Poker Odds Calculation for Texas Hold'em

Learning how to properly count your outs and calculate poker odds is a fundamental requirement of Texas Hold'em. While the math used to calculate odds might sound scary and over the head of a new player, it really isn't as hard as it looks. In fact, most of the time, you only need to know elementary arithmetic to figure out your odds.

Why are Odds Important in Poker

Why are poker odds so important anyhow? Knowing odds is important because it gives you an idea when you are in a good or bad situation. To illustrate:

Let's say you and a friend are flipping a quarter and he gives you 1:1 odds that the next flip will land on heads. You already know that it will land on heads 50% of the time, and it'll land on tails the rest of the time. In this case, he's giving you an even bet, because nobody has a statistical advantage.

Instead, let's say your friend just won $500 playing poker online and is on a lucky streak. He offers you 2:1 odds that the next coin flip will be heads. Would you take this bet? Hopefully you would, because the chances of heads or tails coming up are still 1:1, while he's paying you at the 2:1 rate. Your friend is hoping to ride his luck a little longer, but if he gambles with you long enough, he'll be losing his shirt with these kinds of odds.

The above example is a simplified version of what goes on in Texas Hold'em all the time. This is be summed up in this short principle:

In poker, there are two types of players. The first group are players who take bad odds in hopes of getting lucky. The second group are players who cash in on the good odds that are left by the first group.

Hand Odds and Poker Odds

Hand odds are your chances of making a hand in Texas Hold'em poker. For example: if you hold two hearts and there are two hearts on the flop, your hand odds for making a flush are about 2 to 1. This means that for approximately every 3 times you play this hand, you can expect to hit your flush one of those times. If your hand odds were 3 to 1, then you would expect to hit your hand 1 out of every 4 times.

Odds are given below for hitting a draw by the river with a given number of outs after the flop and turn, and examples of draws with specified numbers of outs are given.

Example: if you hold [22] and the flop does not contain a [2], the odds of hitting a [2] on the turn is 22:1 (4%). If the turn is also not a [2], the odds of hitting it on the river are again 22:1 (4%). However, the combined odds of hitting a [2] on the turn or river is 12:1 (8%). For mathematical reasons, only use combined odds (two card odds) when you are in a possible all-in situation.

Keyword Definitions

• Backdoor: A straight or flush draw where you need two cards to help your hand out.
  You have [A K]. Flop shows [T 2 5]. You need both a [J] and [Q] for a straight.
  
• Overcard Draw: When you have a card above the flop.
  You have [A 3]. Flop shows [K 5 2]. You need a [A] overcard to make top pair. 3 total outs.
  
• Inside Straight Draw (aka 'Gutshot'): When you have one way to complete a straight.
  You have [J T]. Flop shows [A K 5]. You need a [Q] to complete your straight. 4 total outs.
  
• Open Straight Draw: When you have two ways to complete a straight.
  You have [5 6]. Flop shows [7 8 A]. You need a [4] or [9] to complete your straight. 8 total outs.
  
• Flush Draw: Having two cards to a suit with two suits already on the flop.
  You have [A♥ K♥]. Flop shows [7♥ 8♥ J♣]. You need any heart to make a flush. 9 total outs.

To calculate your hand odds, you first need to know how many outs your hand has. An out is defined as a card in the deck that helps you make your hand. If you hold [A♠ K♠] and there are two spades on the flop, there are 9 more spades in the deck (since there are 13 cards of each suit). This means you have 9 outs to complete your flush - but not necessarily the best hand! Usually you want your outs to count toward a nut (best hand) draw, but this is not always possible.

The quick amongst you might be wondering "But what if someone else is holding a spade, doesn't that decrease my number of outs?". The answer is yes (and no!). If you know for sure that someone else is holding a spade, then you will have to count that against your total number of outs. However, in most situations you do not know what your opponents hold, so you can only calculate odds with the knowledge that is available to you. That knowledge is your pocket cards and the cards on the table. So, in essence, you are doing the calculations as if you were the only person at the table - in that case, there are 9 spades left in the deck.

When calculating outs, it's also important not to overcount your odds. An example would be a flush draw in addition to an open straight draw.

Example: You hold [J♦ T♦] and the board shows [8♦ Q♦ K♠]. A Nine or Ace gives you a straight (8 outs), while any diamond gives you the flush (9 outs). However, there is an [A♦] and a [9♦], so you don't want to count these twice toward your straight draw and flush draw. The true number of outs is actually 15 (8 outs + 9 outs - 2 outs) instead of 17 (8 outs + 9 outs).

In addition to this, sometimes an out for you isn't really a true out. Let's say that you are chasing an open ended straight draw with two of one suit on the table. In this situation, you would normally have 8 total outs to hit your straight, but 2 of those outs will result in three to a suit on the table. This makes a possible flush for your opponents. As a result, you really only have 6 outs for a nut straight draw. Another more complex situation follows:

Example: You hold [J♠ 8♣]o (off-suit, or not of the same suit) and the flop comes [9♠ T♥ J♣] rainbow (all of different suits). To make a straight, you need a [Q] or [7] to drop, giving you 4 outs each or a total of 8 outs. But, you have to look at what will happen if a [Q♥] drops, because the board will then show [9♠ T♥ J♣ Q♥]. This means that anyone holding a [K] will have made a King-high straight, while you hold the second-best Queen-high straight.

