STRATEGY

Poker Math – Crunching the Numbers in Poker

Does math go with poker? You bet it does. Poker isn’t a game of chance, it’s a game of probabilities, and probabilities can be calculated with math. What does all of that mean? It means that if you use math in your poker game, the likelihood of you winning more money goes up. Are you interested now? You should be, because most of the world’s best poker players use math to determine whether to play a starting hand or to raise, call or fold in any given situation. In this article, we’ll look at some of the different ways you can use poker math to improve your game.

Why Use Math?

In the old days, professional card players relied mostly on their gut and intuition to decide whether to bluff, call, or fold. While this is definitely a valuable skill to have, modern players are much more advanced. Using statistical probabilities and software, your competition can determine the correct action to take at any point during a hand. Can you be sure you’re taking the right action? If you’re not using poker math, you’re the fish.

Dice and Calculator

Main Poker Odds to Consider

The following are the main ways that players use poker math to help them develop their game:

Pot Odds

Pot odds is a mathematical way to figure out whether to fold or call on a hand you are hoping to draw into. For example, you need another card to finish a flush or a straight. Using pot odds, you can determine whether to stay in the hand or get out based on probability. There are two ways to use pot odds, the ratio method, and the percentage method.

Ratio

This method uses the ratio of the probability you will get the card you need and the probability of you winning the pot to determine your action. Here’s a quick example: You are drawing to a flush on the turn, and you need one card to complete. There are five cards on the table, so there are 47 unknown cards. Of the unknown cards, 9 will make our flush and 38 won’t. That makes the ratio 38:9, or 4:1 reduced.

Next, you figure the ratio of your pot odds. If your opponent has just bet $20 to make the pot $100, you need to bet $20 to stay in the hand. That makes your pot odds $100:$20 or 5:1 reduced. Your card odds are 4:1, and your pot odds are 5:1. If your pot odds are greater than your card odds, you should call. If not, always fold. For this example, you should call.

Percentage

This method is very similar to the ratio method, but you use percentages instead of ratios. In short, if the percentage of the chance of you making your hand is greater than the percentage of pot odds, you should call. If the pot odds percentage is larger, you should always fold.

Getting to grips with odds is a key part of poker math.  It’s down to personal preference whether you prefer dealing in ratios, percentages or fractions.  Some people can even do it all! As long as you find a poker math method that works for you, you’re golden!

Implied Odds & Reverse Implied Odds

Calculating Implied Odds amounts to an educated guess. Your implied odds are the amount of money you expect to get from your opponents if you complete the hand you are drawing to. If you think that the rounds of betting will continue and the pot will grow substantially, you will have greater implied odds, or motivation to stay in the hand. If you think the betting will slow or stop due to the cards on the table, implied odds are low, and it’s probably best not to attempt to draw into the hand because it’s just too expensive.

Reverse Implied Odds is the amount of money you expect to lose if you draw your hand but still lose to a superior hand from an opponent. This number would be the cost of each round of betting to stay in the hand added together or your total loss for the hand. If your pot odds are very good, you can afford to stay in the hand. If your pot odds and reverse implied odds are too great, you should fold.

Expected Value

Poker Math

This term describes how much money you expect to win or lose for a play on average. Good players always make the play with the largest expected value. To calculate expected value, multiply results (in dollar value) by the probability of them happening and adding them together. For example, if you need to draw to a flush and your opponent just bet $50 making the pot $150. You need to bet $50 for a chance to win the pot. Here’s how this particular bit of poker math works out:

EV = hit flush + miss flush
= ($150 x 0.2) + (-$50 x 0.8)
= ($30) + (-$40)
= -$10

With a negative expected value, the correct action to take here is to fold, since every time you call on average, you will lose $10.

Fold Equity

Fold equity is the probability that we can force our opponent to fold when we make a bet. If an opponent folds every time you bet, you have 100% fold equity, because you win every hand. Of course, this won’t happen, so your fold equity is much lower than that. This isn’t a mathematical formula, just a way to describe the tendencies of your opponents. If you have soft opposition, you have a lot of fold equity.

