One of the first and most important examples of 'poker math' that new no-limit hold'em players need to learn is how to calculate 'pot odds.' In fact, when people talk about the 'math of poker,' a lot of the time they are mostly referring to pot odds and how an understanding of them can help you decide whether to bet, raise, call, or fold.

Put most simply, pot odds represents the ratio between what you stand to gain in a hand of poker and what you have to spend in order to get it — that is, the ratio between your reward and your risk when making any given decision during a poker hand.

1. Chances are they won't make their draw (be it a straight or a flush draw, etc). Your goal is to limit their pot odds. The above examples are really just examples of bets. If you are playing no limit or pot limit hold em, the whole thinking behind betting becomes radically different.
2. Pot odds simply involves using the odds or likelihood of winning when on a drawing hand to decide whether or not to call a bet or a raise. Therefore when you are on a flush or straight draw, you will be able to work out whether or not to call or fold depending on the size of the bet you are facing by making use of pot odds.
3. As you can see from these example calculations, calling a flush draw with 2 to 1 pot odds on the flop can lead to a long term loss, if there is additional betting past the flop. Most of the time, however, there is a concept called Implied Value (which we'll get to next) that is able to help flush draws and open-ended straight draws still remain.

Calculating Pot Odds

For example, if there is $80 in the pot and your opponent bets $20, that makes a total of $100 in the middle. That $100 is the reward you can get if you're willing to risk $20 to call the bet. Pot odds are expressed as a ratio (reward-to-risk). In this case you're having to risk $20 to win $100, so your pot odds are 100-to-20, or 5-to-1.

That's the scenario poker players most often describe when talking about pot odds — that is, when facing a bet and deciding whether or not to call or fold. You add the amount of the bet to what is already in the pot to calculate the reward, the bet you need to call represents the risk, and the pot odds 'being given' to call is that reward-to-risk ratio.

Put simply, pot odds means is there enough in the pot to call a bet. The fundamental principle of playing a drawing hand in poker is that you need a pot big enough to call. When we have a flush draw or a straight draw we will make our hand on the next card almost 20% of the time (19% for flush and 17% for the straight draw).

Of course, you can also talk about pot odds after a player raises. Say you decide not just to call that $20 bet described above, but to raise to $80. That would mean your opponent has to call $60 to have a chance at winning what is now $180 in the middle — 180-to-60 or 3-to-1 pot odds.

That might seem simple enough — a little bit of addition and an easy division problem, and you can calculate pot odds.

But why bother? There are lots of reasons.

One big reason why you want to stay generally aware of what your pot odds are — which means keeping track of how big the pot is at all times and being able to compare the pot size to each bet — is that doing so helps you estimate whether or not the pot odds being offered to you are favorable or unfavorable given the situation.

Let's look at three common circumstances in no-limit hold'em in which pot odds can be helpful when making decisions.

Using Pot Odds When Playing a Drawing Hand

Say you are on a flush draw and have with the board showing . There is $120 in the pot, and your opponent has bet $60. You could call to see the river card, but are the pot odds favorable enough for you to make the call?

It's easy enough to see that the reward is $180 ($120 in the pot plus the $60 bet), and so with a $60 risk you are getting 180-to-60 or 3-to-1 pot odds. Is that good or bad?

You believe you probably have to make a flush in order to make a better hand than your opponent's, so that means you have nine outs — the nine remaining clubs — to make your hand. You can see six cards (the two in your hand plus the four on the board), leaving 46 unknown cards, so you can estimate your chance of seeing a club fall on the river to be 9 out of 46, or just over 4-to-1 against.

Compare your pot odds (3-to-1 to call) to the odds you'll make your flush (a little worse than 4-to-1 against). It's clear that calling isn't such a good choice — that the pot odds aren't favorable for calling — because over the long term calling is not a profitable play.

Let's say you were to make this call 100 times. About 20 times you'd make your flush on the river (actually a little less, but we'll round it up). You'd be risking $60 x 100 or $6,000. But your reward would only be $180 x 20 or $3,600. After making this call 100 times and winning only 20 hands, you'd have lost $2,400! (Note: we aren't considering what extra money might be won or lost after the river card, but just the profitability of this particular turn call.)

Pot odds are favorable when they are greater than the odds against making your hand. If the pot odds were 5-to-1 here, it would be a good call with it being just over 4-to-1 against making the flush. But 3-to-1 pot odds are unfavorable when drawing one card to make a flush.

Using Pot Odds to Decide Whether to Call a Preflop Raise

Pot odds can also be compared not just to a specific probability (like drawing to a flush), but also to a more general estimate of your chances in a hand.

