges due to the increase in fold equity we get even when (or especially when) the ace doesn’t hit on the flop. This of course assumes that the opponent is likely to fold second pairs (maybe to two barrels) and that we can read his hand. If there’s any real solidity behind this argument it may be worthwhile cold calling predictable nits with any ragged ace, planning to bluff them if their actions suggest they are not holding TPGK or better.

Looking beyond the ace-rag example for a second, this principle of blockers must be important when the action suggests that the opponent has a very narrow range and we hold a direct or indirect blocker to his range (as in the case of the A being an indirect blocker for the AK when the K hit). Very narrow ranges are normally very strong ranges and associated with a lot of money already having gone in the middle. If our opponent has tried to make a balanced decision to bet for value with 8 combinations and bluff with 4 combinations but we happen to hold blockers for the value but not the bluffing range and that makes this 4 combinations of bluffs and 2 combinations of value that’s a huge factor that we need to include in our decisions.

I guess what I'm thinking is that we should always know when we choose to play a specific hand why this particular hand will be profitable to play against this particular villain. This may be through his tendencies (calling off too weakly etc) or it could be through knowing his range so well that we can tailor our ranges to maximise overlap for blockers so we can bluff or minimise overlap so we hit when the opponent doesn't and similar. And when we decide to play pre-flop we need to know what the plan is going to be on different kinds of flops against this particular opponent.

I can't help but think that aces and kings become blockers for 3betting ranges fairly quickly as well.

I don't think A7o is generally a profitable hand to get involved with.

Great post. I am coming to the view that Ace-rag may be undervalued by players (especially in position with not a lot of limpers), and your reasoning is part of why. Obviously, the Ace reduces the probability of one of our villains having one of the hands that we are afraid of (i.e., pocket aces or ace with a better kicker). The ace is also ahead of any unpaired hand that does not have an ace in it. Further, when an ace hits the board, an opponent's holding an ace with a better kicker is actually usually one of the easier reads in all of poker, so it's not a difficult hand to fold when you are beat.

As far as the math goes, this is my favorite table on this issue:

http://www.lowlimitholdem.com/main/o...outkicked.html

Food for thought.

Nice post Erpel. Cool stuff.

Something to add: Holding A or K gives us a lot of blockers against pretty much any opening range. If that range is wide enough, and villain folds an interesting part of it to 3bets then 3betting when we hold an A or a K is more profitable. Right?

I see your study wasnt about what I just described. I think you were looking for ways to profitably exploit tight ranges, and analysing blockers. Regarding this, it is good to always have in mind while playing how villains range is made up in terms of combinations(given the blockers for any situation). With this in mind, we can find profitable spots to bluff and semi-bluff. Thoughts?

Oh this wasn't supposed to be an exhaustive discussion on any topic - it's just some notes I jotted down as I was thinking about some basic hand combination things - stuff to help me translate percentages into hand combinations and vice versa. Like this:

4 combos is almost exactly 0.3%. 40 combos is almost exactly 3%.

A single pocket pair: 6 combos (0.45%)
All 13 pocket pairs: 78 combos (5.9%)
A single unpaired hand (any suit): 16 combos (1.2%)
A single unpaired hand (suited): 4 combos (0.3%)
A single unpaired hand (offsuit): 12 combos (0.9%)
All broadways (TT+, JT+, QT+, KT+, AT+): 30 combos pairs (2.3%), 160 combos others (12.1%), total 190 combos (14.3%).
All suited aces (A2s+): 48 combos (3.6%)
All suited kings (K2s+): 44 combos because AKs doesn’t count (3.3%)
All suited connectors (AK-2A): 48 combos (3.6%)
Medium suited connectors (T9-65): 20 combos (1.5%)
Medium suited one-gappers (T8-75): 16 combos (1.2%)

There have been some postings with some groups of hands that get included in a range in different orders to arrive at vpip and pfr percentages to help with hand ranges, but to be able to use it I need to immerse my mind in the banal business of counting hand combinations some more. I need to spend some time looking at HUD stats and thinking 27% - that's 9 times 3% which is 40 combos each - so that's 360 combos. That'll include all pocket pairs for sure - 290 to go. All broadways maybe - 130 to go. All suited aces and many suited connectors, one gappers and two gappers - maybe even the occasional offsuit connector or gapper.

I just got to thinking about a few things along the way, such as the effect of blockers on narrow ranges, the fact that your cards and the board are all blockers and such.

The main point in posting it is really just as an example of what you can do when you sit down to study the game. Reading a book is all well and good, but things don't stick properly until you ask your own questions and try to come up with the answers on your own. Some of it may initially go against common wisdom (like the fact that Axo is considered a trash hand) and the challenge then is to reconcile the finding with the common wisdom - something that forums are a good next step for. But it's important to come up with our own questions, basic though they are, and try to answer them. That's an important part of improving as players.

