Mathematics of Poker: Odds and Outs
Introduction
In this article
- The cards that are helpful to you (outs).
- Balancing risk and reward
- Not every out is actually helpful
Before this lesson you should have read:
A draw is a hand that isn't made yet, but would be if another helpful card came into the community. You might have four spades and require the missing 5th spade for a flush. This card is called the out card, needed for your draw to turn into a made hand.
You will often find yourself holding a draw after the flop or the turn and facing a raise.
The question is: is it profitable to call?
This is where the outs, odds and pot odds come into play. They constitute the mathematical basis used to answer this question, and are the subject of this article.
- Outs
An out is any card that would make your hand better. - Odds
The odds are the probability that one of your outs will show up in the community. - Pot odds
The pot odds provide a ratio between the amount you can win and the amount you have to wager to stay in the hand. They also provide a risk vs reward ratio. By comparing the pot odds to the odds, you can determine whether or not it is profitable to continue playing a drawing hand.
Attention: This article is available in multiple versions, customised for each given poker style and format. If you want to read the odds and outs article for the short stack strategy you can find it through this link: Mathematics of poker - Odds and Outs for the Short Stack Strategy.
DOWNLOADS
The starting hand chart as PDF file
Outs - Which cards can help me?
Your outs are all cards that could show up in the community and help your hand. Some out cards are more helpful than others; when you play a draw you are looking for the outs that give you the best possible hand. We will talk more about this later.
   |
    |
Right now your hand looks pretty worthless and has no chance of winning a showdown. If however, an ace or a six land on the board, you would have a straight: a strong made hand.
At this moment your outs are aces and sixes. Now you have to ask yourself, how many outs does that make? Since there are four aces and four sixes in the deck of cards, there are a total of 8 cards that could be dealt on the turn or the river, to give you a straight.
Your Outs
        |
|
  |
Now you are in an even better situation. You don't only have the chance to make a straight with any ace or six, you would also have a flush if another club were dealt.
You can now count the remaining club cards as outs. Since there are 13 clubs and 4 have already been dealt, there are 9 clubs remaining in the deck.
As you've already counted the ace and six of clubs, you have 6 more out cards (3 aces and 3 sixes) for your straight draw. The 9 flush outs and the 6 straight outs give you a total of 15 outs:
Your Outs
       |
-
Flushdraw - 9 Outs
There are 13 cards of each suit in the deck and 4 have been dealt. This leaves 9 outs to complete a flush.
-
OESD (Open Ended Straight Draw) - 8 Outs
Any 4 or 9 would complete an OESD and give you a straight. OESDs always have 8 outs.
-
Overcards - 6 Outs
There are 3 Aces and 3 queens remaining in the deck. Any of those would give you top pair. Overcards give you 6 outs on a draw for top pair.
-
A three of a kind or two pair draw - 5 Outs
There are 2 eights left in the deck that would give you a three of a kind. One of the three remaining kings would give you a two pair. This gives you a total of 5 outs to better your small pair.
-
Gutshot - 4 Outs
A gutshot draw is the weakest straight draw. One card is missing from the middle of the sequence, which means you would need a 'gutshot' to complete the sequence. There are four outs for a gutshot draw, four twos in this case.
Odds - What are my chances of completing a draw?
To calculate your odds, use this simple formula:
This formula describes the odds against you, or the probability that you won't complete your draw. It compares the questions: "How often won't I make my hand?" and "How often will I make my hand?"
This formula shows whether it would be profitable to continue playing your hand, or if you should fold. We will go into further detail in the next chapter.
Let's take another look at our first example:
|
|
Once the flop has been revealed you see five cards, your two hole cards and the three on the flop. This means the turn could bring any of the 47 cards remaining in the deck. 8 of these 47 cards are helpful to you, 39 of them are not.Your odds of hitting an out on the turn are therefore 39:8, or approximately 5:1.
After the turn, there will be 46 unknown cards remaining in the deck. The helpful cards are your outs, which gives us the following formula:
The chart below is used in the Short Stack Strategy and gives you odds for different draws on the flop and turn.
 |
Outs and Odds
|
Pot odds - Should I continue playing my hand?
Now that you know how to calculate your odds, let's take a look at some practical examples.
We will start with our previous example hand:
|
|
Let's assume you have this hand in a NL Hold'em real money game. At this point there are $10 in the pot. Your opponent bets $2. Is it profitable for you to call and see the turn? First we have to look at four factors:
- The size of the pot before the bet: $10
- The amount bet: $2
- Your possible winnings: $12
- The amount you must wager to stay in the hand: $2
The odds of making a straight are 5:1 against you. This means you will only make your hand in one of the six cases.
Should you win the hand, you stand to gain $12. In five out of six cases, however, you will lose $2, assuming you are forced to give up your hand after the turn.
By calling the $2 raise, you will lose $10 and win $12 on average every six times this situation occurs. Your total profit, made up of winnings : loses, would be $12 : $10, or $2 in this case.
It is therefore profitable to make the call in this situation, as you will make $2 every six times this situation arises.
Pot odds = possible winnings : cost of staying in the hand
In this example there are $10 in the pot, plus the $2 raise, giving a total of a $12 potential win. You have to pay $2 to stay in the hand and see the turn card. Your pot odds are therefore $12 : $2, or 6:1.
There is a simple principle behind odds and pot odds:
If the pot odds are higher than the odds against you, it is profitable to stay in the hand. If they are lower, you will lose money in the long run by staying in the hand.
