... 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.