Introductory Mathematics of Poker

So far, we have explored various aspects of card games, learned the rules of poker and some mathematics related to starting hands in poker.

By the end of this article, you will be familiar to concepts like:

• Pot Odds
• Pot Equity
• Outs
• Counting Outs
• Expected Value (EV)

As we have repeatedly reiterated that online poker is all about informed decisions, in this article, we will teach you how to analyze the information available to you and how to make a decision.

What is Poker Math?

Poker math refers to the ability to quickly and accurately calculate odds, probabilities, and potential outcomes during a poker game. Poker math involves various aspects, such as understanding the odds of hitting a particular hand, calculating pot odds, and determining the optimal bet sizing at the poker table. This skill is crucial for making informed decisions and help you win poker games. 

Understanding Pot Odds

In simple words, pot odds means how much reward are you going to gain against the amount of chips you wager to win the pot.

Let's take a basic example:

There are 500 chips in the pot. Your opponent bets 250 in the pot. It's now your turn to act.

Let's do a quick mathematics of your risk vs. your reward.

Right now,

Pot = Initial Pot size + Your Opponent's = 500 + 250 = 750

If you make the call, the pot will be:

Pot + Your Call = 750 + 250 = 1000

This means that you have to take a risk of 250 chips to take down a pot of 1000.

Reward to Risk ratio is 1000:250 =  4:1

This 4:1 or 4 to 1 is called pot odds.

Pot Equity

Pot equity is the measure which lets you know if you can profitably take the risk (make a bet or call a bet) in order to earn the reward (the pot) or not.

In very simple terms, pot equity is your share of the pot based on the probability of you winning the pot.

Let's take another example:

We are on the button in a hand holding Ah Jh

The board (community cards) has: Kh 2c 5h 9d

The pot size is: 2000 chips

The villain opponent bets 500 chips in this pot. Now it's our turn to act.

Let's learn how to calculate equity in this scenario. In order to calculate equity, we need to understand what are outs and how to count them.

Outs and How to Count them

Outs are the cards that help us in improving our hand at any given moment in poker.

Based on the example we are discussing right now, this is how we can approach the hand.

These two are absolute facts that we have information about:

We have 2 cards of hearts in our hand and 2 of the cards are on the board.
Now we know that there are 13 cards of hearts in a deck.

Hence,

13 (total cards) - 4 (The cards we have information about) = 9 (remaining cards of hearts suit in the deck)

So if we get one of these 9 of hearts on the river, we improve our hand and make a nut flush (ace high flush), which is a strong hand.

Also, there are 3 more aces that can beat a pair of kings, which our opponent might have.

Hence, 
9 of hearts + 3 remaining aces = 12 cards

So we have a total of 12 outs, which will help us make a stronger hand than our opponent and win the pot.

How to use outs in order to calculate your pot equity?

Let's revisit the information at hand

1. We are on Turn decision and we are facing a bet of 500 in the pot containing 2000.
2. We are have 12 Out to improve our hand
3. There are 4 cards on the board + 2 in our hands so there are 46 unknown cards and one of those cards will open at the river.

As we have 12 outs, the probability of one of those cards opening on the river is:

Hand Equity = Number of  outs/remaining cards in deck = 12/46 = 0.26 = 26% ..... (1)

So based on the outs, we have a 26% equity in the pot.

Now let's calculate how much equity we really need to call this bet right now.

Pot: 2500 (2000 original post + 500 Opponent’s bet)

We need to call 500 to get a pot of 3000 (2500 + 500 of our call).

Now let's calculate our pot odds.

Pot Odds = Reward/Risk = Pot/Call Amount = 3000/500 = 6:1

Based on this, we can find our Break Even Equity.

Break Even Equity = Size of the call / Size of the pot = 500/3000 = 1/6 = 16.66% .... (2)

In order to make a profitable decision in poker, we need to keep in mind that our Hand Equity should always be more than our Break Even Equity.

When we compare (1) and (2), we realize that our Hand Equity is much higher than the minimum required equity.

Hence, we can call down this bet very profitably.

The important part to understand here is that even if we don't get our card on the river and we lose this particular pot, it still becomes profitable to be calling here every time.

You may wonder how this can be profitable if we are missing the board 74% of them which is almost 3 out 4 times!

Now let's understand the Statistical concept of Expected Value (EV)

Expected Value

Expected value is a statistical term which gives us the exact value of any decision we make over a longer period of time.

EV (Call) = Pot*(Equity) – Call amount

Continuing the Example we have been discussing

EV (Call) = 3000*26% - 500 = 780 - 500 = 280

This means that everytime if you make this call, irrespective of the results of the hand played at that time, you make a profit of 280 chips every time!

Now let's elaborate a little on how the size of the bets affect our decision making.

In this same example, let's explore the scenario where the opponent has made a bet of 1500 instead of 500.

We will calculate pot odds, break even equity and EV again (Our Hand equity still stands as no other changes have occurred apart from the size of the bet made by our opponent)

Pot Odds = Rewards/Risk = (2000 +1500 + 1500)/1500 = 5000/1500 = 3.33 : 1

Based on this, our break even equity.

Break even equity = Call/Pot = 1500/5000 = 0.33 = 33.33% … (3)

As we discussed above, Our hand equity needs to be more than our Break Even Equity to make a profitable call.

When we compare equations (1) and (3), we know that our equity is not enough to make this call and hence, we should fold our hands.

Let's calculate EV now:
EV (Call) = Pot*(Hand Equity) - Call

Continuing with the example that we have been discussing:

EV (Call) = (5000*0.26) - 1500 = 1300 - 1500 = -200

What this means is that every time you make this call, irrespective of the result of the hand, you will lose 200 chips. Hence calling is not the correct decision and we should consider folding the hand against this kind of bet sizing.

We know that it will sound very strange to you in early stages of learning, but let us assure you that as you gain experience on the tables and put these things into practice, you will not find it as strange as it sounds.

And don't be afraid that you will have to do the math in real time. There are some really cool tricks that we will teach you to quickly do these calculations along with a cheat sheet so that you can become a master at counting outs, calculating equities in real time and learn to make +EV decisions as often as possible!

Conclusion

Mastering poker mathematics is crucial for improving your poker game and increasing your chances of winning. You can become a more formidable opponent at the poker table by practicing and understanding key concepts, such as odds of hitting a hand, pot odds, and optimal bet sizing.

Must Read: Understanding Positions in Poker

Poker Math FAQs

What kind of math is used in poker?
Poker involves a significant amount of probability mathematics, which is crucial for making informed decisions and increasing your chances of winning.  

How does Poker Math Help Us?
Math is essential for making different poker strategies, decisions and maximizing your profits at the poker table.

Is poker luck or math?
While luck plays a small role in poker, mathematics is also a crucial aspect of the game. Poker math involves calculating odds, probabilities, and potential outcomes during a game.

Do professional poker players use math?
Yes, professional poker players use math extensively in their gameplay. It is an essential skill for any serious poker expert, and mastering it can significantly improve their performance at the poker table.