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 Geometric Visualizations of Ranges/Equity (warning: long)
Let me preface this with a few things. First, I don't really think of this as a big deal, but I was talking to someone (I forget who, sorry) about it and they suggested I post it to see what people think. Second, this is actually how I think about things because I'm such a math spaz or something and I think about a lot of things with sometimes weird geometrical models because I can do it faster that way. Third, this is aimed towards people who have trouble with the combinatorics involved in thinking about hand ranges and equity calculations. This isn't really something that will help your poker in terms of figuring out ranges or anything like that -- it's just a tool that I find useful, and maybe other people will too.
Holy crap this is long. Well, a lot of it is pictures, but I've got this at around 2000 words or something currently. Sorry for being so long, but I've tried to explain this in the best way possible where I don't sound like a total fucktard. If I sound like a fucktard anyway, then oh well, I've probably had more retarded posts than this before anyway.
Okay so here's the deal. The possible combinations of hands can be represented rather easily using a 4×4 square matrix with the four rows and four columns representing the four suits of each hole card. Then the intersection of a row and column would be one card of one suit and another card of another suit. Here is a picture I’ve put together to demonstrate this idea:
Suppose that we’re examining the hand ace-king. Here we’ll let the columns represent the four aces and the rows represent the four kings. Then the red cell represents the ace of clubs with the king of spades, the blue cell represents the ace of hearts with the king of clubs, and the green cell represents the ace of diamonds and king of diamonds. Obviously there are sixteen possible combinations, although it’s not important that we label the columns and rows in any particular order — knowing that all four aces and all four kings are available are enough for now.
This second figure represents what happens when we’re dealing with a pocket pair and there are only six possible combinations. If this grid represents two aces, then here the red cell represents the ace of clubs with the ace of spades, and the blue cell represents the ace of diamonds with the ace of clubs. All of the impossible or double squares have been filled in with black here.
Now that I’ve laid out the basic visualization for a single set of pocket cards, let's expand to something more useful: how to decide how many possible combinations of certain cards there are based on what cards are on the board and in our own pocket hand.
Suppose we have KJ on a flop of KJ5, and we want to know how many available combinations there are of certain starting hands like AA, AK, QT, JJ, or 55. If we slightly alter how we think about our visualized grids, then we can answer all five of these in a couple of seconds combined without having to think very hard at all. Since I’m always looking for ways to make my thinking more efficient, this was particularly exciting for me.
Let’s start by examining the answers for AA, JJ and 55. Later I'll get to unpaired hole cards. If you remember, our grid for pocket pairs was a sort of triangle made up of six squares in a sort of tilted pyramid, like the figure on the far left in this image:
If you notice, I’ve dropped the labeling that showed which column and row represented each suit. I’ve done this because it’s completely irrelevant to this stage of our thinking.
So we already know that for AA there are six possible combinations of cards because there aren’t any aces in our hand or on the board, and as I mentioned earlier, this is represented in the far left figure in the image. Now, when one of the possible cards comes up on the board or in our pocket hand, we have to eliminate it from our possibilities. Visually, we take away the right-hand column of the left-hand figure, and we’re left with the middle figure which indicates three possible combinations.
The application for this is evident when we try to decide how many possible combinations there are for someone to have the pocket hand 55. We know that there are only three possible combinations of fives now because one of the fives has shown up on the board, and that this is represented by the middle figure in the image above. It follows that when we take away the right-hand column again, we’re left with only one possible combination. This is what our grid looks like when we try to decide how many possible combinations of JJ there are. Once you pause for a moment, it seems fairly obvious that there is only one possible combination of cards for someone to have JJ since there is one jack on the board and another in our pocket cards.
This isn’t such a big deal when we’re trying to figure out the combinations for pocket pairs since there are only four possible scenarios, all of which are fairly trivial (6, 3, 1, or 0), but understanding how to manipulate the visual grid leads us to better understand how to manipulate our grid when dealing with unpaired pocket hands, which I'll get to now.
Whenever we want to manipulate our visual grids for pocket hands that are not paired, we will also be removing columns and rows from the grid, but in a slightly different way. The best way to illustrate this is through example, so here we go.
