Understanding Poker Math

Poker math is such a fundamental aspect of the game that without a basic understanding you’ll be leaking chips every time you play. Fortunately, you do not need to be a math-wiz to grasp the basics. In fact most of the principles require only basic arithmetic, and even the advanced concepts can be understood easily with the help of a poker calculator.

This article introduces the key poker math concepts, starting with the different types of odds. After this you will see the related concepts of outs and equity, which are especially important when it comes to playing draws to flushes and straights. Poker math can be used to help read your opponent’s hand, and I cover 2 ways you can do this in this article. Finally, I have introduced the more advanced concepts including NASH equilibrium and the Independent chip model.

Introducing Odds

If you peel away the layers of complexity involved in poker, you will find the concept of odds at the very heart of this game. Every time you act in poker, you are either taking odds or offering them to your opponents. If you bet $50 into a $100 pot, your opponent needs to call $50 to win $150 more, getting odds or 3-to-1. Instead of assessing his absolute chances of winning the pot, he should assess this in terms of the odds being offered. If he wins more than once for every 3 times he loses (or more than 25% of the time) then this is a profitable call. If he wins less than this, then calling will lose money over time.

Of course, assessing your winning chances is a skill to be learned in its self. What you need to be aware of is that you can’t separate poker thinking from the odds that you take on your bets or offer to opponents.

With multiple streets of betting, the immediate odds (known as ‘pot odds’) are not the only factor. There are many situations where you can win a large amount of chips on future betting rounds. The classic example is when you face a raise holding a small pair before the flop. You are probably behind at that point, though if you catch a set (3-of-a-kind) on the flop, you stand to win a lot of chips.

Calling this raise can be said to have ‘implied odds’, you will hit the set approximately 1 in 9 times (odds of 8-to-1), and so you need to ensure that your chip stack is deep enough to make more 8 times the bet you call before the flop.

The Related Concept of Outs

‘Outs’ are the cards in the deck which will improve your hand. For example, you have 2 spades, and there are two spades and one heart on the flop. You have not seen anyone else’s cards, so there are 47 unseen cards to account for. Since 4 of the 13 spades are out, there are 9 left out of 47. This gives you odds of 47-to-9 (approximately 5-to-1) of hitting a spade on the turn and if you miss this then a similar number (46-to-9) on the river.

As you gain experience at tables, counting outs will become automatic. The real skill comes in comparing the immediate odds and implied odds with the number of outs you have. Bear in mind that when the 3rd spade hits, many opponents will be wary that you have a flush. If you instead hit a straight this is far better hidden – your implied odds will often be higher.

Reading Your Opponent’s Hand

The best poker players are adept at reading their opponent’s hands. This involves estimating their starting hand range, then refining this based on the betting over several streets. It is hard to put someone on an exact hand. However, by the river many people can narrow down the possibilities significantly and even assign weightings based on the likelihood of different combinations.

Math can help with the initial ranges of hands. Some extreme examples will illustrate this point. If a player is super-tight, and only ever raises with 4% of hands, then you can assign the following range:

AA through JJ, Ace-King, Ace-Queen Suited

If they are super-loose and playing 40% of hands, then:  All pairs, all aces, most kings and suited queens and many other suited and connected cards are in their pre-flop range.

Ideally, you will work this out using a poker calculator when away from the tables. Once you have practiced assigning ranges, you can use that same calculator to show how much equity your hand has against the possible range held by your opponent. With this information, you can then use the pot-odds to decide whether you can profitably play the hand.

Card Distribution

Card distribution can help you assess the likelihood of certain hands within a range. For example, there are 6 ways you can be dealt a pair of aces based on suits. If you instead look at Ace-King, there are 16 ways of this hand being dealt based on the suits.

Applying this is a matter of assigning a range of hands – then working out the probability of each hand within that range. For example if you think your opponent would raise with Aces, Kings, Queens or Ace-King only in a certain spot, then you know there are 18 ways he could have a pair, and 16 ways he could have Ace-King. You can then work out your equity against these different hands in proportion to their likelihood, and make your odds decision based on this.

Advanced Concepts in Poker Math

Poker math gets more advanced than this, with concepts like the Independent Chip Model (which handles the changing value of tournament chips), SAGE (a system for heads-up play at low blinds) and NASH Equilibrium, which works on a ‘he knows I would play X, so he plays Y, but if he plays Y, I should adjust again and play Z’ system.

Poker is a game that will take even experts a lifetime to master. These math concepts will help you make a lot of money from players who are not using them. While you are winning, I recommend you take the time to study the more advanced concepts to increase your edge still further.