Standard Lines: The Theoretical Foundations of Backdoor Draws

Introduction

In this article

• How many outs does a backdoor draw really have?
  
• Why raising for a free card isn't helpful
  
• Why backdoor draws are meaningless in No Limit Hold'em
  

Backdoor draws, or runner-runner draws, are draws that need to hit on both the turn and the river in order to improve to a made hand.

Examples:

a) You have A K on a 9 2 6 flop.
This gives you a backdoor flush draw (BDFD).

b) You have Q T on a J 4 6 flop.
This gives you a backdoor straight draw (BDSD).

Backdoor draws aren't particularly strong, but they can make the difference in close call situations. When you are torn between calling and folding on the flop, having a backdoor can be the reason for staying in the hand.

This article will teach you how to accurately estimate the strength (# of outs) of a backdoor draw. The following chart should give you a basic impression of how many outs you can give yourself for various backdoor draws.

Chart 1: Outs for various backdoor draws Draw Outs Backdoor flush draw 1.5 - 2 0-Gap backdoor straight draw 1 - 1.5 1-Gap backdoor straight draw 0.5 - 1 2-Gap backdoor straight draw 0 - 0.5

As you can see, there is a big difference between a regular draw and a backdoor draw. The fact that you need to hit twice means you need to include your investments on both the flop and the turn when calculating your pot odds. But how exactly do you include the additional costs that will incur on the turn in your formula?

In order to do this, you need to develop a deeper understanding of odds and outs. This article will go into detail and explain how the values in Chart 1 were derived.

How large must the pot be in order to call profitably?

Any time your draw is too weak for a value raise and you cannot generate enough fold equity with a semi-bluff, you have to ask yourself how large the pot must be for you to be able to make a profitable call.

When we talk about a decision being profitable, we are really talking about the Expected Value (EV) behind that decision. A decision is always profitable from a mathematical standpoint when the EV > 0. In order to answer our question, we need to determine the EV(call). Unfortunately, the exact EV formula is too complicated to be used during live play. But don't worry, there is a simple solution to this problem: odds & outs. The principle behind counting your outs and comparing them to your odds allows you to estimate the EV of a call within seconds.

Example: You have 4 clean outs (gutshot) on the flop and want to know whether or not you can draw to the turn. The formula, which you may know from the article on Expected Value is written as follows:

EV = Winnings* Probability of winning - Losses * Probability of losing

4 of the 47 remaining cards would give you the winning hand, while 43 would leave you unimproved. Your probability of winning is 4/47, your probability of losing is 43/47.

Winnings = Pot
Losses = Costs
Probability of winning = 4 / 47
Probability of losing = 43 / 47

-> EV(call, 4 outs) = 4 / 47 * Pot - 43 / 47 * Costs

Calling is profitable when you get a positive answer, meaning: EV(call, 4 outs) > 0

You can make a profitable call with a gutshot draw when you are being offered 10.75:1 pot odds or better.

So far so easy. You can use the same formula for any given number of outs (n). We can derive the following general formula for calculating the EV of calling with n outs:

From: EV(call, 4 outs) = 4 / 47 * Pot - 43 / 47 * Costs

We derive that: EV(call, n outs) = n / 47 * Pot - (47 - n) / 47 * Costs

Let's go through this step by step one more time and ask ourselves when the EV = 0, (the point where you break even).

These values provide the basis for the odds and outs charts. The fundamental connection between the size of the pot (pot odds) and the number of outs you have is clearly shown: once you know how many outs you can count on, you know how large the pot must be in order for you to profitably stay in the hand.

Let's assume you have to call a bet in our example. Your costs are therefore = 1. How large must the pot be for you to play? Plug in the values:

This gives our first basic formula:

I Pot = (47-Outs) / Outs

Note: Be sure to include the bet when calculating the size of the pot.

You can change the formula in order to determine how many outs you need to call when you know how large the pot is. Note: the pot size is calculated in (x) bets.

This gives us our second basic formula:

II Outs = 47 / (Pot+1)

You will usually know how many outs you have and be using Formula I to determine whether or not the pot is large enough for you to call. This is the formula you will use most often and is the same formula used to create the odds and outs charts.

We will turn to Formula II in order to determine how strong a backdoor draw is. Formula II tells us how determine the number of outs you have when you know how large the pot must be for you to call profitably.

How many outs does a backdoor draw give you?

We'll start by determining the expected value of a backdoor flush draw. But before we can determine the EV, we need to know the probability of winning with this hand.

Two things have to happen for you to win with a backdoor flush draw:

• 1. You have to pick up a flush draw on the turn.
• 2. Your flush draw must complete on the river.

We begin by calculating the probability of hitting on the turn and river:

P₁ = Probability of picking up a flush draw on the turn
P₂ = Probability of completing on the river.

P₁ = 10/47
P₂ = 9/46

10 of the 47 remaining cards can give you a flush draw on the turn. If one of them shows up, 9 of the 46 remaining cards can give you a flush on the river.

