Postflop: The Mathematics of Poker  Odds & Outs
Introduction
In this article Which cards help you?
 Learning to balance risk and reward
 Not every helpful hand is that helpful
Draws, or drawing hands are incomplete hands which have to be complemented by another community card so that they can be defined as made hands. The strategy from the beginner section doesn't clearly define how to handle these kinds of hands.
In this article you will learn the mathematical basis of poker. You will learn how to figure out the winning percentages of your draw and how to determine whether it is profitable to play the hand or not.
This article is based on the following three central terms:
 Outs
Outs are all the cards which can improve your hand.  Odds
Odds show the probability of one of the next community cards being one of your outs.  Pot odds
Pot odds show the relationship between the potential profit and the bet you have to make. This can be seen as the riskrewardratio. If these are compared to the odds, it is possible to judge the worth of calling a bet in order to complete your draw.
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Extended charts for playing before the flop
The overview on the mathematics of poker  Odds and Outs
Outs  Which cards help you?
Outs are cards which, if dealt as community cards, improve your hand and possibly make it the best hand on the board. The emphasis is placed on 'making it the best hand', which we will discuss a little later.
EXAMPLE A 
At first, you have a seemingly worthless hand. You can't win a showdown with this hand. However, you do have the chance of making a strong hand, namely a straight, if either an ace or a six is dealt on the turn or the river.
These cards, the ace and the six, are your outs (they are still in the deck). The question is: how many outs do you have in total? The answer is relatively easy if you think about how many aces and sixes there are in a card deck. In each case there are four cards of one value, which makes a total of eight outs. Only one of these eight cards has to be dealt in order to improve your hand.
Your Outs
EXAMPLE B 
This situation is even better. Not only will every ace or six help you make a straight, but every club will give you a flush.
The number of outs has therefore increased. On the one hand, you can allow for all remaining club cards in the deck to be your outs. There are 13 cards of one suit in a deck, four clubs have already been dealt, so a total of 9 (134=9) outs remain to make your flush. The eight outs mentioned in the example above are added to this.
From the eight outs mentioned in Example A, two are subtracted, namely the ace and the six of clubs because they have already been taken into account in the flushdraw outs. This makes a total of 15 (9+6=15) outs:
Your Outs
FURTHER EXAMPLES 

Flushdraw  9 Outs
There are 13 cards of one suit in a deck; four have already been dealt. Therefore 9 outs remain which would complete the flush.

OESD (openended straightdraw)  8 Outs
Any 4 or 9 complete the OESD to make a straight. Therefore an OESD always has 8 outs.

Two Overcards  6 Outs

A pair as a three of a kind or a two pair draw  5 Outs
2 eights are left in the deck which would make three of a kind. One of the three remaining kings would make a two pair. This adds up to a total of 5 outs.

Gutshot  4 Outs
A gutshot draw means that you have a chance to make a straight if the missing 'inner card' of your straight is dealt. There are exactly four cards which would do this; here it would be any deuce. You therefore have 4 outs with a gutshot.
There are 3 aces and 3 queens left in the deck which would make a top pair. You therefore have 6 outs in this example.
Odds  How likely is it that I complete my draw?
So what does the term odds actually refer to? It is a commonly used term representing the probability of completing your hand.
This type of ratio is called Odds against, because it shows the probability of not making your hand. It's the number of times you do not complete your draw against the number of times you do. What does your ratio look like? These odds display your probability in order to ease the process of determining whether it is profitable to continue playing the hand or not, as you will see in the next chapter.
Let's look at the first example one more time:
You already know 5 cards after the flop: your two starting cards and the three flop cards. Any of the remaining 47 unknown cards can still be dealt (52 cards are in a deck in total). 8 of these 47 cards help you to complete your draw, while the other 39 (478=39) cards would not complete your draw and are therefore unhelpful. In short, your odds from the flop to the turn are 39:8, which is the same as about 5:1
You know 5 cards on the flop, so there are 47 (525 = 47) unknown cards. Consequently, after the turn card is placed on the board, there are 46 unknowns after the turn. The helpful cards are your outs. From this logic we can derive the following equation:
A standard scenario for the application of odds from the flop to the turn occurs if you are holding a draw in a freeplay position (in the Big Blind where you don't pay any extra to see the flop).
Outs and Odds

