The Poker Player’s Manifesto: - Conditional Probability

In our previous discussion about knowing our enemy, we focused on creating a dossier of facts about our opponents that we can constantly reference to help us better assign ranges to them. We should be making mental notes of anything that sticks out as unique about them, and moving forward, we can reflect on these hands we have seen them play before to better help us predict how they will play hands in the future. It starts as simply as that person raised, I have seen them raise hands as bad as this hand, and I have observed them to raise about this percentage of hands in this spot, so therefore, they can have any hand as bad as this hand or as good as aces. Then everything they do after that narrows that range down to smaller and smaller sets of hands, and eventually, we get the best picture we can get about what their hand is and how our hand stacks up against their range of hands.

Poker players these days are well-versed in the concept of ranges. I even hear amateur players discuss the concept. Even if they don’t know the term range, they still are narrowing down what they think you have as the hand moves along. Fortunately, people are not talking about the concept of conditional probability. Since poker is a game of incomplete information, the game is governed by probability theory. Common knowledge, right? Since poker is a game of people though, expertise in conditional probability is what sets the great poker players apart from the good ones.

Here is how probability theory and conditional probability merges on a poker table. Player A opens for a raise, and we believe that this player is raising 15.1 percent of the time from this position. This gives that player a range of 7-7+, J-10 suited+, and K-10 offsuit+ or so. The probability of a player having a hand as strong as jacks or A-Q offsuit is 4.2 percent. Since this player has already raised, we know that they have a hand in the top 15 percent of all hands, and the probability of that hand being J-J+ or A-Q+ is 4.2/15.1, or 27.8 percent of the time. Simple probability theory tells us this, and we should react accordingly. How we go from here depends on the situation and the player, but at least we have a clear picture of what hands they may have and how often they will have the strong side of their range.

Suppose we can gather more information about this player than their opening range? Now we are getting into the realm of conditional probability. After doing the math problem, “probability of A-Q, jacks, or better given they raised,” we consider all the other information available to us. We should always be calculating the static probabilities and skewing them based on current givens. That player raised, but given it is the bubble, their range is probably skewed to the tight side or weak side, depending on their stack and the players behind them. If their range is skewed to one side, then the probability of them having a premium hand is also skewed.

The most underrated use of conditional probability at the poker table revolves around physical tells. Everybody who has sound poker fundamentals understands ranging and is probably pretty decent at it. Only the best poker players though can use conditional probability to determine that this time my opponent has the strong side of his range, and this time my opponent has the weak side. The best at conditional probability assign a completely new range to their opponents that may sometimes deviate greatly from what their range should be on paper.

Let me give you an example of how I used conditional probability to play one of my more entertaining hands to ever make it to TV. It was deep in the 2014 World Series of Poker main event. I had just acquired a tell on an opponent about the preflop strength of his hand. I basically knew he had a big hand after he looked at it, thought he had a really big hand after a player went all in before his turn, and confirmed the tell when he snap-called the shove with a monster. The very next hand, he reraised me before the flop, and that tell was not there. He didn’t feel weak, but I knew with high confidence that he was not nutted. Now, instead of his range being x because he three-bet me, his range is x given y, where y is, “his range is likely missing the best and worst hands.” That information comes strictly from tells, and, like all incomplete information in poker, tells have degrees of strength. This means that although I am pretty sure he doesn’t have aces, every once in a while I will be wrong and he has aces.

It’s good to be in a spot where I am pretty sure he doesn’t have a big hand, but I still could have a big hand. Jumping on what I perceive as an exploitable weakness, I decide to reraise. I expect my opponent to fold often, call sometimes, and shove a very small percentage of the time when I am wrong about the strength of this hand. I can get away with this based on my perception of my opponent’s perception of me at that given time. While he knows I am capable of spazzing, he knows that usually I have the goods in this spot. When he elects to call, I am okay with this because I can pretty comfortably narrow his range down to an above average pair and the occasional ace-queen, knowing that the better hands would have shoved and the worse hands would have folded. Now that I know what he has, and I know that he thinks I have a better hand, this hand of poker is going to be easy to play. I end up bluffing his jacks on the turn, I held 6-4 offsuit, good for a pair of sixes.

Without tells, I would have never made this move. Tells are a big example of dynamic information that is a product of the human element of a game that is largely mathematically based. We know that 4.2 percent of the time you will be dealt J-J/A-Q+. Conditional probability helps us know that right now is when that 4.2 percent is probably happening. The best at this are the best at poker.

Next issue we will talk just about tells. Until then, play well and have fun. ♠

Bryan Devonshire has been a professional poker player for nearly a decade and has more than $2 million in tournament earnings. Follow him on Twitter @devopoker.