We therefore need to know the probability of hitting on both the turn and the river.

P_total = Probability of picking up a backdoor draw and completing on the river.

This is derived by simply multiplying both values.

The probability of holding a flush on the river when you have a backdoor door flush draw on the flop is therefore 4.16%.

The probability of losing is 1 - 0.0416 = 0.9584, or 95.84%.

Now that we have these values, we can determine the EV of a backdoor flush draw.

Winnings = Pot
Losses = Costs
Probability of winning = 0.0416
Probability of losing = 0.9584

This gives us our third formula, which is used for calculating the EV of a backdoor flush draw:

III EV = Pot * 0.0416 - Costs * 0.9584

But this isn't quite enough - we want to know how large the pot must be for a call to be profitable (EV>0).

If you must pay 1 SB to stay in the hand on the flop with a backdoor flush draw, there must be a little over 23 SBs in the pot for your call to be profitable.

Think back to Formula II. This formula tells you how many outs you need to call a bet in a pot that is x bets in size.

II Outs = 47 / (Pot + 1)

This formula allows you to translate the pot size into outs. You now know how many bets must be in the pot in order to call a bet with a BDFD - 23.02222. Plug this in to the formula to determine how many outs you can give yourself for your backdoor draw.

A backdoor flush gives you app. 2 outs according to this model.

The problem with this simple approach, which is used in the book Weighing the Odds, is that it neglects the fact that you are often not all-in on the flop and will often be confronted with another bet on the turn. The next section in this article will address the implications the future costs that can occur on the turn and the number of outs you can give yourself.

What effect does a turn bet have?

In the last section, we said that you could give yourself up to 2 outs for a backdoor flush draw, but this is was only true when we left out the possibility of incurring further costs on the turn.

You usually have the best hand when you hit your draw on the turn; hitting a backdoor on the turn isn't enough to win. You have to hit again on the river to win. And experience has shown that you will usually have to pay to get there, which means our formula is not exact enough.

Let's take a look at an imaginative No Limit game in order to get a better idea of the crux of the matter:

Situation: You are deep stacked and have nothing but a backdoor draw.

Assumption: Under certain conditions, you will never be able make a profitable call on the flop, no matter how good the pot odds are. A call could be -EV, even if you are getting 1,000:1 for your money.

Reason: Your opponent can bet any amount he wants in No Limit Hold'em. He may be giving you unbelievable pot odds on the flop, but you have no idea how much he will bet on the turn. The problem: you can't complete your backdoor draw without calling on both the flop and the turn. Your opponent's turn bet can be so large, that your call on the flop becomes -EV, no matter what odds you were getting. The reason: You can't view your decision on the flop as a single call, because it is the first of two calls which must be made when playing a backdoor draw.

If you and your opponent are both deep stacked and you know your opponent is going to push all-in on the turn, your backdoor draw is already worthless on the flop. There is no use in calling on the flop with great odds only to fold when you land a draw on the turn because you can no longer afford to stay on your draw.

This example might seem a bit artificial, but the fact of the matter is that it calling with a backdoor draw is almost never profitable in NL.

In order to evaluate the EV of a flop call with a backdoor draw, you must include a variable for the action on the turn in the EV formula. Assume you have a BDFD on the flop and are facing an opponent who always bets on the turn. In order to determine whether or not it will be profitable to stay in the hand, you need the basic formula found in the article on Expected Value.

EV = Payout₁ * Probability₁ + Payout₂ * Probability₂ + ... + Payout_n * Probability_n

There are three ways our example hand could play out, which gives us three possible payouts and three possible probabilities.

• You miss on the turn.

  Payout₁ = - Flop call
  Probability₁ = 37/47

• You hit on the turn, but miss on the river.

  Payout₂ = - (Flop call + Turn call)
  Probability₂ = 10/47 * 37/46

• You hit on both the turn and the river.

  Payout₃ = Pot + Turn call
  Probability₃ = 10/47 * 9/46
  The payout consists of the money in the pot plus your opponent's turn bet.

After plugging these values into the formula we get the following equation for the EV of calling with a BDFD:

IV EV = - Flop Call * 37/47 - (Flop call + Turn call) * 10/47 * 37/46 + (Pot + Turn call) * 10/47 * 9/46

You need 4.1:1+ pot odds on the turn in order to continue playing if you do pick up your flush draw. This will only be the case if your opponent's bet is not larger than 1/3 the size of the pot. Let's assume your opponent bets 1/4 of the pot. We can now define the price of calling on the turn as 1/4 * (Pot + flop call). Note: Include your opponent's bet when totaling the size of the pot.

