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Charlie Charlie 9 9 bronze badges. With weighted probabilities, you can do something like the following a real generic example : You are facing an all-in bet on the turn. DrfbagIII Dr. DrfbagIII 1, 5 5 silver badges 13 13 bronze badges.

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Pot odds Implied odds EV - Expected Value used for making advanced decisions using probability Combinations this is used for probability with multiple cards to come Game theory pretty advanced but useful for certain spots Binomial Distribution used for odds with multiple players. What you should know: Pot odds. Odds of hitting a certain amount of outs. Optional: How much equity certain backdoor draws add.

The Science Of Suited Connectors - Poker Quick Plays

Raymond Timmermans Raymond Timmermans 2, 2 2 gold badges 5 5 silver badges 18 18 bronze badges. What about fold equity and the like? Can you give me an example? I dont think it should be in this list, but I could be wrong. That's really why I'm asking. Ice heard of other mathematical principles but I can't recall them and thus can't research and learn them, but they do exist. Charlie well perhaps there are some principles, but I don't think they are too relevant for most player, for most games.

Why A Math-Driven Approach To Poker Is Powerful

Charlie go to two plus two and buy some books. The mathematics that can be used in poker is a huge field. Whatever you learn about it helps your game. The books are twenty bucks and less for the most part, and almost instantly pay for themselves.

The Mathematics of Poker

Mitchell Faas Mitchell Faas 3 3 bronze badges. Do you have any links for reading? I don't really have any online sources. For standard poker maths, Pokerstar's academy is pretty good. John William's The Compleat Strategyst is a dated but decent introduction to basics. Paparazzi Calculus is of extreme importance in moving from few to many hands. It is what drives forth the law of large numbers and without it probability theory doesn't exist.


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Though you won't be doing calculus at the table, it will be absolutely vital if you plan on developing your own mathematically sound strategies and even more so if you're searching for game theory perfect plays. In fact, at that point you'll need real analysis, which essentially builds calculus from counting numbers. I have references for game theory used in poker. Calculus and real analysis are used in game theory. Sign up or log in Sign up using Google.

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Unicorn Meta Zoo 8: What does leadership look like in our communities? If you don't want to see all the maths, skip ahead. The open circles in the decision tree represent points where different random events can happen. We'll call these random nodes. In the AKQ game, the random nodes represent John and Tom's cards: the first random node at the top represents John receiving one of the three cards and the second row of random nodes represents Tom receiving one of the remaining two cards.

The probability of each card being dealt is given along the arrows. The solid nodes are decision nodes, where one of the players must choose an action. Nobel laureate John Forbes Nash made fundamental contributions to game theory. The additional amount that John wins at the end of the game on top of what he would have won had they shown their cards straight away his ex-showdown winnings is given at the bottom of each leaf of the decision tree. By multiplying out the numbers along the arrows of each possible path through the decision tree, we can calculate the probability that the game takes this path.

So the probability that John has A and Tom has Q and bluffs is Multiplying this probability by the number of the corresponding leaf gives us the value of the game to John.

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In our example this is From this value a computer or a human for a small decision tree like this can work out an optimal strategy. The dotted lines connect decision nodes where the decision is the same — these are called information sets. As you can see, the AKQ tree has nine nodes, ten leaves and two information sets. In order to work out the amount that John can expect to win on average if the game were played many, many times in addition to what he would win at showdown his ex-showdown winnings , we have to add up the results we get from the individual paths through the decision tree.

This gives. If John calls with a fraction of his Kings greater than so that is positive, then. Similarly you can work out that if John calls less often than this, so that is negative, Tom can maximally exploit John by always bluffing. The only way that John can defend himself against exploitation is to use. This is his unexploitable calling bluff catching frequency. We can also rearrange the expression for to get. In this game each player is dealt a card from a full deck, there are three rounds of betting, and a card is dealt on the table between each round.

Introduction to the Mathematics of Poker

The players try to make the best three card hand. The decision tree for this game has about three billion nodes, and it's the largest game for which the unexploitable strategy has been calculated see the further reading list below for more information. The simplest poker variant that people actually play for money is two player Fixed Limit Texas HoldEm.

Each player starts with two cards, there are four rounds of betting and a total of five cards are dealt on the table between each of these rounds. The decision tree for this game has about 10 18 nodes that's 1 with 18 zeros after it, or a billion billion.


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Although there are computer algorithms that can calculate a good approximation to the optimal strategy, the exact solution has yet to be found. The most popular variant of poker nowadays is No Limit Texas HoldEm, in which the players can bet any amount, which adds another layer of complexity, but even this is simpler than Pot Limit Omaha, a game that is growing in popularity, in which each player is dealt four cards. In addition, poker is usually played by more than two players at a time, sometimes with as many as ten. Once you start to think about the strategic complexity of these larger games, it really becomes quite mind boggling