Say for example you're playing $1/$2 no-limit hold'em and get dealt in the big blind. A player raises to $7 from the button and it folds to you.

First off... what are your pot odds here? There is $10 in the middle (the $1 small blind + the $2 big blind + the $7 raise), and you have to call $5 to stay in the hand. That's 2-to-1 pot odds.

Now, think about the prospect of playing out of position. It's a hand without a lot of potential that is almost certainly worse than whatever the player on the button who raised has. Unless you flop a couple of diamonds or perhaps trips or two pair, you're not likely to feel good about going very far with this hand. Are these 2-to-1 odds favorable?

No, they aren't. You could quantify this perhaps, noting how you with two suited cards you flop a flush draw about 11% of the time, you flop two pair about 2% of the time, and you flop trips about 1.3% of the time — that adds up to around 14% good flops, meaning it's worse than 6-to-1 against your seeing a good flop. That's just an estimate, really, but is obviously way worse than the 2-to-1 pot odds, so folding is in order.

What if a player raises to $7 from early position and five other players including the small blind call before the action reaches you in the big blind with your ? Now there's $44 in the middle and you have to pay $5 to see the flop. Those are almost 8-to-1 pot odds, which are in fact greater than the odds against your flopping something good — you might consider calling.

Using Pot Odds to Decide Whether to Call a Suspected Bluff

Pot odds can also be relevant when deciding whether or not to call what you think might be an opponent's bluff.

You've reached the river with your and the board shows . Your opponent raised before the flop and you called, and you called his bets on both the flop and turn. Now there's $100 in the middle and he's betting $50, giving you 3-to-1 pot odds to call.

You suspect strongly he could be bluffing, but you think it's possible he might have something like aces, kings, jacks, ace-queen, or king-queen and have you beat. While it's not feasible to calculate exactly the likelihood he's bluffing, you might be able to make a rough estimate — say, that he's probably bluffing at least a third of the time here.

That would make it 2-to-1 against your tens being best, making 3-to-1 pot odds favorable for you — a profitable call to make.

Conclusion

There are many other applications of pot odds in no-limit hold'em, but you can't take advantage of them until you start to become comfortable figuring out pot odds as a hand is playing out.

This is often easier to do when playing online poker, where the betting amounts and pot sizes are shown as numbers. But even when playing live, you can with practice become increasingly at ease keeping track of what's in the pot and calculating pot odds until it becomes second nature to you.

And once you do, you can then use pot odds to help direct your decision-making in a variety of contexts.

Also in this series...

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Poker odds tell you how often an event fails to occur. In other words, how many times you’re going to lose versus how many times you’re going to win. A simple to way to use poker odds to your advantage is to compare them to pot odds and determine whether you’ll be able to play profitably or not.

What Are Pot Odds?

Pot odds are used to determine whether your decision is profitable or not. In fact, most of the poker theory revolves around the basic concept of pot odds. You always have to consider risk and reward and, no matter what, you always make decisions based on poker odds to some degree.

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Calculating Pot Odds

Calculating pot odds is about learning whether it’s beneficial to risk $X in order to win the pot when your probability of winning is Y%. The amount in the pot to the amount you have to call is your pot odds. In other words, the ratio of pot to involuntary bet.

Here’s the calculation: (Pot + Bet) / Bet = Pot Odds

Example:
With $2000 in the pot, your opponent bets $1500. What are your pot odds? Here’s the calculation: ($2000 + $1500) : $1500 = 2.3 : 1

So your poker pot odds are 2.3 to 1. The easiest way to understand what this actually means is to translate the ratio into a percentage. Here are two ways of doing it:

1. The pot odds ratio is 2.3 : 1, so you calculate the sum of those numbers (in this case, 2.3 + 1 = 3.3) and then calculate 100% : 3.3 = 30,3%.
2. Another way of translating the odds ratio into a percentage is to calculate the sum of the numbers in the ratio (2.3 + 1 = 3.3) and then calculate (1 : 3.3) * 100% = 30.3%.

The percentage tells us the minimum amount of time we need to win the pot in order to turn a profit. So in the example above, you need to win at least 30.3% of those situations in order to justify your call. If you win less than 30.3%, calling would be unprofitable.

And that is the basic idea of any poker hand: to figure out whether you win the hand often enough to call, bet or raise profitably.

Example #2: You’re on the button. There’s $500 in the pot. Your opponent bets $400. You suspect your hand beats the opponent in 40% of these situations, so how to figure out whether you can call profitably?

