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January 2 — “Poker Stove in your head”

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. Oftentimes, people are intimidated by percentages not realizing that finding 30 percent 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 how 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.

First, let’s go over the simple formula of finding the 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 (two cards to come), you take the number of outs and multiple by four. If you are all-in on the turn, (with one card to come) you multiple the outs by two. 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 percent. That is your opponent’s equity versus our own hand. Let us say we are up against a set. We have two outs to improve our hand and our equity is eight percent. So let’s 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 percent equity and against the set we have eight percent equity. So we can simply average the two to get our equity needed, right? 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 from one of my subscribers. First, let’s 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 A-K or 3-2 offsuit, is 4 × 4=16, 12 of which are unsuited and four which 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 A-K on an A-8-7 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 A-K on an A-8-7 board and we are trying to figure out how many other combos of A-K there were? Well, now two aces remain and three kings for a total of six combinations.

Let’s say that we have A-A and the board is J 10 2. How many combinations of J-10 are possible? Well, there would be nine, 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, 7-6 so, in that case, you have to look at the distribution of suits on the board. If the board is J 7 6, there are three combinations of 7-6 suited. If the board is rainbow, then there are only two combinations of 7-6 suited.

There are nine combos of a way to be dealt a pair and, if one card is on board, three combinations of sets. So if you were trying to count the combinations of sets on a 9-7-3 board, for each set there are three combos, so here we would have nine combinations of sets.

Let’s 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 7 6. We think that our opponent has squarely a 15-out draw (60 percent equity) or a set (92 percent 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 three 15-out combinations in the form of 9 8, 5 4 and 9 6. There are nine 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 9:3 ratio of sets to draws, so instead of evaluating 12 different terms we can actually reduce this down to 3:1 and evaluate four terms. This makes it much easier when doing the math in real time. So here we would add (8+8+8+40)/4 = 16 percent 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 7-6 suited that our opponent had. But the point is, if you want to 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 8 7 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 ( 10 9, 9 6, 5 6) = 50 percent w/ three combos

2 pair (8 7 and 8 7) =50 percent w/ two combos

Now, more complexly, we have the five combos of sets.

88= 0 percent equity (one combo)
77= 8 percent equity (one combo)
22= 16 percent equity (three combos)
(16+16+16+8+0) / 5 = 11 percent

Now because we have five combos at 50 percent and five combos at 11 percent, we can reduce the ratio down to 1:1 and simply average the two to evaluate our equity —(50+11)/2= 30.5 percent.

Lastly, let’s look at a common preflop scenario, especially in no-limit tournaments. When evaluating your equity with a medium pocket pair, like 8-8 or 9-9, if we assume the Villain only puts it in with Q-Q+ and A-K, we have 18 combos of overpairs to our hand (our equity is 18 percent) and 16 combos of A-K (our equity is 54 percent). We could treat this as a 1:1 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 A-K. That means he has 18 combos of 18 percent and 12 combos of 54 percent. So how do we evaluate that? We reduce the ratio down to {3(18) + 2(54)} / 5 = 32.4 percent

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. It’s not magic, it’s nothing special, but it’s a good plan to learn the approximate equities of opposing hands vs your own. ♠

Follow Bart for daily strategy tips on Twitter @CrushLivePoker and @BartHanson. Check out his poker training site exclusively made for live cash game play at CrushLivePoker.com where he produces weekly podcasts and live training videos.