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Combinatorics – Paired Boards
Flop Textures:
- Monotone flops occur about 5% of the time
- Two-Suited flops occur about 55% of the time
- Rainbow flops occur about 39% of the time
- Flops w/ No Pairs occur about 82% of the time
- Flops w/ One Pair occur about 17% of the time
- Flops w/ Trips occur about < 1% of the time
On paired boards, we will need to be able to understand the combinatorics behind the following hand strengths.
- Quads
- Full House
- Trips
- Two Pair
Let’s start with the flop (Ah Ac Kd rainbow).
There are no flush draws and we will ignore straight draws.
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- How many combos of Quads are possible?
- For quads, there is only 1 combo. The only hand is AdAs.
- How many combos of Full Houses are possible?
- AK — There was 16 combos of AK pre-flop however with two aces and one king on the flop are acting like “blockers” by reducing the number of combos. Use the counting rule to get (2 aces left and 3 kings left) or 6 combos of the nut full-house on this board. If a third player has trip aces against you, then you only have (1 ace and 3 kings) or 3 combos to have AK.
- KK — There was 6 combos of pocket kings to start however with one on the board, there are only 3 combos left after the flop. The 3 hands are KsKc, KsKh, KcKh.
- How many combos of Trips are possible?
- AQ – A2 — For each hand of “Trips” or 3 of a kind, there are 16 combos pre-flop however we again need to reduce this because two aces are on the flop. Our hand would need one of the two remaining aces (As and Ah) and then a non-ace or non-king (Q, J, T, 9, 8, 7, 6, 5, 4, 3, 2) to qualify our hand as Trips. Make it easy by using the example of AQ has these combos (As and Qh, Qd, Qs, Qc) and (Ah and Qh, Qd, Qs, Qc) or a total of 8 combos per AX. Thus (8 x 11) = 88 combos of any hand of Trips.
- How many combos of Two Pair are possible?
- KQ — This hand represents Two Pair (aces and kings with some kicker). With one king on the flop, this reduces the starting 16 combos to only 12 combos for the range (KQ). If they call with any king then there are (12 x 11) = 132 combos for the range (KQ – K2).
- QQ — Even though QQ would be a questionable hand to play on AAK flop, there is some merit to knowing the number of combos. There are 6 combos for any pocket pair that doesn’t hit on the flop. If your opponent is calling with any pocket pair QQ – 22 then there are (6 x 11) = 66 combos of Two Pair.
- Additional Thought: consider another flop like 556 and think about how you might play these four hands of two pair (over pair AA, under pair 22, weak two pair 62, strong two pair A6) on that board.
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3 Replies
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I love that we are looking at the science of combinatorics here – it looks really complicated but honestly folks if you spend a little time practicing your ranges and internalizing the ways suited, unsuited, and paired hands present as portions of those ranges, the actual combo counting will get much much easier in real time. Like anything worth having, it WILL take some hard work and practice off the felt and in the lab. But it REALLY pays of at the table when you don’t have to start from scratch on every street.This is great stuff @ARW please keep them coming!
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Very True,
Learning about combinatorics will help you bluff more effectively, pick off bluffs, and simply make better decisions post-flop.
If you study hard enough, you may catch @SteveFredlund bluffing on the Jack high dry flops with Q9 when he has position and the betting lead. I didn’t forget Steve. Keep trying though, I love the hunt!!!
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Hey, come on now @ARW – calling me out like that! It’s why I get paid when I actually have it!! LOL
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