When we talk about draws or drawing hands, we are talking about incomplete hands that need help from the coming community card(s) in order to turn into a made hand.

The strength of a draw depends on the number of cards that can help. These cards are referred to as outs. Another factor that determines the strength of a draw is the so-called 'fold equity.'

Fold equity is your probability of winning the hand by making everyone else fold. Your fold equity is generally higher when you are facing fewer and more cautious opponents, simply because they are much more likely to fold if you bet or raise.

You have to understand the mathematics of poker in order to play a draw correctly. This is how you determine how likely you are to complete your draw and whether or not it will be profitable to stay in the hand. This article will introduce you to the mathematics of poker and teach you ...

• ... how to calculate your probability of completing a draw.
• ... a strategy for playing drawing hands in the early, middle and late phases of a tournament.

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An overview of the mathematics of poker - Odds and Outs [http://resources.pokerstrategy.com/Editorial/en/Data/ps_chart_en_outs_und_odds.pdf]

Outs - Which cards are helpful?

Outs are the cards that can improve your hand, possibly making it the best hand. The emphasis should, as you will see, be placed on the "making the best hand."

Your hand looks pretty worthless at first glance; you won't be able to win a showdown with what you've got. An ace or a 6 on the turn or river would, however, give you a strong made hand.

These are your outs (A, 6). They are the cards that could give you a made hand. Now you have to ask yourself how many outs you have. The answer is pretty simple. Since there are four aces and four sixes in the deck, there are a total of eight cards that can help you. If one of the eight aces or sixes in the deck shows up on the turn or river, you will have a strong made hand.

Your outs

This time you find yourself in an even better position. In addition to the aces and sixes that would give you a straight, another club would give you a total of 5 for the flush.

This means you have more outs. You can count each of the remaining club cards as an out. Since there are a total of 13 and four have already been dealt, you have 13-4=9 outs for a flush.

You also have outs for a straight, but you can't count the ace and six of clubs, since you already counted them towards your flush draw. This leaves you with 6 outs for the straight, giving you 9 flush outs + 6 straight outs = 15 total outs:

Your outs

• Flush draw - 9 Outs

  There are 13 cards of each suit in the deck. Four have already been dealt, meaning 9 remain as outs for a flush.

   

• OESD (open ended straight draw) - 8 Outs

  Any 4 or 9 would complete the OESD. An OESD therefore gives you 8 outs.

   

• Two overcards - 6 Outs

  There are still 3 aces and 3 queens in the deck that would give you top pair. Having two overcards gives you 6 outs.

   

• A pair with the chance of improving to two pair or three-of-a-kind - 5 outs

  There are still two eights in the deck that would give you three-of-a-kind. One of the three remaining kings would give you two pair. This gives you a total of 5 outs when you have a small/mid pair.

   

• Gutshot - 4 outs

  A gutshot draw is a type of straight draw. It means you are missing one card in the middle of the numerical sequence. You need a 'gutshot' to complete, metaphorically speaking. In the example, any 2 would complete the straight. A gutshot gives you 4 outs.

Odds - What is the probability of completing your draw?

What exactly are the odds? The odds are simply a way of saying how likely a draw will complete.

This formula refers to the odds against you, since it gives you the probability of not making your hand. It's a question of, "How often will I miss vs. How often will I hit?" The answer gives you your odds.

This formula is a conventional way of determining whether or not it is profitable to stay in the hand with a draw. We will go into further detail in the next section.

But before we go on, let's take a quick look back at our example:

You have seen 5 cards once the flop has been revealed (3 on the board and two in your hand). The turn card can therefore be any one of the 47 cards remaining in the deck (52 total cards - 5 you have seen). 8 of these 47 cards would complete your draw; the other 39 leave you with nothing. Your odds from the flop to the turn are therefore 39:8, or roughly 5:1.

You've seen five cards, meaning there are 52-5 = 47 unknown cards remaining. Once the turn has been revealed, there will be 46 unknown cards remaining in the deck. This give us the following formula:

A standard situation for using this formula is found when you are in the BB and hit a draw after checking.

