Math is Easy

Hand Combinations

A hand combination is any two cards that you are dealt, such as JT spades. This is one of my favorite math topics in poker because learning hand combinations, or simply “combinations”, will automatically make us better hand readers. It allows us to narrow Villain’s hand range based on his actions, our cards and the community cards. With that said, let’s get some facts out of the way.

This may seem intimidating, but it’s really not as hard as it looks. Let’s go over a quick example to illustrate this.

There are 4 aces left in the deck. Based on the list from above, there are exactly 6 combinations of AA. If we manually count them out, we have

That’s simple enough. Now let’s go over a slightly more complicated problem and see how things change when we hold one of the cards on the board.

We’ll need to make some adjustments because we hold one of the cards on the board.

  1. Sets–7 combos: 3 combos for a set of eight, 3 combos for a set of 9 and 1 combo for a set of kings (you have one king and there’s one king on the flop).
  2. 2 pairs – 2 combos: Since we don’t think he’s going to be calling our UTG raise with 98 offsuit, we can subtract those combos from his range. There are 4 combos of 98 suited preflop. On the flop, he can only have 2 combos for 98s (98 clubs and 98 spades). We also remove K9s and K8s from his range because TAGs don’t call UTG raises from the blinds with those hands.
  3. AA – 3 combos: Again, there can be 6 combos of AA, but since we have one of the aces in our hand, he can only have the following 3 combos.
  1. AK – 6 combos (3 aces left x 2 kings left = 6 combos): A pair + kicker usually has 12 combos. However, we hold a king and an ace so that takes away several combinations.
  2. Straight draws – 4 combos of JT suited (JT clubs, JT spades, JT diamonds, JT hearts): We can safely assume that Villain will not call an UTG raise with JT offsuit from the blinds. I left out T7s and 76s because he is less likely to call an UTG raise with those hands from the blinds in a heads-up pot. Even if he does, he will more likely check-call.

If we know he never check-raises with a straight draw here, then that decreases the chances that he has JT and increases the chances that he holds two pair or better. Thus, the correct play is to fold. Another option is to call and see what he does on the turn. If he bets again, then we’re likely behind and can fold. If he checks, then we can check behind for a cheap showdown. If you’ve seen villain overvalue his KQ and check-raise here, you can call the flop more liberally.

As you can see, hand combinations allow us to focus on villain’s hand range to figure out how likely he is to have a certain hand. This does not replace hand reading in any way. It should be a great supplement, however, to dissect your opponent’s range.

Hand Ranges

A hand range is a group of holdings that one player is likely to have based on previous history and the actions in a hand. Let’s look at some examples.

His hand range here is very strong, which means that a lot of his holdings include TT+ and AK. Versus that range, our AK is a 40 percent underdog. Since we know he’s a TAG, he is unlikely to be doing this with AQ/AJ or 99. So, we stand to lose money in the long run if he shows up with this range every time in this situation.

His range is much weaker compared to that of our previous opponent because of his maniacal, aggressive nature. He might be the type to 5-bet all-in with AJs and 55. There are tons of mediocre hands that we either dominate or are 50/50 against. He could do this with TT+ and AK, but a larger portion of his range consists of AT+, 55+ and KQ-type hands. We have 48 percent equity against this range. With the dead money out there, we have more than enough odds to call profitably.

We did not try to put Villain on a specific hand, but rather on a range of hands he might have had. This is how you should be thinking in poker. It’s often very difficult to put a player on an exact holding, but it’s much each easier to estimate his hand range.

Hand ranges also allow us to analyze a situation in a broader perspective. For example, if our maniacal Villain from the previous example 4-bets us and shows up with AA, it doesn’t mean that we made a bad read. We know that AA is only a small part of his range. Sometimes he will show up with a good hand, but most of his range consists of much weaker holdings. As you can see, you estimate a player’s hand range based on his image, tendencies and history.

Hand ranges are dynamic and can change after any given action. Consider the following example.

Unless he is going for an unlikely check-raise, we can safely assume that he rarely has KK, 99, 22, and K9+ in his range. He would have bet those hands to try to extract value from a weaker Kx, TT-QQ, or 9x. Checking allows us to control the pot size and check behind with our weak hands. So his hand range has weakened, while ours has gone up relative to his holdings.

A good generalization to keep in mind is that if you think your opponent’s range is stronger than your hand, you want to fold more often. And if you think his range is weaker than your hand, you want to call or raise more.

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