Combinatorics in Lay Terms (Part 1)

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Combinatorics in Lay Terms (Part 1)

Most hero calls, mad bluffs and wild folds performed by the top regular players happen because of certain moments when an opponent's range mostly consists of hands of the same category. To learn this technique, you should look at all deals from a combinatorics perspective. Your excitement about the top regular player's various actions should disappear once you have finished reading this article.


When someone is making a bet, there is always an ideal ratio between value bets and bluffs. No matter the way that you play and your range whenever you call or fold, you have zero chance to win. This ratio depends upon the site of the bet in relation to the pot, and the amount of money you've put into the pot in various streets, but usually it falls inside of a limit between forty and sixty percent.



In some cases, however, this ratio can radically alter. For example, if it moves closer to the bluff side, you have a chance of winning by making a call. Conversely, if an opponent rarely bluffs, then folding is the best winning strategy (given that every dollar that is not lost is equal to a dollar won).


When an opponent makes a pre-flop raise, he has a certain range of hands (part of which are value hands, others are bluffs). When the flop is dealt the value-bluff ratio of the range changes in relation to the make up of the flop, the range itself pre-flop, and the percentage of your opponent's continuation bet. Then the turn is dealt and the ratio changes again, then the river and the same thing happens.

The common tendency is that the amount of value combinations in an opponent's range certainly decreases from pre-flop to the river. For perfect play, your opponent should maintain a proportion of bluffs to value hands.


In actuality, even the best opponents are often incapable of maintaining this ratio in certain spots. This makes it our main goal to monitor such situations, when either the amount of value bets in our opponent's range is significantly decreasing, or the amount of bluffs stays broadly the same.


For an illustration of this, let's examine a recent deal from XA at NL100 against a good, aggressive regular player:



At the moment of the deal my opponent's 3-bet was around 20 percent. I thought he would contbet with almost all his range on the flop. A queen on the turn would be the best card for a bluff, as it makes almost my entire range reasonably weaker (all my hands that had been top pair became, two pair, and two pair became three, and so on). That's why I thought the likelihood of a bluff on a queen would be very high.

The river was a seven, and my opponent went all-in. Let's consider the combinatorics:


What can he value bet with on the river?


It's unlikely to value bet in this way with hands such as K9 or TT, as he could have had a value bet on the flop and turn. Even a value bet on the river looks doubtful with QJ, so let's look at all hands starting from KQ+


  • KQ (12 combinations)
  • AQ (9 combinations)
  • KK (6 combinations)
  • АА (3 combinations)
  • Q9 (6 combinations)
  • 22 (3 combinations)
  • 33 (3 combinations)
  • 77 (3 combinations)
  • 99 (1 combination)
  • QQ (3 combinations)


In total: 49 combinations on value.


Now let’s look at the bluffs.


My opponent's 3-bet pre-flop is about 20 percent, which means he has about 225 hand combinations on the flop. 49 of these are value hands by the rover, leaving 176 possible combinations.


On the flop he is likely to contbet with all bluffs. On the turn, bluff frequency will remain very high, as it is on the turn were it's best to bluff, so let's estimate this frequency as 80 percent. Then this means there are 140 combinations remaining by the time of the river.


Some of these hands, such as QJ or TT or 98 will be of some value, and there is no sense for him to raise by an overbet if he has them. We cannot be sure of the precise 3-bet range of the opponent so we can count the exact number of combinations, because he will 3-bet from time to time, for example with QJ. Sometimes he will call as well, therefore it goes that not all twelve QJ combinations will reach the river – probably around five or six. The same thing happens with all the other hands.

I don't know how to count the number of combinations that have some value on the river, meaning that it was foolish of him to make an overbet raise. I think there will be about 40 combinations.


So, the total amount of his combinations:

  • 49 combinations of strong value
  • 40 combinations of small value that the opponent will not overbet raise 
  • 100 potential bluff combinations


Judging by the pot odds, we need 36 percent on the river for a zero call. Knowing that, we can count the minimum amount of bluff combinations in his range to make a zero call.


X is the minimal amount of bluff combinations that are necessary for our zero call.


X/(49+X) = 36%

X = 27


In other words, in our opponent's raise range we need at least 27 combinations that we will beat.


From the calculation above we know that he has 100 potential bluff combinations on the river. This means that we need him to bluff with a higher frequency of 1-in-4 in order to make a profitable call.


It's obvious that a good, aggressive regular player will bluff here much more often than 1-in-4. That's why it looks like quite an easy call to make.


To put it another way, we need to call on the river because too many potential bluff combinations get in the opponent's range from pre-flop until the river, and for a profitable call we simply need a very small frequency of such bluffs from his side.


If our opponent maintained the same proportion of value bets and bluffs in his range on all streets, for example, made a bluff contbet on the flop in only 50 percent of cases, and kept bluffing on the turn also in 50 percent of cases, this call would not be so easy to determine. Then, only between thirty and forty potential bluff combinations would reach the river, and we would need a higher frequency of bluffs – say, 70 percent – to make a profitable call.


In reality, even a very good regular player will not be able to maintain a proportional amount of value bets and bluffs in his range on this board, so the call will be easy enough:



I will give one further example where the opponent's range has clearly moved to the bluff's side, so that you can fully understand what I am talking about.

Next article: Combinatorics in Lay Terms (Part 2)