Outs und Odds Outs Odds Flop-Turn or Turn-River Odds Flop-River Examples 1 46:1 22.5:1 Backdoor flush draw (two cards of the same suit on both the turn and the river) 2 22.5:1 11:1 A pocket pair turning into three-of-a-kind 3 15:1 7:1   4 11:1 5:1 Gutshot 5 8:1 4:1 A pair improving to two pair or three-of-a-kind 6 7:1 3:1   7 6:1 2.5:1   8 5:1 2:1 OESD 9 4:1 2:1 Flush draw 10 3.5:1 1.5:1   11 3.5:1 1.5:1   12 3:1 1:1 Flush draw + Gutshot 13 2.5:1 1:1 OESD + pair 14 2.5:1 1:1 Flush draw + pair 15 2:1 1:1 Flush draw + OESD

Pot odds - Can I play my hand profitably?

Now that you know how to determine the probability of completing a draw, it's time to address putting this to practical use.

We'll stay with our example hand:

Let's look at a concrete situation in a NL cash game. You are facing a single opponent on the flop with our example hand. There are currently $10 in the pot. Then your opponent bets another $2. Should you pay the $2 to see the turn card?

• Pot size before bet: $10
• Size of bet: $2
• Possible winnings: $12
• Cost of staying in the hand: $2

The odds of hitting on the turn are roughly 5:1 against you. This means that you will complete your straight in one of six cases.

Now, assuming you win the pot every time you complete your draw, we can say that you will win $12 one in six times, and lose $2 the other five times, assuming you fold after missing on the turn.

If you call the $2 bet, you will on average lose $2 five times (a total of $10), and win $12 once. Your profit, made up of winnings - losses, is therefore $12 -$10 = $2.

It is therefore profitable to call your opponent's bet in this situation. You will, on average, win ($2 / 6 =) $0.33 every time you do so.

This is where the so-called pot odds come into play. They give you the ratio of possible winnings : cost of staying in the hand and are a way of describing your cost-benefit ratio.

Going back to our example: There are $10 in the pot, then your opponent bets another $2. There are now a total of $12 in the pot that could be won. You have to pay $2 to stay in the hand and see the turn card. Your pot odds are therefore $12:$2, or simply 6:1.

As the numbers let on, there is a simple rule:

How would the situation change if your opponent bet $4 instead of $2? Your possible winnings would increase to $10 + $4 = $14, but your pot odds would decrease to $14:$4, or 3.5:1. Calling your opponent's bet would no longer be profitable and you would have to fold your hand.

In other words: You would win, on average, $14 one in six times, while losing $4 five in six times ($20). Every six times this situation repeats itself, you make a total of $14 winnings - $20 losses = -$6. On average, you'll lose $1 for every repeat of this gaming situation.

We used monetary values for one simple reason: The pot odds answer the question, "How much would I win/lose if I were to repeat this decision in this situation thousands of times?"

Of course, when it comes to tournament play, you can't just buy more chips if you lose a hand. There are a number of other strategic factors that come into play, as well as the principle 'I want to defend my chips.' This means that a decision that is correct according to the pot odds may be strategically false in a tournament. The next section will show you the other factors that influence how you play a drawing hand in tournament poker.

The early phase

The following factors influence your play in the early phase of a tournament:

• The number of opponents in the hand

  The more opponents in the hand, the less aggressive you should be. If everyone checks in front of you, you should check behind and take a free card, especially if anyone is still sitting behind you. Your draw also loses strength as the number of opponents in the hand increases.

• Position

  You generally shouldn't bet if you are first to act. You should also have a plan in case a) you miss the turn, or b) an opponent raises on the flop.

• Fold equity

  Fold equity is the probability of winning the pot by forcing all other opponents to fold. Theoretically, you always have some fold equity, but be careful in the early phase, especially in tournaments with low buy-ins. There will be a lot of weak players at the table with no idea of pot odds looking for action and ready to call your bet with any two cards.

• The pot size

  The pot will rarely be very large in the early phase of a tournament, meaning it is rarely worth playing a draw aggressively.