For whatever reason, let’s say we want to know how many combinations an opponent could have of QT. Figure 1 is what our visual grid of QT looks like preflop. Since all of the queens and all of the tens are available for use, we have a full 4×4 grid with 16 squares, which represents 16 possible combinations. Now suppose we have a flop of Q83, then Figure 2 is what our new visual grid of QT looks like. Since one of the queens is no longer available, we remove it from our visual grid, only leaving a 3×4 grid behind, or twelve combinations.
With more complicated examples, the use of visual grid representations of pocket card combinations is equally as simple. Suppose we have KQ preflop, the flop comes KJT against one opponent, and for whatever reason we would like to know how many combinations of KJ there are. Figure 3 shows what our visual grid of KJ looks like preflop since we hold one of the kings. However, figure 4 is what our visual grid of KJ looks like after the flop. Since another king has come off on the flop, we remove another column from our visual grid, and since a jack has come as well, we remove a row. This leaves a 2×3 grid for KJ, indicating that six possible combinations are left.
Now this use of grids is really just a quicker way for me to think of many hand combinations at once since I can just glance at the board and my own cards and know how many combinations of different hands there are. After you get used to thinking about this, you’ll start to know the more common spots by memory without having to use this visual device, but I think that imagining these grids is a fairly easy way to get in the habit.
So how do you really use this stuff for something more useful? You apply it to your thinking when you’re comparing your hand against ranges, like when you're trying to calculate your equity against a range of hands. Extremely complicated examples probably aren't important here, but I'll do something a little more realistic than my simple examples before.
I'm going to show some pictures and stuff, and the explanation of this example might seem sort of long, but the equity calculation can be figured to a pretty good accuracy in just a few seconds using this idea, it just takes some getting used to. Anyway, here we go.
Also, another note. In the following example, I'm going to use the color green for the units that we "win" against and red for the units that we "lose" against. Don't worry if that makes no sense, I explain it more just a little lower.
Suppose we have AKo on a flop of A93 rainbow, and we think our opponent's range is AA, 99, 33, AQ, AJ, A9. We know that we're basically drawing dead to AA/99/33, the grids for which look like the following:
So that's seven units (or hand combinations) that we lose against since we're pretty much drawing dead to them all.
Moving along, against AQ/AJ we are a pretty big favorite. Villain has three outs twice, or about 12% equity with either of these hands, and we know that the grids for each of these are 2x4 since there are two aces that we've seen (and we haven't seen any queens). So when we think of these 2x4 grids, we want to think of them as being about 12% "losing" and 88% "winning". We do this with approximation: we know that 1/8th is 12.5%, and that's close enough for us, so for each of AQ and AJ, we'll think of one of the units as being red, and the other seven as being green, as below:
Finally, against A9 we have the crap end of the stick again, since it's us that is drawing to three outs twice, putting us at somewhere around 12.5%. The grid for A9 is 2x3 since we've seen two aces and one nine, so we'll think of one of the six units as being "winning" and the other five as being "losing". (Note that 1/6 is really 16.7%, but this will be close enough for our purposes). Here is what the grid for A9 looks like:
So now that we've figured our equity with each hand (which we've put into groups that have similar equity for quick calculation), we simply count the "winning" units and compare them to the "losing" unites to have a good idea of our equity. There are 14 "losing" units, and 15 "winning" units, meaning that we have a very, very slight edge against this range. So what does PokerStove say?
Board: Ah 9c 3s
Dead:
equity win tie pots won pots tied
Hand 0: 51.111% 50.72% 00.39% 131067 999.00 { AKo }
Hand 1: 48.889% 48.50% 00.39% 125325 999.00 { AA, 99, 33, AQs-AJs, A9s, AQo-AJo, A9o }
In this example, I didn't include anything suit-specific for consideration because it's just an example of the idea in motion, but with some practice it comes fairly quickly. I pulled the range out of my ass, but you will have to decide on ranges for yourself based on what you know about your opponents and what has happened so far in the hand and hands before.
If it seems overly complicated, it's not really. It's just a visual representation of an equity calculation. In all honesty, if you're used to doing equity calculations on any sort of regular basis where your intuition for it has been built up, I doubt you'll find much use for this since when situations arise you'll already have a good feeling for where you're at.
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Originally Posted by Robb
, J
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, we're drawing to 11 