So, the only card that can really help you is the [7], which gives you 4 outs, or the equivalent of a gut-shot draw. While it's true that someone might not be holding the [K] (especially in a short or heads-up game), in a big game, it's a very scary position to be in.

How to calculate hand odds (the longer way):

Once you know how to correctly count the number of outs you have for a hand, you can use that to calculate what percentage of the time you will hit your hand by the river. Probability can be calculated easily for a single event, like the flipping of the River card from the Turn. This would simply be: For two cards however, like from the Flop to the River, it's a bit more tricky. This is calculated by figuring out the probability of your cards not hitting twice in a row and subtracting that from 100%. This can be calculated as shown below:

The number 47 represents the remaining cards left in the deck after the flop (52 total cards, minus 2 in our hand and 3 on the flop = 47 remaining cards). Even though there might not technically be 47 cards remaining, we do calculations assuming we are the only players in the game. To illustrate, here is a two-overcard draw, which has 3 outs for each overcard, giving a total of 6 outs for a top-pair draw:

However, most of the time we want to see this in hand odds, which will be explained after you read about pot odds. To change a percentage to odds, the formula is:

Thus, to change the 24% draw into odds that we can use, we do the following:

How to calculate hand odds (the shorter way):

Now that you've learned the proper way of calculating hand odds in Texas Hold'em, there is a shortcut that makes it much easier to calculate odds:

After you find the number of outs you have, multiply by 4 and you will get a close estimate to the percentage of hitting that hand from the Flop. Multiply by 2 instead to get a percentage estimate from the Turn. You can see these figures for yourself below:

As you can see, this is a much easier method of finding your percentage odds. But what about ratio odds? This is still done using this formula:

However, we can rephrase this equation so that your brain might process it a bit more easily:

Using 100 divided by the whole percentage number, such as 24%, we can easily see that 100/24 is equal to about 4. We minus 1 from that and get a rough estimate of our odds at about 3:1. Let's try this all the way through with an example:

You hold: A♣ J♠
Flop is: 5♣ T♦ K♦

Total Outs: 4 Queens (Inside Straight) + 3 Aces (Overcard) - Q♦ or A♦ = 5 Outs

Percentage for Draw = 5 Outs × 4 = 20%

Odds = (100 / 20) - 1
     = 5 - 1
     = 4:1

Again, 4:1 odds means that can expect to make your draw 1 out of every 5 times. If the 1 out of 5 doesn't make a ton of sense to you, think about the 1:1 odds of flipping heads or tails on a coin. You'll flip heads 50% of the time, so 1 out of every 2 times it'll come up heads.

Pot Odds and Poker Odds:

Now that you know how to calculate poker odds in terms of hand odds, you're probably wondering "what am I going to need it for?" That's a good question - this is where pot odds come into play.

Pot odds are simply the ratio of the amount of money in the pot to how much money it costs to call. If there is $100 in the pot and it takes $10 to call, your pot odds are 100:10, or 10:1. If there is $50 in the pot and it takes $10 to call, then your pot odds are 50:10 or 5:1. The higher the ratio, the better your pot odds are.

Pot odds ratios are a very useful tool to see how often you need to win the hand to break even. If there is $100 in the pot and it takes $10 to call, you must win this hand 1 out of 11 times in order to break even. The thinking goes along the lines of: If you play 11 times, it'll cost you $110, but when you win once, you will get $110 ($100 + your $10 call).

The usefulness of hand odds and pot odds becomes very apparent when you start comparing the two. As we now know, in a flush draw, your hand odds for making your flush are 1.9 to 1. Let's say you're in a hand with a nut flush draw and it's $5 to you on the flop to call. Do you call? Your answer should be: "What are my pot odds?"

If there is $15 in the pot plus a $5 bet from an opponent, then you are getting 20:5 or 4:1 pot odds. This means that, in order to break even, you must win 1 out of every 5 times. However, with your flush draw, your odds of winning are 1 out of every 3 times! You should quickly realize that not only are you breaking even, but you're making a nice profit on this in the long run. Let's calculate the profit margin on this by theoretically playing this hand 100 times from the flop, which is then checked to the river.

Net Cost to Play = 100 hands * $5 to call = -$500
Pot Value = $15 + $5 bet + $5 call

Odds to Win = 1.9:1 or 35% (From the flop)
Total Hands Won = 100 * Odds to Win (35%) = 35 wins

Net Profit = Net Cost to Play + (Total Times Won * Pot Value)
           = -$500 + (35 * $25)
           = -$500 + $875
           = $375 Profit

As you can see, you have a great reason to play this flush draw, because you'll be making money in the long run according to your hand odds and pot odds. The most fundamental point to take from this is:

If your Pot Odds are greater than your Hand Odds, then you are making a profit in the long run.

