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 Ah    AhKc    -AhKd-    AhKh     AhKs

 As    AsKc    -AsKd-    AsKh     AsKs

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      -Ac-       Ad       Ah       As

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Solid poker players know that putting your opponents on a range and acting according to that range is one of the most important aspects of playing winning poker. One of the most common questions that beginning players will ask of more experienced players is, "How do you put a player on a range?" The answers will vary greatly, from simply, "It just comes with a lot of experience," to more in-depth answers like, "Well, put them on a range pre-flop, then take board texture and the player's tendencies in previous hands and narrow down the range. Then look at the turn, etc, etc." It can get a little complicated when the whole process is verbalized and many people tend to get lost along the way, even if they think they "get it."

One of the things many players get stuck with is considering the number of possible combinations of hands in a given player's range. This is where blockers come in to play. For instance, if we are dealt Ace-King and are 3-bet, inexperienced players might say, "Well, he must have AA or KK and I'm beat." Whereas an experienced player, taking into consideration the player and situation, would understand that it's exactly half as likely that he has AA or KK since we hold both and Ace and a King (as opposed to if we held anything except an Ace or a King).

For those who are still struggling to really grasp the concept of hand ranges and hand combinations, there is hope! There are a lot of people who are just more visual learners and may have some difficulty completely grasping concepts solely by reading about them or having someone explain them. So for all the visual learners out there, here's a great tool for visualizing hand combinations and ranges.

We know that there are 16 possible combinations for any given non-paired hand, such as Ace-King for example. These 16 combinations can be represented on a matrix, using four columns across the top to represent the Ace in each of the four suits, and four rows across the side to represent the four possible Kings. The four-by-four matrix would look like this:

The intersection of each row with each column would represent a different suit of each hole card. So again, across the top is the Ace in our hand and down the side is the King in our hand. The matrix will look similar to this given the fact that we do not have a pocket pair and we only know the cards in our hand (we haven't seen a flop or anyone else's cards).

So in this picture, the red block would represent the hand Ace of clubs, King of spades. The green block would be Ace of diamonds, King of diamonds, and the blue block would be Ace of hearts, King of clubs.

Now what about paired hands? We might recall that there are only six possible combinations of being dealt a pocket pair. With that in mind our grid will look much different, as you can see in this figure. Imagine now we have pocket Aces. Here, the blue block represents Ace of diamonds, Ace of clubs, whereas the red block would be Ace of spades, Ace of clubs. All of the blocks that represent impossible combinations (same suit) or repeated hands have been blacked out.

So, now that you can see how the matrix works, let's look at how we can put it to use to help us visualize hand combinations.

Let's say, for instance, that you have T9 on a T94 flop. We might want to know how many combinations of overpairs there are, how many sets, and how many strong draws. We'll start with the sets (pocket pairs) and overpairs first since they'll be easier to visualize.

So AA, KK, QQ, and JJ all start with 6 combinations. You'll recall that this is represented by the first figure in the series of images shown below. Notice that we don't need to show the suits because firstly, they're always the same, but also because the actual suits don't matter. The grid itself represents the possible number of combinations.

The middle figure represents the number of possible combos of 44 that our opponent could have. We know he started with six possible ways to make 44. Now that there is a 4 on the board, we have to take one of our suits out of the matrix, thus eliminating 3 possible ways that he can hold 44 in his hand. It's easiest to visualize if you remove the one farthest to the right, leaving 3 blocks, as shown in the middle figure. This tells us that there are 3 possible combinations of 44 left.

Finally, how many combinations of TT or 99 are left? Well, it may be obvious that there's only one of each, since we hold both a 10 and a 9 and there is also one of each on the board. In our visualization we take away 2 suits, leaving one block - one possible combination each of TT and 99.

Some examples may seem more complicated, yet they will become quite simple once you understand how to manipulate the grid.

We'll look again at AK and how many possible combinations of Big Slick your opponent could hold in his hand. We know that all of the combinations make up a complete matrix, as shown in the leftmost figure. Four across the top and four across the side make for 16 possible combinations of AK pre-flop (assuming we don't hold an Ace or a King).

