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Poker Hand Distributions

By Tony Guerrera

The lipstick camera profoundly influenced the world of poker. People watching poker on television can see the drama unfold before them with an information set that’s 100% complete. Well, at least it’s 100% complete with respect to the hole cards everyone is holding – the information set regarding the very important (but often over-looked) metagame is far from complete. If you’re used to watching poker on television, it’s easy to be trapped into thinking that poker is a game of match-ups happening between specific hands. It’s tempting to watch a guy call all-in preflop with QQ against a guy with AA and say “he’s only twenty percent to win; that was a horrible call.”

When you’re actually engaged in battle, it’s usually impossible to know precisely what your opponents are holding (and reciprocally, it’s often impossible for your opponents to know precisely what you’re holding). It’s also usually impossible to know precisely what your opponents will do (just as, reciprocally, it’s often impossible for your opponents to know precisely what you’ll do). Really, poker is a game of distributions: hand distributions and action distributions.

Hand Distributions

A hand distribution is the set of all possible hole cards that an opponent might hold. If an opponent raises from under-the-gun at a fullhanded table, that player might be on a distribution looking something like [AA-TT, AK-AQ]. If an opponent limps in late position after a few callers, the flop is 9heart6heart2diamond, and he raises after a few people have bet, his hand distribution might be something like [TT, 99, 66, 22, A9, AheartXheart].

Hand distributions can be as narrow as one specific set of hole cards (e.g. a very tight player 4-bets you preflop, announcing his AA to the world). Or they can be as wide as “any two.” In practice, they are usually somewhere in the spectrum in between. Every action an opponent makes corresponds to an assignable hand distribution. The closer your assigned hand distribution is to an opponent’s actual hand distribution, the better your results will be. Therefore, how can we put our opponents on accurate hand distributions?

It all starts from the first hand. When you first sit at a table, it’s tempting to think that you’re sitting with a bunch of clean slates…players you know nothing about. However, you actually have lots of knowledge to draw upon. You have all the hands of poker you’ve played in your life. From all the players you’ve played before, you can come to the table armed with a default player profile, where certain betting patterns and tells correspond to certain hand distributions.

If you’re really experienced, you might come to the table armed with multiple default profiles that you can assign based on certain stereotypes. For example, if I go to the casino, I assign different default profiles based on a wide range of parameters including talkativeness, age, and what I refer to as the ADD (attention deficit disorder) factor (is the player reading, watching TV, or ??? while playing). That’s right! Before I play a single hand, I have a rough idea of what I’m about to face.

Though default profiles are important when you first sit at a table, you need to study your opponents closely, so you can make appropriate modifications based on specific, observed behaviors. Suppose that the default hand distribution you assign to under-the-gun (UTG) raisers is [AA-JJ, AK], but you see that a player raised UTG and showed down KJ. Now, you can say that his UTG raising distribution is [AA-JJ, AK, KJ]. But really, since a raise with KJ indicates that he’ll probably also raise with AQ, AJ, and KQ, we can extrapolate and say that his UTG raising distribution is [AA-JJ, AK-AJ, KQ-KJ], and probably something more like [AA-99, AK-AJ, KQ-KJ]. And if we want to extrapolate and say that players are typically more aggressive in late position, we can say that when he raises in late position in an unraised pot, he is probably on a distribution looking something like [AA-77, AK-A9, KQ-KT, QJ-QT, JT]. (Of course, you want to verify this read with actual information gathered on this player’s late position play…stuff like showdowns and preflop raising frequency in unraised pots)

Note that it’s much easier to adjust reads from tight to loose than it is to adjust them from loose to tight. Suppose you put a UTG raiser on [AA-JJ, AK], and you see him raise UTG with AA. Does this mean that [AA-JJ, AK] is the correct UTG raising distribution? Who knows! To adjust a read from loose to tight, you generally need to play a large quantity of hands. A player raising with [AA-JJ, AK] is raising about 3.02% of his hands from UTG (about 1 out of every 33 UTG hands which, at a 9-handed table, will take 297 hands to observe). If he’s only raising [AA-QQ], then he’s raising about 1.36% of his hands from UTG (about 1 out of every 74 UTG hands which, at a 9-handed table, will take 666 hands to observe). The only way to detect this difference is to play tons of hands, and even at that point, you’d need to be playing online with the help of a heads-up display like Poker Ace HUD.

Action Distributions

An action distribution refers to all the different possibilities for what an opponent will do. For example, if you bet into an opponent, perhaps he’ll fold 50% of the time, call 30% of the time, and raise 20% of the time. When you are figuring out what your best course of action is, considering your opponents’ hand distributions isn’t enough. You also need to take their action distributions into account.

Suppose you’re at the river, and your opponent checks to you. And further suppose that you have the nuts. At this point, it’s obvious that you have your opponents beaten. However, knowing your opponents’ hand distribution relative to your hand isn’t enough to determine your best course of action. To determine how much you should bet, you need to know how your opponent will react to each possible bet you can make. If you want to optimize your profits, you should only consider acting after seriously considering all possible bets.

Now suppose that you have an overpair on a draw-heavy board. You bet out-of-position (OOP) on the flop and the turn, and your opponent calls both bets in position. The river doesn’t help any suspected draws, and you largely suspect that your opponent has a missed draw. If you bet into him, he’ll fold. But if you check to him, he’ll make a pot-sized bluff. The most profitable action here is to check and induce the pot-sized bluff. Again, knowing how your hand stacks up relative to an opponent’s hand distribution isn’t enough. You need to know an opponent’s action distribution to choose the most profitable line of play.

Putting It All Together

Most players try to figure out what their opponents have. Good players are aware of hand distributions. Great players go the extra mile and consider the combination of their opponents’ hand and action distributions. With hand distributions and action distributions properly defined, we can move on and define straightforward players and tricky players. Straightforward players have action distributions that correspond very closely with their hand distributions. Tricky players have action distributions that don’t necessarily correlate with their hole cards.

Regardless of whether your opponents are straightforward or tricky, you’ll come out on top if you keep on top of the distributions.

Tony Guerrera is the author of Killer Poker By The Numbers and co-author of Killer Poker Shorthanded (with John Vorhaus)


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