Separating the possible from the probable
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“Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth.” The words of Arthur Conan Doyle, as spoken by his most famous creation Sherlock Holmes, have a special resonance for poker players, as they describe a model for uncovering the truth that disregards emotion, desire, ego, and relies instead on logic alone.
Unfortunately, Doyle’s model doesn’t go quite far enough for poker players. Poker is a game of partial information, so when you’re trying to determine the contents of your opponent’s hand, there are really only a few combinations you can eliminate outright as impossible. For poker players, then, the key distinction isn’t between impossible and possible, but rather between possible and probable.
This sounds like a small distinction, and in many ways it is. There are plenty of contexts where you can even get away with using the concepts interchangeably. Poker, however, is not one of them. In fact, the ability to separate the possible from the probable is arguably one of the key skills that separates solid recreational players from professionals.
Let’s take a look at a sample hand that illustrates the concept. This hand is taken from tournament play on Full Tilt Poker. You have about 40BBs and are in the SB. The table folds to you and you complete with
From here on out, we’re going to evaluate our opponent’s hand from two perspectives: the possible and the probable. So, let’s start preflop.
Possible: All combinations that don’t involve one of our hole cards (1225).
Probable: All combinations that don’t involve one of our hole cards and aren’t premium hands (AA-JJ, AK), as most opponents will raise the SB limp with strong hands in this spot (1185).
Obviously, that’s a pretty small gap, but as the hand progresses and you get more information to work with, focusing on the probable and not the possible becomes critical. For many players, the possible becomes an enabler of bad behavior, allowing them to construct justifications for terrible plays that a consideration of the probable would never permit. Let’s return to our sample hand to see this in action.
The flop gives you top pair and has some draws:
Ks Th 5s
You bet 650 and the BB calls.
Possible: All combinations that don’t involve one of our hole cards or the flop and aren’t premium hands (1021 combos).
Probable: TT, 55, combinations with two spades, a king, a ten, a five, an open ended straight draw along with AJ, some underpairs and some complete air looking to take it away on a later street (approximately 500 combos)
Nothing earth-shattering there, but we’re starting to see the significant difference that emerges between the probable and the possible as the hand progresses and you get more information, even in a generic situation. That’s the first major flaw in thinking in terms of the possible instead of the probable: Thinking in terms of the possible doesn’t fully utilize new information. Let’s move on to the turn.
The turn is the 3d, for a board of Ks Th 5s 3d.
You bet 1250 and your opponent calls.
Now that we have new information, our possible set is based on our probable set from the previous street, and our probable set is a refinement of our possible set based on that new information. Note that there are some situations where you might want to insert additional hands into the possible set based on new information, but that’s the exception rather than the rule.
Possible: TT, 55, combinations with two spades, a king, a ten, a five, an open ended straight draw along with AJ, some underpairs and some complete air looking to take it away on a later street (approximately 500 combos).
Probable: This second bet should chase out most of the air and the weaker draws, along with most fives, the underpairs and some weaker tens. That leaves us with kings, some tens, some spade draws, QJ and a few weaker fives and draws and the occasional air (approx 325 combos).
While there’s a good amount of fuzziness involved in determining a probable range for our opponent, considering the probable instead of the possible still moves us much closer to our ultimate goal of correctly determining our opponent’s hand. Let’s return to our example.
The river pairs the board with the Tc, for a final board of Ks Th 5s 3d Tc. You check and your opponent bets 2900.
Possible: Some spade draws, QJ, a few weaker fives and missed draws, air, all kings, some tens, TT, 55 (approx 250 combos).
Probable: All kings, some tens, some spade draws, QJ and some weaker fives, missed draws and air, TT, 55 (approx 250 combos).
Those ranges appear, by all accounts, to be the same. To understand how they’re actually very different, let’s pause for a moment and think about how this hand looks to someone only considering the possible. From that point of view, your opponent could be making this bet with a very wide range that includes missed spade draws, missed straight draws, and even some smaller pairs that are now turning their hands into bluffs. That seems to suggest a relatively simple river call.
It’s important to note that there’s nothing about the above assessment that’s untrue. All of those things are possible. However, in poker what’s possible is a bit of a Rorschach test – when you focus on it, you’re likely going to only see what you want to see. In this example, even decent players are going to skew their assessment to favor a distribution that allows them to win the pot. That highlights the second critical flaw in considering the possible: you leave yourself vulnerable to making decisions based on how you’d prefer things to be rather than how things likely are.
The difference in the above ranges is, of course, in their weighting. The possible is arranged in the order of hands you’d like your opponent to have, while the probable is arranged in the order of the hands your opponent is most likely to have from a strict distributive perspective. That’s the final flaw in relying on the possible: You risk ignoring the fact that some combinations of cards are far more or less likely than others. To understand the impact distribution can have, consider the following about our example:
- Two kings remain in the deck, and 44 cards remain for the kings to combine with. That results in 87 combinations. We assumed earlier that your opponent would likely raise preflop with KK and AK, removing 9 combinations for a total of 78. Of those combinations, you beat 16 and tie with 6.
- Two tens remain in the deck, so 87 combinations.
- 11 spades remain in the deck, resulting in 55 possible spade combinations. 10 of those combinations involve a ten, so 45 missed spade draw combinations exist.
- There are 16 ways to make QJ. One of those is QJ of spades, so 15.
- There are three ways to make 55 and one way to make TT.
That leaves us with 145 combinations you’re losing to and 76 you’re beating, along with 6 ties. Even with a healthy allowance for air and missed draws, you’re still losing on this river a lot more often than you’re winning. All of the sudden, the relatively simple river call seems a lot less simple.
The point of this article isn’t really to determine whether or not you should be calling on the river – there are certainly other factors you’d need to consider before making that decision. Rather, the point is to illustrate how different a hand can look to you based solely on the manner in which you evaluate the information you receive during the hand. Considering only the possible results in decision-making bordering on pure guesswork, ignorant of new information and over-reliant on ego and emotion. Considering the probable results in guesswork as well, but an educated version steeped in deductive reasoning and an appreciation of the mathematical realities of distribution.
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