Pot odds  Can I play my hand profitably?
As you can now determine the probability of completing your draw by making use of odds, the only question that hasn't been answered is how to apply it practically in a game.
Let's make use of the old example again:
We are looking at a true situation from a real money Fixed limit game. You are on the flop with one opponent and you are holding the cards shown above. The pot is now at $5. Your opponent bets $1. Is it worthwhile to call this bet and pay $1 to see the turn card?
 Pot before the bet from your opponent: $5
 Bet from your opponent: $1
 Possible profit for you: $6
 Bet (which you have to call): $1
As we now know, the odds of hitting your straight on the turn are roughly 5:1 against you. This means you'll complete your hand one in six times. Let's assume that you'll definitely win the hand if you hit one of your outs. So you'd win $6 one in six times, whereas the other five out of the six times you would lose $1. This is assuming that you would give up your hand on the turn if it doesn't improve.
On average, if you call the $1 you would lose $1 five out of six times. In total this is $5. However, you'll win $6 once so the net profit, which is calculated as the profit minus the losses, is: $6  $5 = $1. It is therefore profitable in the long run to call your opponent's bet in this situation. As a rule you win $1 in every repetition of this situation.
In this example the pot odds and pot chances come into play. They represent the relationship between the possible profit and the bet which has to be paid and are therefore an expression of the benefit/cost ratio.
In this situation, the pot is at $5. In addition, the $1 which the opponent bet is added to the pot, resulting in a total pot size and possible profit of $6. You have to pay $1 to stay in the game to see the turn card. The pot odds are now $6 : $1 or 6:1.
Like the numbers 6:1 and 5:1 suggest, a simple rule applies:
Since 6:1 is bigger than 5:1, this situation is profitable.
What would happen if instead of the flop, this situation took place on the turn? Your opponent bets double the amount ($2). On the one hand the possible profit would rise to $5 + $2 = $7. On the other hand, your pot odds for calling the bet would be $7 : $2, which is a ratio of 3.5:1. It would therefore be unprofitable to call the bet. You'd be best advised to fold your hand in this situation because you will lose in the long run.
To explain the calculation: you win $7 one out of six times, and lose $2 five out of six times. This means you lose $2 five times = $10, and win $7 one time. On average you lose $3 with every repetition of this situation.
Discounted / modified Outs
Let's go back to the topic of outs and slightly modify the introductory example:
In the part about outs, you have figured out that you have eight outs: any ace or six will make a straight.
Your Outs
What would happen if you are encountering an opponent who is holding the following cards?
The ace of hearts and the six of hearts would give you a straight, however these would give your opponent a flush and therefore the better hand. These two cards are no longer of value to you, so we calculate only six outs instead of the original eight. These are called discounted or modified outs.
Your discounted Outs
You obviously don't know your opponent's cards, but there is a certain chance that he is holding two hearts. You cannot assume that you have eight clean outs to make your straight. You have to subtract the number of outs by the number of cards that give your opponent the better hand.
In this example, it is very unlikely that you can give yourself the full number of outs due to the large number of opponents in the hand. The way your opponents play the hand could give away whether they are holding a flushdraw or not.
Another reason to discount the outs is that an opponent could be holding a hand such as:
Hence, the four sixes aren't clean outs because they would give your opponent a stronger straight. Only four outs therefore remain.
Your discounted Outs in this example
It is necessary to realistically discount your outs, in order to make a correct estimate of the odds in an incomplete hand. You can almost never give yourself every out, especially if you are playing against several opponents. It is always possible that they are holding a better incomplete hand. They could even be holding your same hand. Many things can happen in a poker game leading to you losing the hand, even though you hit one of your outs.
You always have to ask yourself the following question: which of my outs actually makes my hand the best hand? If you are holding an OESD and there is a flushdraw possible, you would only be able to give yourself 6 discounted outs by default, rather than the total 8 outs.
Especially on the lower limits, players like to play suited cards because of the chance of making a flush. You therefore need to be relatively strict with yourself and consistently subtract two outs, especially against several opponents.
If you are faced with the question of how many outs you can give yourself, you have to answer the following question first: which better hands are possible and how likely are they? The more opponents on the table, the more likely these hands are. In the lower limits, connected cards like 87, 54 or 86 and suited cards are frequently played, so you need to take this into account.
Conclusion
 Odds are the relationship between: unhelpful cards : helpful cards.
 Pot odds are the relationship between: possible profit : input (bet which has to be paid).
Essentially, a draw can be played profitably if your pot odds are better than the odds for your hand. These are situations in which you can win more by completing your draw than you can if you don't complete your draw.
It is essential for your longterm profit to have understood and learned the concept of odds and pot odds and with them the mathematical basis for poker. Knowing when to bet and when it is worthwhile to call a bet are fundamental strategic elements of the game. Take the time to really grasp this material, because it will certainly help push your game and your bankroll forwards.
LINKS  

#1 Odieone, 19 Apr 08 16:45
Excellent rewrite on the outs and odds. I was just thinking that the one on the SnG section was better than the FL one. SnG is more complicated but it had the pictures.The New one on FL is awesome.
#2 mouse89, 06 Oct 08 14:15
ok#3 mrsnok, 27 Jan 09 11:38
there's a typo on page 4, where it says:"In the described situation, the pot is at $10. "
Actually, it's 5$.
#4 Pirokunn, 31 Jan 09 00:41
mrsnok, you are right  there's a contextual mistake, butI'm sure, that the pot size in the described situation is actually
6$ (5$ preflop + the 1$ bet from the opponent), not 5$ ;)
I notified the PS team :P
#5 Pirokunn, 31 Jan 09 00:45
sry, my bad, mrsnok is 100% right :]#6 DCaldeira, 07 Sep 09 22:48
nice#7 krumvirek, 15 Aug 10 22:31
very good pictures#8 rbdflyboy, 01 Oct 10 22:21
Excellent presentation. Most people don't give me the pot odds to call a bet if I am in a drawing situation. Most players bet between 66%100% of the pot to potect their hands. So 90% of the time I'm folding after the flop or preflop. Just too expensive to keep going. I've learned two of many important lessons: Live to fight another day,and, if I haven't made my hand by the turn...fold. The reason, if I'm going to coin flip, I'll do it preflop.#9 danoosh, 27 Feb 12 14:54
thx