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Poker Mathematics

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Poker is a skill game but also a game in which the ability to read situations and adversaries is needed to take advantage in each hand you play.

It is also a game of mathematics. That means that you should be able to calculate the odds of either you or your opponent winning the hand in any situation.

The articles in the following table are the most common uses of mathematics in poker:

Title Level Strategy Utility
Pot Odds 1 - Beginner Mathematics 9/10
Implied Odds 1 - Beginner Mathematics 7/10
Poker Equity 2 - Intermediate Mathematics 7/10
Fold Equity 2 - Intermediate Mathematics 7/10

Is the use of poker mathematics the way to become a winning player?

This is a common question; it is possible to win the game without incorporating poker mathematics into the plays you make?

Using poker mathematics will not certainly lead you to be a winning player but it can help you to weight the odds and hence to improve your game giving you an advantage over players that do not use them.

Despite almost all players use mathematics in their game, they don't realize that they are using them.

The use of mathematics in poker

Where is mathematics employed in online poker strategy?

Using mathematics for drawing hands

Players use to apply Mathematics when they are on a draw such as a flush or straight draw.

If you are on a draw and your opponent makes a bet, you have to decide whether or not you should call to try and complete the draw by the next card, or fold and let your opponent win.

In this scenario a player that knows how to apply poker mathematics will always recognize whether or not to call, while a player that has no knowledge of mathematics won't be sure.

Players that are not familiar with poker odds will make an educated guess on whether or not they should call. If the bet is large, they may feel that it is too expensive to try and catch the right card, but if the bet is small they will prefer to call.

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Besides, a player that can apply poker mathematics properly will be able to act accordingly once the pot odds they are getting on the hand were worked out. Pot odds take into account the amount your opponent has bet in relation to the pot, and the probability of completing your draw to inform you about whether or not you should call or fold.

Expected value

There are other situations that incorporate mathematics in a loosely way, nevertheless still incorporate them. The situation below is an example of 'Expected Value'.

What should you do if you are on the river holding bottom pair and your opponent bets $4 into a $10 pot? If you have no concrete evidence from the way the hand played out about whether or not your opponent has a better hand than you, you can use mathematics to determine whether or not you should call.

Example of using mathematics in poker:

First, you should estimate the possibility that your opponent is bluffing and holds a worse hand than yours. Let's say that:

  • Our opponent is a little tricky and bluffs 1 time for every 3 times he has the best hand on the river.
  • This means that there is a 1 in 4 chance that we will have a better hand than our opponent.
  • Then, there is 3 in 4 chance that we do not have the best hand.
  • Thus, for every 3 times we lose, 1 time we will win (3-to-1).

So, if we call and actually have the best hand we will win $14 once, but if we call and have the worst hand we will lose $4 three times. As a result: if we call every time, we lose $12 (3 times $4) and win $14 after 4 hands. This means that we would be making a net profit of $2 if we call on the river every time, therefore we should make the call.

The previous paragraph probably didn't make a lot of sense the first time you read it, but trust us; it isn't as hard as we made it sound. A simpler way to interpret what we just said is that you should have better winning odds than the pot is giving you. In the above example we had to call $4 to win a $14 pot, which is $3.5-to-$1. Our winning odds are 3-to-1, which means we have better odds of winning than the odds in the pot.

If you're having trouble to understand the drawing hands and odds, try using the handy SPOC tool.

A few quick suggestions

A key point to remember when using mathematics in poker is that the decisions you make will not have an effect on your winnings in the short term.

Just because you work out the odds in a particular hand and make the correct call, does not mean you are going to win the hand. It also does not mean that you have made the wrong decision if you lose the hand. If you keep making the right decisions based on odds you will be making more money in the long run, so try not to let results in the short-term have an effect on you making the correct plays.

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In the beginning, learning how to use mathematics in poker can be a difficult process, but we highly recommend that you keep trying, as it will help you to become a long-term winning poker player. After a while things will start to 'click', and you understanding of pot odds and expected value will become a lot easier.

Further information on poker mathematics

If you're looking for more information on these topics (odds and everything else), read The Mathematics of Poker by Bill Chen. It's not an easy read, but you won't find more information on poker mathematics anywhere else.

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