**Practical use of Combinatorics**

For anyone not familiar with this, combinatorics is a big word for a very simple concept in poker. If we can put someone on a range of say AA, KK, QQ, AK then it looks like 75% of the time they’re going to have a made hand where in actual fact only slightly more than half of the time are they going to have better than A high if no A or K hits on the flop.

We work this out by realising that there are 16 ways we can make up AK: AcKc, AcKd, AcKh, AcKs, AdKc....etc

AA has 6 combinations: AcAd, AcAh, AcAs, AdAh, AdAs, AhAs

KK and QQ also have 6 combinations.

So instead of 3 out of 4 hands being big pocket pairs (75%), it’s actually 18 out of 34 which means they’re about 53% to have a pocket pair.

We can do the same thing with the benefit of card removal post-flop, let’s say a villain raised pre-flop and the flop hit ATT, especially the ace hits the pre-flop raiser’s range really hard. Well, how hard does it really hit?

If we assume that from whatever position it was he raised in whatever table dynamics, that his range is something like AT+, KJ+, 22+, 56s-JTs. The A’s in his range are greatly reduced by the board, AK-AJ only have 12 combinations each and AT only has 8 due to card removal (the quick calculation for this is say for AK, you multiply the number of unseen A’s by the number of unseen K’s, 3x4, even I can do those heavy calculations pretty quick).

AT 8

AJ 12

AQ 12

AK 12

KJ 16

KQ 16

22 6

33 6

44 6

55 6

66 6

77 6

88 6

99 6

TT 1

JJ 6

QQ 6

KK 6

AA 3

56s 4

67s 4

78s 4

89s 4

9Ts 2

JTs 2

Pair of aces or better: 52 combinations

KK or less: 110 combinations

What does this mean?

It means that if he cbets 100% of the time that if you’re not floating 100% of the time, you must not like money very much.

If he cbets 50% of the time... well, I have no idea how you should proceed based on combinatorics, that’s kind of the reason for this thread to hopefully create some discussion about this stuff to hopefully make this more clear.

Keep in mind that as soon as he takes an action that he wouldn’t make with 100% of his range his range has to change after that action.

**The actual hands in his range don’t change, just the frequency that they will show up.** For example the tens in his range include only 13 combinations out of 162 but if he bets into us all the way to the river and shoves those 13 combinations become a very significant portion of his range. However we never completely lose even 56s from his range but its value reduces to almost nothing, it will show up at showdown occasionally but the frequency that he decides to

bluff three streets with it is presumably low.

The nearest I can come to a conclusion about combinatorics is that it’s incredibly useful but the more actions that take place, the lower its accuracy becomes and the harder it becomes to calculate in real time. The best method seems to be estimation, in the above ATT board example, assume TT not to exist until given significant evidence that he has it, the same to a lesser degree with various other Ts. By the time you get to the river, well, I’ve made enough dumb decisions based on this stuff that to be honest I think the most you should usually apply it is to consider that unpaired pocket cards are much more likely than paired and otherwise ignore combinatorics and focus instead on the rest of your poker decision making considerations like board texture and player tendencies.

Any thoughts?