Let's take a look at a flop. We see A95 turned over and again want to know how many combinations of AK our opponent can hold. Now we need to remove one Ace from our matrix, because it is no longer possible that he holds that card in his hand, since it's on the board. Now our matrix looks like the figure on the right. Three Aces across the top and four Kings down the side make 12 possible combinations of AK that our opponent can hold.

Even as the hands become a little more complicated, using the matrix visualization makes it simple to "see" how many combinations of a particular hand our opponent could hold. For example, let's say we hold QJ and see a flop of QT7 and want to know how many combinations of QT our opponent may hold. We can start with what we know about his possible holdings of QT. Since we hold one Queen in our hand, we have to remove it from the matrix. This is shown in the first figure. Since we also now know there is a Queen on the flop, we need to remove another one from the possible combinations he might have. There is also a Ten on the flop, so we need to remove one "row" of Tens from his possible combinations. What we're left with is the figure on the right. Two possible Queens across the top and three possible Tens down the side, (2x3) make for six possible combinations that our opponent may hold of QT.

Now we can start to put it all together to help figure our equity against the ranges that we are putting our opponents on using the hand combinations matrix. Our starting hand is AK and we see a flop of A93 rainbow. We have a pretty good read on our opponent and know that in order for him to continue on this flop, his range is going to be something like AA, 99, 33, AQ, AJ, and A9. We know how to visualize a matrix for each of these hands. What we need to do is look at the number of combinations that dominate us and the number that we have dominated, and then make a determination as to whether or not we have enough combinations of winning hands to continue (assuming our opponent stays in the pot). So the hands that have us completely crushed are the sets AA, 99, and 33. They are represented by these figures.

To make the visualization easier, let the figures in red be the hands that have us dominated (hands we lose to) and the hands we dominate will be in green (hands we win).

Another hand that we believe could be in his range, A9, is one we would be in bad shape against. Since we hold an Ace and there's one on the board, we take 2 Aces off the matrix and one 9. Once again (as in the example above) we're left with a 2x3 matrix, or 6 possible combinations of our opponent holding A9. Since we do have some outs against this range, a King on the turn or a King on the river, we do show some equity. A draw to 3 outs on the turn or river gives us a little more than 12% equity (the actual number is slightly higher, but we'll use the rule of 4 and 2 to simplify). So we'll say in our matrix that 1 out of the six possible times our opponent holds A9 we will "suck out" and win (again, simplifying - 1/6 is actually 16.7% to win).

So far we've visualized all the hands that have us dominated. Now let's look at the two hand combinations that will continue in the hand that we are well ahead of, AQ and AJ. As a reminder, we started with a full matrix and removed two Aces (the one in our hand and the one on the board) and we're left with a two-by-four matrix, for 8 possible combinations of each hand. Now, as we did in the A9 example, our opponent is drawing to three outs twice and has somewhere around 12% equity in the pot. Since 1/8 is 12.5%, we can use that and plug it into our matrix. That shows that 1 out of 8 times that our opponent holds either AQ or AJ, they will catch up and we will lose the pot.

Now we just add up all of our "green" or winning combinations and all of our "red" or losing combinations and we can get a very good estimation of how likely we are to win the hand. There are 15 green blocks and 14 red blocks, so we could say that 15 times we win and 14 times we lose. This graphically shows us that we have a very small edge in equity versus the villains range for continuing on this board. Where's the proof? We can look at PokerStove and see what our real equity is in the given situation.

Board: Ac 9d 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 }

As you can see, PokerStove confirms our estimation. We are a slight favorite in the hand versus this particular range. This is, of course, a simplified example and the actual range you might put someone on could include many more hands and, therefore, more combinations.

It may seem complicated at first, but once you get the visualization down it becomes quite simple. You'll find that you can "see" the combinations in your head when you're trying to narrow down ranges on given boards. The next, and most important step, is to act according to the range you put your opponent on.