PokerPAL: Ranges and Combinatorics
Thursday, 23 February 2012
Once you have worked out an opponent’s range, using software tools or otherwise, you will then need to calculate the probability of them holding a specific hand or type of hand, such as a flush draw, one pair, etc. This can be useful pre-flop and post-flop but as more variables are added, it takes longer to calculate so it’s recommended that you familiarise yourself with using combinatorics while reviewing past hands.
We can start by taking a common example of a tight player who only 3-bets with a merged 2% range. This would include the very top hold’em starting hands, A-A, K-K, Q-Q and A-K. A common misconception is to think that these four holdings have the same probability of occurring, ie 25% of the time. This leads to the idea that the opponent has a made hand – a pair – 75% of the time. This is incorrect as I will explain:
There are 16 ways to make up A-K: AdKd, AdKc, AdKh, AdKs, AcKc… etc
A-A has 6 combinations: AhAs, AhAc, AcAs, AdAh, AdAs, AdAc (similarly for Q-Q and K-K – and all pairs)
So, there are 18 total combinations that are pairs (3x6) out of a grand total of 34 combinations including the A-K. This gives a total probability of the opponent holding a pair as 18/34 = 53%. A significant drop from the 75%!
Post Flop Usage
Imagine if you raise from the button with Q-Q against an opponent in the big blind who only calls with medium to small pairs and suited connectors T-Js and below. He would 3-bet stronger hands and fold everything else. We also know that he is a passive player who only calls with flush draws.
So, the opponent calls our PFR and checks to us on a flop of Td3c4d. We C-bet and again he elects just to call. Before we see the turn, we should already have a plan of what to do on the next card and a fair idea of where we are in the hand, knowing what we do about our opponent. By using combinatorics we should be calculating the probability of which hands he could be holding and how this affects our turn play.
To do this we firstly need to calculate the total combinations of each of his pre-flop holdings. For the pairs, we shall say T-T and below, each having six combinations, giving a total of 6 x 9 = 54. For the suited connectors, we have T-J, T-9, 9-8.... 3-2 which gives nine possibilities in one suit, so total the in all suits is 9 x 4 = 36, giving us a total of 90 combinations. (You don’t need to be 100% accurate here; we are just trying to make quick estimates.)
Here comes the fun part. Given that our opponent has just called the flop C-bet, we can eliminate all the hands that missed the flop. Also, it’s fair to say that he would raise his sets and two-pair hands given the wet, draw-heavy board. (Something like 5d6d or 6d7d may also be a raising hand.) If the turn brings the 2d and he checks to us again, are we scared of the flush or do we bet again for value?
Once again, we redefine his range and calculate the probability: possible pocket pairs, 9-9, 8-8, 7-7, 6-6 and 5-5 (five combos); possible top-pair hands, JsTs, JcTc, JhTh, Ts9s, Tc9c, Th9h (six combos); possible flush-draw hands, 8d9d, 7d8d, 6d7d, 5d6d (four combos); possible straight draws, 5s6s, 5c6c, 5h6h (three combos). We will assume that gutshots and everything else would be folded on the flop. Of the original 90 combinations, only four flush draws remain and this represents 17% of his possible holdings on the turn (4/18). So it is clear that we bet again for value and even consider three streets of value on the river.
Think about what would still be included if we add in Ax suited and more broadway cards to the opponent’s prep-flop calling range.
Applying Combinatorics to Last Month’s Example
Quote from last month: “For example, you raise with AcJs from the button, the BB calls and flop comes Jc8s2c. Your C-bet is then called and the turn brings a 9d. This card rarely completes your opponent’s hand but can help him, thus giving you more reason to bet.”
You will probably find that most online players defend their big blinds a little wider than the last example, so for this one we can assume that they call with medium to low pairs, suited connectors and one-gappers, suited and unsuited broadway cards and suited aces.
Once the nine is dealt and the opponent checks to us, there will be a lot of hands that will fold to another bet, some that are still behind and will draw again and then some that will be in front and may call or raise.
Hands that would no longer continue facing a double barrel would be pocket pairs such as 6-6 and below, overcards like K-Q and second-pair hands like A-8. You rarely get any further value from these and you don’t want to give them a free card to outdraw you, so it’s OK to fold these out by betting.
What we are most interested in are the hands that we will extract value from versus the hands that have us beat. Hands that are now ahead include; 9-9 (three combos), J-9 suited (two combos), Q-T suited (four combos) and Q-Tos (12 combos), giving a total of 21 combinations.
Hands that are behind and will call a turn bet include: T-T (six combos); K-J; Q-J; J-T (24 combos); straight-draws-non-clubs; 9-T; 8-T (four combos); flush draws; KcQc; KcTc; 9cTc; 8cTc; 7c8c; 6c7c; and 7c9c (seven combos), for a total of 41 combos. Hopefully you will be able to see that there are twice as many hands that will pay extra value compared to those that will raise you and make you fold. Don’t worry about the times that some of the hands outdraw you because you can balance this against the thin value-bets you make on the river when they don’t.
When you start narrowing down your opponent’s range and probabilities of certain hands, you will find yourself playing against their range rather than concentrating on what you are actually holding. This will make it very hard for your opponent to exploit your play.
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