Combinatorics in Lay Terms (Part 2)

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Combinatorics in Lay Terms (Part 2)

NOTE: If you haven't read part one of this article yet, I strongly advise you to read it first, then return to read this one.

If in the last deal, our opponent had a huge range of potential bluff combinations on the river, then in this one he almost does not have any value combinations of raise on the turn.


How is this so?


First, imagine the frequency at which our opponent can check behind on the flop with value. At a rough estimate, it will be extremely low, but let's take it as one out of three for the purpose of performing the calculations.


Then we donk-bet on the turn, and even if our opponent has hands such as KQ or 22 set in check-behind range, would he raise right now, getting a ton of bluffs from my side? Most players will just call with these hands with a frequency that's close to one hundred percent. However, let's take it as one out of four for the purposes of the calculations.


Eventually, my opponent opened from UTG, and here is his approximate range of open hands:


It's about 15 percent of the hands. That seems like quite a lot, but nowadays most players will open all these hands from UTG, thinking they are playing them in the correct fashion.


15 percent of hands equals about 160 different possible combinations.


The value on this board is:


AK - 12 combinations

KQ - 12 combinations

22 - 3 combinations

77 - 3 combinations

= 30 combinations in total.


Then, the total percentage of contbets is important. Naturally, this will vary depending on position, the number of opponents and the board. All in all, I assume that most regular players will contbet on such a board in one hundred percent of cases, however, let's take something balanced for the calculation, say 65 percent.


We at the top decided that our opponent will check value one out of three times, even though this is quite a pessimistic assumption.


Next comes the turn. To make the calculation even more pessimistic we'll allow our opponent to always check with 66 on the flop, i.e. that they always come to the turn in full – also let's take one 67s combination.


Therefore we have 56 combinations in total left on the turn before the raise, 14 of which have value.


We also decided that our opponent will raise on the turn with a one in four frequency. Then, there are 3.5 combinations in raise range on the turn, which our opponent value raises, with 42 potential bluff combinations.


If we raise him on the turn, the upshot is we risk 196k in chips in order to win 195k. This means that we need our raise to work out in more than 50 percent of cases in order for it to become profitable.


In other words, if our opponent has 3.5 value bet combinations and 42 potential bluff combinations, we need him to bluff in more than 8 percent of cases in order to make the raise profitable.


It's obvious that the most doubtful reasoning always starts with “it is obvious”. That's why an unknown opponent is more than likely to steal the pot on the turn in the middle of a tournament – much more often than in 8 percent of cases. I also think the solution becomes relatively easy.



Here is another example – something that happened in ME during winter MRT.


Blinds are 200/400, the stake is about 40k. An Italian limped in 400 from UTG (I had 3-bet him in position and he started limping), fished, then raised 1,000. I am on CO with 55 and call 1,000. The most straightforward regular player in the world on the button with 3,300. Everyone before me folded.


I thought that my opponent's range would be JJ+ and AK, and nothing more. If that as the case, then it is profitable for me to come in at this range with a value set.


The flop A52 comes in, I check, and my opponents checks behind(!).

The turn was a Q. I bet 4,500, and he raises 12,000. I now have about 32,000 in stakes, and after a moment of speculation I fold.


The reason I folded against a straightforward opponent is that I was sure he would bet on the flop with AK, especially against me, and there are JJ-KK left in range of his check, some of which have showdown value. Then the turn is out and QQ comes, and there is not much reason for him to bluff with JJ and KK. Most people are not prone to hands on the turn with supposed showdown value into a bluff.


So, I just folded fives knowing there were not many combinations aside from QQ which allowed him to play in such a way. My opponent then showed me his QQ openly.


Finally, one last bonus for you:



Literally several minutes ago, probably the most incredible fold in all poker history happened. Phil Galfond even wrote about it in his Twitter.

“I’ve just seen the most insane fold in my life. A guy showed and folded eights on Js 8 7s 8s Ks board. After that I’ve been interviewed three times. All three times I’ve been asked about this deal, and not even once about my own self”.

It was like this: Tom Dwan opened to 32k, Smirnov called his 88 from small blind. Finally, businessman John Morgan sitting on big blind, also played call. Flop came out this way: Js 8 7s.

Smirnov bet 50k, Morgan immediately called, Dwan folded his cards. 8s came on turn.

Smirnov bet 200k, Morgan called at the same time.

According to Mikhail, Morgan looked particularly excited on turn. On river Ks were dealt and Smirnov overbet 700k into 600k pot. After a short speculation Morgan declared all-in for 3.4k in chips left in Mikhail’s stake. The latter showed for of a kind and folded it.

”It was quite an easy fold for me”, the hero of news comments afterwards on his decision. He thinks that if his opponent had had kings, he would have overcome Dwan yet on preflop, as he was active and raised a lot before. “So, kings definitely drop out”, Smirnov presumes. He also has his own perspective on jacks.

In Mikhail’s opinion, the businessmen would not “stuff” his full house with jacks on river with such steadfast confidence. Upon his words, Morgan bet all-in to him looking like it was the easiest decision in his life, and before that moment he was playing though aggressively, but utterly neat and tight.

According to Mikhail Smirnov’s logic, John Morgan had a Ts9S hand.

What about buy-in size, Smirnov does not consider it to be too significant factor for making a decision. Moreover, Morgan was seen to be upset after the end of the deal, and it gave Mikhail even more confidence about the correctness of his fold.

Phil Galfond honestly admitted that he would not be able to fold four of a kind in such a situation. “I would not be able to sleep normally afterwards. But all in all I think fold is possible here. After a good speculation one can see that John Morgan could have not so many combinations except straight flush here”, the professional shared his view.




This is the article from the second part of the book ”Online MTT Success Conception”.