What if the opponents bet $4 instead of $2? There would now be $14 in possible winnings. Your pot odds, however, would now stand at $14 : $4, or roughly 3.5 : 1; it would no longer be profitable to stay in the hand.
Lets take a closer look. In one of six cases you would win $14. In five of six cases you would lose $4. This means that every six times this situation occurs, you will win $14 while losing $20, giving you a total profit of -$6.
Discounted / Modified outs
Let's return to the subject of outs by making a few changes to our sample hand:
|
 |
We determined that you have 8 outs in this position, the 4 aces and sixes that would give you a straight.
Your Outs
|
This changes, however, when an opponent has the following hand:
  |
The ace and six of hearts would give you a straight, but either one would also give your opponent a flush and thereby the better hand, meaning these two cards would not help you. The 6 remaining outs that would truly give you the best hand are called discounted outs.
Your Discounted Outs
  |
In a situation like this one, the more players involved in the hand, the more likely it is that someone will have a flush draw and you will have to discount your outs. Of course, you shouldn't automatically assume your opponent has a flush draw every time two cards of the same suit show up in the flop, but it is always a possibility. Getting a good read on an opponent can help you decide whether or not it is necessary to discount out cards.
Suppose an opponent has the following hand:
  |
Now four of your outs, the sixes, are no longer helpful, as they would give your opponent a higher straight. This leaves you with only four outs that would indeed give you the best hand.
Your Discounted Outs in this case
   |
In order to accurately determine your odds it is necessary to be realistic when discounting your outs. You will often have to discount a number of out cards, especially when several opponents are still involved in the hand.
Any one of your opponents might have a better unfinished hand, or need the same outs as you for an even better hand. There are a number of ways to hit your outs and still lose the hand.
It is important to ask yourself the question: Which of my outs will definitely give me the best hand? If you have an OESD, for example, and a flush draw is possible, you can only count on 6 of the 8 possible out cards.
A lot of players on lower limits like to play suited hands, which means the chances of an opponent having a flush draw requires you to discount two outs when the board is suited, especially if several opponents are involved in the hand.
When asking yourself which outs you can truly count on, you should start by asking yourself what hands could possibly beat yours and how likely they are. The more players involved, the more likely such a hand will show up.
Case Studies
|
You are BU
|
|
 |
   |
|
 |
|
Your calculations start on the flop. You have to pay $0.75 to see the turn. There are $1.10 + 2 x $0.75 = $2.60 in the pot. Your pot odds are $2.60 : $0.75 or roughly 3.5 : 1.
You’re holding a high flush draw, for which you have a total of 9 outs. Four additional outs come from the gutshot draw for a straight. Since you’ve already calculated the spades-out for the flush draw, you only count 3 outs for the gut shot. A quick glance at the table will tell you that you need pot odds of at least 3:1, in order to be able to see the next community cards with 12 outs. Since the pot odds are better than the required odds, you can go right ahead and call.
The same calculation applies on the turn. However, now you only have pot odds from 2.1:1. You still got 12 outs left, however you cannot give yourself the full 12 outs anymore, as the pair in the community cards doesn’t make a full house probable, but possible. Even with the full 12 outs left, you’d need pot odds 3:1. Your decision is made. You fold.
|
You are BB
|
|
 |
  |
|
You do have a flush draw, but since it is weak, you can only count on 6-7 outs. Since the SB made a large bet against four players, it is likely he will bet on the turn as well. In order to call you would need 7:1 pot odds, but are only getting 2:1.
|
You are SB
|
|
 |
  |
|
In this case you can count on all 9 flush draw outs. You can also expect your opponets to raise on the turn. In order to call and see the turn card, you would need 4 : 1 pot odds. Unfortunately you are only getting 2.3 : 1.
|
You are SB
|
|
|
   |
|
Due to the flush draw on the board you need to discount 2 outs and are left with 6. You don't have to discount the three queens, as the BB is unlikely to have AK or K9. All in all, you are getting 2.5 : 1 pot odds. In order to call, you would need 7 : 1 odds, so you fold. Had other players called to see the flop, you might have attained the necessary pot odds to stay in the hand.
|
You are CO
|
|
 |
 |
|
 |
|
You bet on the flop with an OESD; your opponent raises. You have 8 outs and two over cards, which allow you to add another 2-3 outs. The pot odds are approximately 4.5 : 1. A quick look at the charts shows you that you can call.
Unfortunately the turn card is not helpful. Your over cards are no longer that strong and you have to discount your outs. You only have 9 remaining outs that would be certain to give you the best hand. You would need 4 : 1 pot odds to stay in the hand, but your opponent is only giving you 2.4 : 1. This is not enough to consider calling.
Conclusion
Odds are the ratio between: unhelpful cards and helpful cards.
Pot odds are the ratio between: possible winnings and costs of staying in the hand.
It is profitable to play a draw when the pot odds are higher than the odds. In such cases, you win more on average when you make your hand than you lose when you don't.
In order to make a profit in the long run, it is essential for you to understand the concepts of odds and pot odds: the mathematical basis of poker. Knowing when it is profitable to make a call and knowing how much you have to bet in order to make an opponent's draw unprofitable, are fundamental elements of a strategic player's game.
Taking the time to master these concepts will take you and your bankroll a good step forwards.
There is, however, more to the subject of odds than covered in this article. Namely Implied Pot Odds that play a key role in the NL Hold'em Big Stack Strategy.
Go to this article: The mathematics of poker - Implied Pot Odds