I don't think Axo is profitable to play from all positions and against all opponents, but I think if you DO decide to play it you should know what kind of opponents to play it against, what kind of flops to make moves on and when to fold it post-flop. The only way to figure that out is to study a lot, and until we feel confident we've done that and can apply the answer reliable to situations we encounter we should fold it.

The reason premium hands are premium is that they have so much raw equity and potential that we don't need to understand what it is derived from. But not understanding what it's derived from allows us to play them poorly - mainly by overplaying them. Even with premium's we're better off if we've studied and understand exactly where the hands perform well and where they don't perform so well.

Originally Posted by LawDude

Great post. I am coming to the view that Ace-rag may be undervalued by players (especially in position with not a lot of limpers), and your reasoning is part of why. Obviously, the Ace reduces the probability of one of our villains having one of the hands that we are afraid of (i.e., pocket aces or ace with a better kicker). The ace is also ahead of any unpaired hand that does not have an ace in it. Further, when an ace hits the board, an opponent's holding an ace with a better kicker is actually usually one of the easier reads in all of poker, so it's not a difficult hand to fold when you are beat.

As far as the math goes, this is my favorite table on this issue:

http://www.lowlimitholdem.com/main/o...outkicked.html

Food for thought.

Yeah, I've been thinking how horribly underplayed some hands are recently. Like top pair. People will often feel inclined to bet or make calls on more than one street with a second pair because you can't fold all the time, but if they have top pair with a weak kicker they'll worry about being outkicked. It makes no sense. You have the ace as a blocker and top pair is better than second pair.

Similarly people are often worried about having the baby flush or the baby straight. I think a large part of it is emotional. It's emotionally burdensome to have a baby flush or a baby straight and get beaten by the bigger flush or straight. But that's just something we need to cope with. It's certainly true that when we are in that situation where we are effectively outkicked we lose a big pot, but if that causes us to play baby straight and baby flush draws cautiously we can lose much more. With draws to the effective nuts (which straights and flushes often are - at least in the sense that they are hands we are typically happy to stack off with) we need to be working on building a pot. Even if we have considerably less than 50% equity the fold equity and implied odds will make it profitable to do so. But if we decide based on losing to a bigger straight or flush one time in ten that we want to build small pots with our straights and flushes that's 1 time we lose a small pot and 9 times we win a small pot opposed to 1 time we lose a big pot, 2 times we win a big pot and 7 times we win a small pot because people fold to our aggression. Numbers grabbed out of thin air with not much more substance than that. Also I'm talking strictly river when we hit our flush as I'll just pretend the fold equity and hand equity will make the pot building exercise more or less a wash.

Not a very good or very well put together example (and proceed with caution etc) but something also to think about. Certainly better to be cbetting with a draw than with total air.

5 spades, great post. Have you tried applying it to a wider range of hands? Without actually doing the work, for flops were we hit our ace it still seems like the number of hands that beat us shrinks just as much as for a tight range, due to the affect of added hands like KT. If this is true, then fears of being outkicked might also be baseless against looser opponents. Thoughts?

Originally Posted by JKDS

5 spades, great post. Have you tried applying it to a wider range of hands? Without actually doing the work, for flops were we hit our ace it still seems like the number of hands that beat us shrinks just as much as for a tight range, due to the affect of added hands like KT. If this is true, then fears of being outkicked might also be baseless against looser opponents. Thoughts?

Thoughts?

My thoughts are that I need to do many of those kinds of analyses on different kinds of flops to build more data points from which to draw conclusions as well as ingrain an intuitive understanding of the relationship between different kinds of hand ranges and different kinds of flops. And that everyone else reading this needs to do the same for themselves.

The goal of the thread is not to provide final answers, but to provide thought and guidance. When someone posts that he spent some time in study or that you should spend some time studying they're not talking about just reading an article or a book. It's this kind of thing. I'm saying that if you're not independently asking yourself these kinds of questions and working out the answers then you are not studying poker.

Robb's post from last year about how he worked out how he wanted to continuation bet once more reminded me of this. Everyone who read the post praised it. But very few of the praises indicated that people had taken the trouble work things out similarly for themselves and it's always true that those who do the puzzling out themselves get a deeper understanding than those who just read an article on the subject.

Never be afraid to go back to basics and never be afraid to share even though you think your thought process may be flawed. Through the act of sharing a thought process you rethink the whole thing and try to put it into a logical framework that will work for someone coming to it cold - and this juggling of concept is actually highly likely to improve your own fundamental understanding of even very basic things.

I agree with you completely Erpel. Hope this thread gets some people to study this stuff(and anything else) on their own.