Equity and Drawing Hands

Your total hand equity is: Total Equity= fold equity + hand equity
Where fold equity is the chance our opponent will fold and hand equity is the chance our hand will win at showdown. If you have a strong hand and an opponent who likes to fold, your total equity is huge. If your hand is weak and you have an opponent who likes to play out every hand, your total equity is very low. Higher equity means you should call, while lower equity means you should fold.

You can use reference cards to determine the equity of each hand you are dealt as a percentage. Using this number, you can figure out if you should be betting or folding. For example, if you have a 33% chance of your hand winning the pot, but you will be supplying 50% of the total pot amount from your bankroll, you’re wasting your time, and you should fold. If you have a 55% chance of winning a hand at the showdown and four players stay in the hand, you are only contributing 25% to the pot, so you should raise and bet aggressively.

Hand Combinations

There are over 1,300 different starting hands or combinations that you can be dealt in poker. Determining combinations is a relatively easy task, and it can be some pretty useful information. To begin, let’s look at starting hands. There are sixteen different combinations for you to receive any two cards in the deal. By comparison, there are only six combinations for you to get a pair. That means you are three times more likely to be dealt an unpaired hand every time.

There are two methods of determining possible combinations. Using paired hands or unpaired hands.

To figure out the possible combinations of an unpaired hand, multiply the number of available cards by the number of available cards. For example, if you need an Ace and a King, and there are four Aces and two Kings left in the deck, the math is 4 x 2 = 8 combinations.

To figure out combinations for a paired hand, multiply the number of available cards by the number of available cards minus one, then divide by 2.

Using information from hand combinations can be a helpful tool in determining other players’ strengths, weaknesses, and tendencies. It can give you a big advantage!

Winning Margins

Some Common Odds

If you’re new to using poker math, it can be tough to figure out where to start applying these principles and formulas. The best place to start is the deal, then look at the flop and the river. Use the following probabilities to help you determine whether to play or fold your starting hands:

Probability of…
Holding a pocket pair 17-1 (5.9% )
Being dealt pocket Aces 221-1 (0.45%)
Having Ace-King Suited in the hole 331.5-1 (0.3%)
Flopping a set with a pocket-pair 8.51-1 (11.76%)
Flopping two pair (without a pocket-pair pre-flop) 48-1 (2.02%)
Making a Flush by the river (flopped 4 to a suit) 1.9-1 (35%)
Making an open-ended straight by the river 2.2-1 (32%)
A full house or better by the river (flopped three of a kind) 2-1 (33%)

Poker Math Conclusion

All card games are really games of probability, and poker is certainly no exception. With a fixed limit of 52 cards, there are only so many outcomes for any situation. That’s why using poker math for your game can really improve your results. While relying on your gut and intuition are always going to be a valuable tool for any poker player, using solid mathematical probabilities to guide your play can improve your results long term.

Just start slow and build your poker math skills as you get better at using them. Your bankroll will thank you!  We hope you enjoyed our guide to poker math, and we look forward to hearing your comments!

Suggested Reading Material and Links

Poker Odds, Implied Odds, and Pot Size – Pokerhack.com

[amazon text=The Mathematics of Poker&asin=1886070253] – Bill Chen
This book is written for those poker players who don’t have a strong background in math. It shows you how to use quantitative analysis and game theory to improve your poker results.

Position and Realizing Equity – Pokerhack.com

[amazon text=Poker Math That Matters&asin=061539745X] – Owen Gaines
Using real world situations and quizzes to test your knowledge as you progress, this book shows you how to apply poker math to your game, specifically No Limit Texas Hold’em.

Build Your Bankroll From Zero – Pokerhack.com


I am a poker enthusiast who also enjoys sailing, rugby, and gaming. A relative newcomer to the world of online poker, I will be putting our strategy guides to the test as I start the Bankroll Challenge: how far can I get without depositing a single penny? Keep an eye out for updates.


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