In poker, there are a lot of odds and numbers you can learn about how often various things happen. You probably know how often you'll be dealt pocket rockets. Maybe you know how often your pocket pair turns into a set on the flop.
Maybe you're not quite sure about the chances of three pair against straight draw and a flush draw on the turn in Omaha. And, frankly, how often do you use these numbers when it's your turn to act and the table bully rattles away about your private parts?
Not much calculations at the table
Truth is, most players probably do very little actual calculations in the middle of a hand. Calculations are fine in between games, to set things straight and find things out and educate the intuition.
But in practical decision making, the brain is believed to use a set of very simple rules rather than actually performing advanced probabilistic calculations.
Very few of us reason in exact numbers like "he raised me and the probability of that was 23% and now I'm only 13% to win the pot in a showdown but there's also a 9% probability that he folds if I reraise, so with pot odds of 2.7:1 and implied odds of 7:1 I have a call if and only if I can see him swallow hard."
In the midst of things that's not how the brain works, mostly. Instead, we observe a range of things that happen at the table, interpret their meaning and somehow weigh them together. Some of this is done unconsciously, but good players typically move more of this processing to the conscious levels of their minds.
Reversed probability a natural tool
The basic principle in this process is called reversed probability. It's the very basis for the part of probability theory that deals with reasoning and decisions. We'll try and explain it without going into the mathematics.
If a player raises on the flop, does this indicate that he's got a strong or weak hand? To find the probable answer, let's look at things the other way around: how likely is he to raise if he has a strong hand, and how likely if he's got a weak hand?
You don't need to give any specific numbers here, all you need to do is compare those two likelihoods (or your opinion of them). If he is more likely to raise with a strong hand than with a weak hand, then his raise indicates that he's got a strong hand.
It doesn't say so for certain, of course, but it points our mind in that direction. That's how reversed probability works.
Bringing more factors into the picture
When the opponent puts in the raise, he's leaning forward with his elbows on the table. How do you interpret this behavior? Again, let's turn things around. How likely is he to be leaning forward like that if he has a strong hand compared to if he has a weak hand?
His position is no absolute tell, many times you've seen him sit just like that and fold his hand. But still you feel that he would more often be leaning back if he had a weak hand than a strong hand, so you think his forward position is more likely with a strong hand than a weak hand. The observation of his body position adds to the probability that he's strong.
You also notice that he's wrinkled up his forehead. When you first started playing with this guy you used to take that as a sign of weakness, but now you know that that's just the way he looks when he's involved in a hand.
Turning things around again, would he be more likely to wrinkle his forehead with a strong hand or with a weak hand? Your answer is that it's just as likely with both kinds of hand. So comparing these two likelihoods gives no indication in either direction. The observation is irrelevant to your conclusion.
Observing and adjusting
The other part in our inductive reasoning is the view of things that we had before making the observation. In the theory it's called prior probability, or just prior. This is the parameter that we'll be adjusting each time we make an observation.
In the examples above, when the flop comes you already have a view of how the land lies. Depending on the opponent's behavior before the flop, you've constructed an opinion about the probable strength of his hand, much in the way described above. This opinion is your prior probability on the flop.
If the player in the examples above limped in before the flop, chances are that his hand is not very strong. If he raised or re-raised, your probability for him having a strong hand is higher.
Even if you make a couple of observations indicating a strong hand, your final judgment still depends on your prior. If his play before the flop made you almost certain that he's weak, the high likelihood of a strong hand conveyed by these two observations may not be enough to convince you that he's strong. The low prior probability holds you back.
On the other hand, if the opponent acted strong before the flop as well, the new observations might push your belief close to certainty, maybe close enough for you to make a big fold.
Testing many hypotheses
Of course, in real poker we're not only asking whether the opponent is strong or weak. All kinds of questions run through our mind: Could she have the ace? Is it possible that the river helped her? Would she really have called the flop with an inside straight draw?
But all these questions get their answers through the process of reversed probability described above. Since the idea is very simple, we can perform plenty of such deliberations in just a few seconds. It's second nature to us, and it's probably how we cope with most everyday decisions.
Involving the card probabilities
Often, the prior is based on card probabilities. Say that the flop comes 9-5-5 and you have pocket nines. You bet out and an opponent moves all in with a huge overbet. Of course, with that move you must fear him having the pocket fives. He'd be more likely to move in on that board if he has quads than he'd if he has an overpair in his hand, for example.
But when you consider his play in this hand up until now, you realize that the probability of him having exactly 5-5 is very low. That is, prior to the action on the flop, which you have now observed and are trying to analyze.
Since he raised from early position (say), you put him on AA to 99, AK, and AQ. Usually 55 is not even part of the distribution, but if you put it in, it's just one possibility out of 63. (Six each of AA-KK-QQ-JJ-TT, no 99, sixteen each of AK-AQ, and then one single 5-5 since two fives are already out.)
So, even if his all in move indicates 5-5, your prior probability says "very unlikely", and you conclude that you probably have him beaten. Even a pure bluff is probably more likely than 5-5, and then there's 9-5 and overpairs that you can beat as well. Maybe a flush draw semi-bluff is in his range.
Of course, if you have doubts about your hand's chances, you'd also bring in pot odds and implied odds into the picture, as well as the range he thinks you're playing - your percieved range.
Conclusions
Now, like we said above, it's not about the numbers. When you're in a hand, it's not so much about numbers and exact hand ranges. Very few players would actually see the figure 63 before their eyes when facing that all in.
But you certainly need to have some kind of feel for the relations involved in this kind of equations. You need to consider how often your adversary would be holding various kinds of hands when making different moves.
By doing this actively rather than just sub-consciously, you'll be more aware of our decisions and what they are based on. You may discover situations where you've been thinking illogically, and enhance your reactions in front of the moves of your opponents.
/Charlie River
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