Posted Dec 27, 2015

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Owner and Lead Pro

Professional Cash game trainer Bart Hanson has been producing strategy content for over fifteen years. He first started on Live at the Bike! back in 2005, then moved on to host "Cash Plays" on Poker Road, then "Deuce Plays" on Deuces Cracked and then to CrushLivePoker in 2012.

In his career as a professional poker player, Bart Hanson has:

-6 WSOP Final Tables

-Over 15 years of experience at the table

-Over $1,000,000 in tournament earnings

-Multiple appearances on ESPN and Poker Night in America

-4th place finish in 2019 WSOP Monster Stack

Poker Stove in your head

Recently, over the last few months at my live training site, CrushLivePoker.com, we have been dealing heavily with the math of combinations and quick ways that we can calculate our hand’s equity in your head. Fortunately for me, I have always had the luxury of being able to do simple arithmetic quickly. Often times people are intimidated by percentages not realizing that finding 30% of 150 is as simple as multiplying 15 by 3.

PokerStove has been a very valuable tool over the last ten years as an aide in evaluating equity in ranges. But PokerStove is not a magical thing--all it does is take a weighted average of hands in a range—something that you can do in an approximate fashion if you understand combinations work and how to figure out averaging. Most importantly this math in our head is done in an approximate fashion using the rounding of numbers.

Firstly, let us go over the simple formula of finding mean or an average. This is perhaps one of the simplest concepts in the math we use in poker. Average= (S/N) where S is the sum of the numbers in a set and N is the number of terms. So if we want to find the average of 50, 30 and 10 we simply add the numbers together (50+30+10=90) and divide by 3, the number of terms. In this case we get 30.

Another element in this process of doing “Stove in our head” is learning the shortcut for equities of a hand based upon outs. Simply, if you are all-in on the flop (2 cards to be seen) you take the number of outs and multiple by 4. If you are all-in on the turn, (with one card to come) you multiple the outs by 2. So for example, if you are facing a hand that has 15 outs against you on the flop we simply multiply 15 by 4 to get 60%. That is your opponent’s equity versus our own hand. Let us say we are up against a set. We have two outs two improve our hand and our equity is 8%. So let us say that we are up against an all-in on the flop and we think that our opponent has either a 15 out draw or a set. That would mean against the 15 out draw we would have 40% equity and against the set we have 8%. So we can simply average the two, right, to get our equity needed? Wrong!!

When we are evaluating a weighted average we must take into consideration the different amount of combinations a certain hand may have. I always had difficulty with this until I learned a very simple technique form one of my subscribers. Firstly let us start with the number of combinations of a single unpaired hand you can be dealt. There are four suits of each ranking so the total number of holdings whether it is AK or 23os is 4x4=16, 12 of which are unsuited and 4 are suited. I never had trouble with this concept but always was confused because of board considerations like trying to evaluate the combinations left of AK on an A87 board. But it really is not difficult. Because one of the aces is accounted for there are three aces left and four kings. All we need to do is multiply four times three and we get the correct answer, which is twelve. What if we held AK on an A87 board and we are trying to figure out how many other combos of AK there were? Well, now two aces remain and three kings for a total of 6 combinations.

Let us say that we have AA and the board is J♦ T♥ 2♣. How many combinations of JT are possible? Well, there would be 9 as there are three of each card left. However many times with ragged connected cards people only play them if they are suited, like say 67s. So in that case you have to look at the distribution of suits on the board. If the board is 6♣ 7♣ J♥ there are three combinations of 67 suited. If the board is rainbow then there are only two combinations of 67 suited.

There are 6 combos of a way to be dealt a pair and if one card is on board 3 combinations of sets. So if you were trying to count the combinations of sets on a 973 board, for each set there are three combos so here we would be 9 combinations of sets.

Let us look at a common situation. We hold an overpair and we get a lot of heat in the form of a check raise all-in on a board of J♣ 6♦ 7♣. We think that our opponent has squarely a 15 out draw (60% equity) or a set (92% equity). How would we do this “stove in our head” situation? Firstly we need to count the combinations of each holding. On this particular board there are 3 15 out combinations in the form of 8♣9♣, 5♣4♣ and 9♣6♣. There are 9 combos of sets. One of the things that can make it very easy for us when we are trying to do this quick math in our head is that we can reduce the combos. Here we have a 9 to 3 ratio of sets to draws so instead of evaluating 12 different terms we can actually reduce this down to 3 to 1 and evaluate 4 terms. This makes it much easier when doing the math in real time. SO here we would add (8+8+8+40)/4 = 16% equity.

One of the things that people do not realize is that there are usually many more combinations of made hands than there are draws. In the previous example we did not even account for the three combos of 67 suited that our opponent had. But the point is if you want to do evaluate equity in an accurate way you have to understand this combinations work.

Let’s take a look at another example. Lets say we have 7♠ 8♠ on a board of 8♥ 7♥ 2♣. The action is such that we think we are up against a 15 out draw, a set or 2 pair.

15 out draw ( 9♥T♥, 9♥ 6♥, 5♥ 6♥) = 50% w 3 combos 2 pair (7♣ 8♣ and 7♦ 8♦) =50% w 2 combos Now more complexly we have the 5 combos of sets. 88= 0% equity (1 combo) 77= 8% equity (1 combo) 22= 16% equity (3 combos) (16+16+16+8+0) / 5 = 11% Now because we have 5 combos at 50% and 5 combos at 11% we can reduce the ratio down to 1:1 and simply average the two to evaluate our equity--(50+11)/2= 30.5%.

Lastly let’s look at a common preflop scenario, especially in NL tournaments. When evaluating your equity with a medium pocket pair, like 88 or 99, IF we assume Villain ONLY puts it in with QQ+and AK we have 18 combos of overpairs to our hand (our equity 18%) and 16 combos of AK (our equity 54%). We could treat this as a 1 to 1 ratio and simply average the two numbers (18+54) /2 = 36 to get our equity ---
OR
if we wanted to give the Villain a more conservative range we could say he only shoves 12 of the 16 combos of AK. That means he has 18 combos of (18%) and 12 combos of (.54%). SO how do we evaluate that? We reduce the ratio down to {3(18) + 2(54)} / 5 = 32.4%
The key is to sum the equities and divide by the reduced combos. This gives us our average equity. This is all that Poker Stove does. Its not magic, it's nothing special. But its good plan to learn the approximate equities of opposing hands vs your own.