1. ($500+$400):$400 = 2.25:1
2. 100%:(2.25+1) = 30.8%
3. You would make a profitable call in this situation.

To become a winning player, the goal should be to learn how to calculate poker odds during actual play, not just afterward (although calculating them at any point is a good start). This is surprisingly easy, as most of the time you’re going to be in one of these situations, so memorize them and you’re well on your way:

Implied Pot Odds

Implied odds are used estimate the money that can be won (or lost) during the later stages of a hand. They consider what happens on future streets and give you a more realistic idea of your odds of winning. Calculating implied poker odds requires more skill and experience because you need a good idea of how the opponent is going to react in the future. Should you predict an opponent’s actions inaccurately, you’ve also made inaccurate calculations, which in turn affects your bottom-line negatively.

Simply put, you calculate whether it’s a good decision to commit money into the pot based on how often you’re going to fold in the future (and lose no additional bets) and how often you’re going to hit your draw and make more money in the future (and win additional bets).

So the idea is the same as with pot odds except you also consider future bets. It’s easier to calculate implied poker odds in fixed limit Hold’em since you know how much your opponents can bet. In no-limit, need to correctly predict an opponent’s bet sizes. In pot-limit, you know the maximum your opponent can bet.

Example: You’re on the turn. You check and your opponent bets. There’s $30 in the pot (including your opponent’s bet) and you have $10 to call.

You get 3:1 pot odds.

If you call now, you expect the opponent to call your bet on the river each time you hit your draw, but only when the bet is max. 75% of the pot. You, on the other hand, are going to check/fold each time you fail to hit your draw.

You’re on a flush draw, and therefore you have 9 outs out of 46 remaining cards, meaning 5.1:1 odds of hitting your draw.

By simply looking at pot odds, you would either fold or raise (hoping to bluff your opponent out of the pot). But what about implied odds? As mentioned, the opponent calls your bet 100% of the time, so how much is the minimum you have to be able to bet (and get called) in order to make the play profitable?

Obviously you haven’t got very good pot odds at all. What about implied pot odds? The actual challenge when determining implied pot odds is to calculate the amount of money that you think you can likely win, and the amount of money that you absolutely have to win in order to make profit.

Pot Odds To Call Flush Draw

You have 3:1 odds and you need better than 5.1:1 odds, so you need to improve your odds by at least 2.1:1. In this case, 2.1 multiplied by the amount to call ($10) is $21. By getting your opponent to call your bet when it’s more than $21 every time you hit your draw, you would play profitably.

And how do you make sure of that the math above works? Here’s how:

0.836*10 = 0.164(30+x)
8.36 = 4.92+0.164x
-0.164x = 4.92-8.36
-0.164x = -3.44
0.164x = 3.44
x = 3.44/0.164 = 21

You multiply the amount you have to call by the times you fail to hit your draw and also multiply the amount in the pot and the money you need to win to justify a call (“x”) by the times you do hit your draw. Then you simply figure out what “x” is.

75% of the pot, which the opponent calls every time, is $30. You would make a profitable play here, since the opponent had to be willing to call $21 or more.

If the example above was hard to understand, please re-read it until you get it. The conditions were simplified for the sake of the example (the opponent always calls when bet is max. 75% of the pot, for instance–this is rarely the case).

Value of nut draws:
Nut draws are more valuable mainly because of two reasons:

1. A player might end up in situations where he and his opponent both hit their draws, and if you’re the one with the nut draw, you’ll be able to make a lot of money out of that hand.
2. It can go the other way around, and you’ll be in huge trouble if both of you hit your draws, but your opponent’s got the nut draw.

Overestimating implied odds:
Many players often learn about implied poker odds and start justifying every other situation with “plus, I’ve got implied odds” without doing the math. You need to go through the calculations 100s of times before you can make educated estimates on your implied odds, and even still, you should keep doing them until the day you stop playing poker for real money.

Determining correct implied odds relies on how well you can analyze opponents. Unless you have a good idea of what the opponent is going to do, use basic pot odds. Implied odds work only if you can make an educated prediction on how your opponent is going to react.

Can you sell it?
The success of your implied odds decisions also depends on your skills to extract money from your opponents. Players have different success rates in different situations (because of their image, for example). While one player might be able to get his opponent to commit the needed $21+, another player might not. You truly need to understand your capabilities.

Odds, Math, and Poker

Pot Odds For Flush Draw

Poker isn’t just about math but it’s all about odds. What are the odds of your opponent folding, calling or raising? What are the odds of you hitting your outs? What are the odds of your opponent hitting his outs? On some level, you’re always thinking about odds.