Even though you may be faced with a gut shot straight draw at times - which is a terrible draw at 5 to 1 hand odds - it can be worth it to call if you are getting pot odds greater than 5 to 1. Other times, if you have an excellent draw such as the flush draw, but someone has just raised a large amount so that your pot odds are 1:1, then you obviously should not continue trying to draw to a flush, as you will lose money in the long run. In this situation, a fold or semi-bluff is your only solution, unless you know there will be callers behind you that improve your pot odds to better than break-even.

Your ability to memorize or calculate your hand odds and pot odds will lead you to make many of the right decisions in the future - just be sure to remember that fundamental principle of profitably playing drawing hands requires that your pot odds are greater than your hand odds.

Poker Odds from the Flop to Turn and Turn to River

An important note I have to make is that many players who understand Hold'em odds tend to forget is that much of the theoretical odds calculations from the flop to the river assume there is no betting on the turn. So while it's true that for a flush draw, the odds are 1.9 to 1 that the flush will complete, you can only call a 1.9 to 1 pot on the flop if your opponent will let you see both the turn and river cards for one call. Unfortunately, most of the time, this will not be the case, so you should not calculate pot odds from the flop to the river and instead calculate them one card at a time.

To calculate your odds one card at a time, simply use the same odds that you have going from the turn to the river. So for example, your odds of hitting a flush from the turn to river is 4 to 1, which means your odds of hitting a flush from the flop to the turn is 4 to 1 as well.

To help illustrate even further, we will use the flush calculation example that shows an often-used (but incorrect) way of thinking

Example of Incorrect Pot Odds Math
You Hold: Flush Draw
Flop: $10 Pot + $10 Bet
You Call: $10 (getting 2 to 1 odds)

Turn: $30 Pot + $10 Bet
You Call: $10 (getting 4 to 1 odds)

Long-Term Results Over 100 Hands
Cost to Play = 100 Hands * ($10 Flop Call + $10 Turn Call) = $2,000
Total Won = 100 Hands * 35% Chance to Win * $50 Pot = $1,750

Total Net = $1,750 (Won) - $2,000 (Cost)
           = -$250 Profit
           = -$2.5/Hand


Example of Correct Pot Odds Math
You Hold: Flush Draw
Flop: $30 Pot + $10 Bet
You Call: $10 (getting 4 to 1 odds)

Turn: $50 Pot + $16 Bet
You Call: $16 (getting about 4 to 1 odds)

Long-Term Results Over 100 Hands
Cost to Play = 100 Hands * ($10 Flop Call + $16 Turn Call) = $2,600
Total Won = 100 Hands * 35% Chance to Win * $82 Pot = $2,870

Total Net = $2,870 (Won) - $2,600 (Cost)
          = $270 Profit
          = $2.7/Hand

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 profitable even with seemingly 'bad' odds. The draws that you want to worry about the most are your long shot draws: overcards, gut shots and two-outers (hoping to make a set with your pocket pair). If you draw these hands using incorrect odds (such as flop to river odds), you will be severely punished in the long run.

Implied Value

Implied Value is a pretty cool concept that takes into account future betting. Like the above section, where you have to worry about your opponent betting on the turn, implied value is most often used to anticipate your opponent calling on the river. So for example, let's say that you have yet another flush draw and are being offered a 3 to 1 pot odds on the turn. Knowing that you need 4 to 1 pot odds to make this a profitable call, you decide to fold.

Aha, but wait! Here is where implied value comes into play. So, even though you're getting 3 to 1 pot odds on the turn, you can likely anticipate your opponent calling you on the river if you do hit your flush draw. This means that even though you're only getting 3 to 1 pot odds, since you anticipate your opponent calling a bet on the river, you are anticipating 4 to 1 pot odds - so you are able to make this call on the turn.

So in the most practical standpoint, implied value usually means that you can subtract one bet from your drawing odds on the turn, as it anticipates your opponents calling at least one bet. In some more advanced areas, you can use implied odds as a means of making some draws that might not be profitable a majority of the time, but stand to make big payouts when they do hit. Some examples of this would be having a tight image and drawing to a gut shot against another tight player. Even though this is a horribly bad play (and hopefully you don't have to pay much for it), it can possibly be a positive play if you know your opponent will pay you off if you hit your draw - because he won't believe you played a gut shot draw. For many reasons, I do not recommend fancy implied odds plays like these, but mentioned it more so that you can recognize some players who pull these 'tricky' plays on you as well.

Conclusion - Poker Odds

Knowing how to figure out your odds in Texas Hold'em is one of the most fundamental points in becoming a solid poker player. If this poker odds page was a bit difficult to understand, don't worry. Keep playing, bookmark this page and come back when you need another brush-up on how to properly apply odds. It takes a while to learn how to calculate them properly and to memorize them as well.