• Your odds
  You can call if the pot odds are right, assuming that it won't cost you a significant portion of your stack. Just remember, there is no point in calling with the right pot odds if it is going to cost you too much of your stack.

• Your outs

  The more outs you have, the more readily you can call.

A piece of advice: Don't risk too many chips! Take the safer path and take a free card instead of playing your draws aggressively.

The middle phase

The following factors influence your play in the middle phase of a tournament:

• Your fold equity

  Fold equity plays a decisive role in this phase, it may even be at its highest. It all depends on the previous action. If one or more opponents has shown strength, you will have less fold equity.

• The pot size

  The pot will usually be so large that you won't be calling, but rather folding or going all-in.

• Your outs

  The more outs you have, the more aggressively you can play your draw.

• The number of opponents in the hand

  The more opponents in the hand, the more chips in the pot. An aggressive attack can work depending on the size of the pot.

• Reads / Your impression of your opponent's playing style
  

  Reads are information about how your opponent plays. They become important in the middle phase of a tournament when it comes to estimating your fold equity. An opponent looking for action is more likely to call with a weak hand, meaning you have little fold equity. Your fold equity is higher when facing a cautious opponent.

A piece of advice: You can consider playing aggressively depending on the ratio of pot size : your stack size.

The late phase

The following factors influence your play in the late phase of a tournament:

• Your outs

  The more outs you have, the more readily you can go all-in.

• Fold equity

  You generally have less fold equity in the late phase due to the general chip situation.

• Your odds

  The odds hardly play a role anymore, since you are basically playing push/fold. And if it's not an all-in bet, it's usually so large that you will never have the right pot odds to call.

• The pot size

  The pot will usually be large enough to go all-in directly.

• Who has the initiative?

  If you have the initiative (meaning you are the aggressor), you should go all-in with a strong draw, as long as you have some fold equity. If your opponent has taken the initiative and gone all-in, you should only call with a very strong draw.

A piece of advice: Since the general chip situation is usually pretty critical for everyone, you should give yourself less fold equity. A desperate opponent is much more likely to push his chips in the middle.

Examples

Stack sizes:
Button (t1500)
Hero (t1500)
BB (t1500)
UTG (t1500)
UTG+1 (t1500)
MP1 (t1500)
MP2 (t1500)
MP3 (t1500)
CO (t1500)

Pre-flop: Hero is SB with Q, K.
UTG calls t20, 2 folds, MP2 calls t20, 3 folds, Hero completes, BB checks.

Flop: (t80) 5, T, J (4 players)
Hero checks, BB checks, UTG bets t60, MP2 calls t60, Hero calls t60.

Comments:
You hit an OESD on the flop and have 8 outs. The probability of completing on the turn is 17%; your odds are app. 5:1. There are t200 in the pot and it will cost t60 to call, giving you app. 3:1 pot odds.

However, with 3 opponents in the hand you can't necessarily depend on the A and 9, since they could give one of your opponents a flush. For this reason, you give yourself 6 outs.

You would complete your straight on the turn one in seven times. Since, however, these 60 chips only make up a very small portion of you stack and you can certainly expect to win more if you do complete, you can call.

Stack sizes:
MP2 (t1740)
CO (t5705)
Button (t1340)
SB (t345)
BB (t1430)
UTG (t1470)
Hero (t1470)

Pre-flop: Hero is MP1 with A, K.
1 fold, Hero raises to t90, 4 folds, BB calls t60.

Flop: (t195) Q, 3, 4 (2 players)
BB checks, Hero bets t120, BB raises to t1340, Hero ?

Comments:
Your opponent will usually have top pair in such a situation. If, for example, he has QJs, you are slightly favored with 52:48. If your opponent does, in fact, only have top pair, you have 14 outs against him with your flush draw and overcards.

You will hit one of these outs 52% of the time. This means that calling your opponent's all-in is only profitable when he never has a hand better than top pair. This is an assumption you simply can't make, which is why you should simply fold.

Stack sizes:
UTG (t1730)
CO (t2280)
Button (t2728)
SB (t10952)
Hero (t2310)

Pre-flop: Hero is in BB with 7, 8
UTG folds, CO raises to t200, Button calls t200, SB calls t150, Hero calls t100

Flop: (t800) 9, 9, T (4 players)
SB checks, Hero bets t200, CO raises to t2080, Button folds, SB folds, Hero ?

Comments: 
You can justify this pre-flop call, but you should only make this type of call if you are confident in your post-flop game. You hit an OESD + flush draw on the flop. This gives you a total of 15 outs, meaning you will complete the flush or straight by the river 55% of the time.

One thing to keep in mind: You are facing three opponents and one of them could easily have a (better) flush draw, as well. Any 8, J or K could also give someone a straight.

On the other hand, you have a strong draw and the pot is large enough - in relation to your stack size and current standing in the SnG - to consider using a different strategy than you see in this example.

If your opponents aren't too reserved, your best option would probably be checking and responding to a bet with an all-in. If your opponents are passive, you could bet t400-t600 with the intent of calling if your opponent goes all-in. Your t200 bet in this example would not be appropriate in such a situation.

Stack sizes:
UTG (t680)
UTG+1 (t2579)
MP1 (t1739)
MP2 (t2196)
CO (t3403)
Button (t1740)
SB (t5773)
Hero (t1890)

Pre-flop: Hero is in BB with 4, 5
UTG folds, UTG+1 folds, MP1 calls t200, MP2 folds, CO folds, Button folds, SB calls t100, Hero checks

Flop: (t600) 2, 3, J (3 players)
SB bets t300, Hero raises to t1690

Comments:
By no means a desperate situation. You hit an OESD on the flop, which will complete by the river 32% of the time. One of your opponents, the one with the biggest stack, bets half the pot on the flop.

The other opponent is behind you. There is no reason for you to go all-in, since your bet won't generate much fold equity against the SB, who has already signaled strength. Folding would be the correct move in this situation.

Stack sizes:
CO (t2355)
Button (t5030)
SB (t3205)
Hero (t2910)

Pre-flop: Hero is in BB with 7, K
CO folds, Button calls t200, SB calls t100, Hero checks

Flop: (t600) A, 9, T (3 players)
SB checks, Hero bets t400, Button raises to t800, SB folds, Hero?

Comments:
It would have been better to simply check here on the bubble with your flush draw. It costs another t400 to stay in a pot of t1800 after the button's raise, which gives you roughly 4.5:1 pot odds.

You will complete your flush on the turn 1 in 5 times, so it would seem as if you have the right pot odds to call. But this is the bubble phase and you shouldn't be risking any of your chips with speculative play.

You will lose a fair portion of your stack the 80% of the time you don't hit on the turn. Going all-in against the aggressor and big stack would not generate much fold equity, either, which is why you should simply fold.

Conclusion

A quick summary:
Odds are the ratio of: non-outs : outs
Pot odds are the ratio of: possible winnings : cost of staying in the hand

When the pot odds are better than the odds, meaning you would win more chips when you complete than you would lose when you don't complete, it is profitable to play a drawing hand.

Other factors come into play when you are in a tournament. From a strategic standpoint, you have to ask yourself: Even if I have the right pot odds, will staying in the hand cost me too much of my stack? And remember the principle of tournament play: Protect your stack and avoid unnecessary risks.

If you want to play profitable poker, you will have to understand the concepts of odds and outs, as well as the mathematics of poker. Knowing when it is profitable to call and how much you have to bet to make your opponent's draw unprofitable (not giving him the odds) is a major part of strategic poker. Take the time to understand the content of this article, your bankroll will thank you for it.

LINKS
Printable Versions: Overview of Odds and Outs [http://resources.pokerstrategy.com/Editorial/en/Data/ps_chart_en_outs_und_odds.pdf] [http://www.adobe.com/de/products/acrobat/readstep2.html] Related Articles: How Should You Play Speculative Hands? [/strategy/sng/774/] Advanced Strategy for Post-flop Play [/strategy/sng/778/]

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