After plugging in the numbers and a bit of calculating we get:

EV = 10/1081 * Pot - 1071/1081 * Flop call

Calling is profitable when EV > 0. After multiplying both sides by 1,081, we arrive at:

10 * Pot - 1071 * Flop call > 0

And finally:

Pot > 107.1 * Flop call

So, even if your opponent only bets 1/4 the pot on the turn, you still need 107:1 pot odds to call. A backdoor draw is clearly worthless in NL Hold'em, unless you are in an all-in situation on the flop or know that your opponent will give you a free card on the turn. We will now return our attention to Fixed Limit.

The size of the flop call is 1 SB and the size of the turn call is 2 SBs in FL. When we plug these values in we arrive at the following:

Calling is profitable when EV > 0.

You therefore need app. 29.3:1 pot odds to call with a backdoor flush draw on the flop knowing that you will have to pay 1 Big Bet on the turn. Our previous calculation said that, when guaranteed a free card on the turn, the pot would have to be 23 SBs or larger in order to call profitably on the flop. 6 additional SBs must be in the pot in order for you to call profitably when your opponent bets on the turn.

We can now use Formula II to calculate the number of outs you can give yourself now that we know that the pot must be 29,2444 SBs large in order to call profitably.

A backdoor flush draw gives you 1.55 outs when you will be required to call another bet on the turn.

The action on the turn costs you app. 0.4 outs. This may sound absurd - why should the action on the turn influence the number of outs that you have? But you should have realized by now that 'outs' is a synonym for 'necessary pot odds for a EV(call)>0.' And the fact that your pot odds are influenced by the action on the turn should be enough to make the connection.

We can take a look at another interesting aspect by rewriting the EV following. As we said:

EV = 45/1081 * Pot - 28/23 SB

45/1081 corresponds to a probability of 0.0416, or 4.16%. The odds of losing are therefore 0.9584, or 95.84%.

In order to arrive at the simple EV formula: EV = Winnings * Probability of winning - Losses * Probability of losing, we turn the 28/23 SB into 1.2703 * 0.9584.

Which gives us:

EV = 0.0416 * Pot - 0.9584 * 1,2703 SB

This isn't a new formula, it's just a new way of writing the same formula that makes it easier to include future costs on the turn. Let's compare the two formulas two each in two examples.

• Your opponent checks the turn:
  EV = 0.0416 * Pot - 0.9584 * 1 SB
• Your opponent bets the turn:
  EV = 0.0416 * Pot - 0.9584 * 1.2703 SB

The additional costs incurring on the turn total to app. 0.27 SB. There are two reasons why they are so much less than 1 Big Bet.

• You don't have to pay 2 SBs on every turn; you only pay if you pick up your draw.
• When you do pay 2 SBs on the turn, 20% of it belongs to you (due to equity).

The average costs on the turn are therefore only 0.27 SBs. This leads to an important conclusion:

A free card raise on the flop is based on the idea that you pay an additional SB* on the flop in order to avoid paying 2 SBs on the turn. Your turn costs are, however, only 0.27 SB. If you miss on the turn, there's nothing a free card can do to help. There is no sense in trying to buy free card on the flop for 1 SB when you will only have to pay 0.27 SB on average. You also risk facing a 3-bet (or donk bet on the turn) if you try raising for a free card. Raising for a free card can backfire, and even if it works, it doesn't help you at all.

Of course, this shouldn't mean that you should never raise the flop with a backdoor draw. There are a number of reasons for raising on the flop. If you can generate sufficient fold equity or have an additional draw which makes your hand strong enough for a value raise, do so. Just don't raise hoping to get a free card.

* You don't actually pay one full SB, since you have pot equity. A BDFD gives you 46% equity on the flop heads up, 29% equity 3 handed, 21% 4 handed, etc. The costs also vary from 0.92 SBs heads up to 0.84 SB 4 handed. The costs of raising for a free card are still higher than the average costs (0.27 SB) you will incur on the turn.

A quick review

What have you learned so far?

The decisive question behind every decision in poker is, "What is the EV?" Any time you have a draw which is too weak for a value raise, and when you cannot generate sufficient fold equity, the question is whether or not calling will be +EV. If you know how many outs you have, you can easily tell how large the pot must be for you to call profitably.

Pot = (47-Outs) / Outs

When you have a backdoor draw, you don't know how many outs you have. You can, however, use the EV formula to find out. The pot size can be translated into outs:

Outs = 47 / (Pot+1)

In a NL game, your opponent can bet so much on the turn, that you can't make a profitable call on the flop, matter what odds you are getting. Additional costs also come into play in FL. Since you have to hit both the turn and the river, any bet on the turn ultimately reduces your odds.

You can give yourself 1.55-2 outs for a backdoor flush draw when you are heads up and your opponent bets on the flop. This means there must be between 23.02-29.3 SBs in the pot after your opponent bets. 23.02 are enough if you know your opponent never bets on the turn, but you will need 29.3 if he is certain to bet. This, of course, is all theory. When you put it to practice, you'll find that the truth is somewhere in the middle.

So far, we haven't looked at any practice-oriented examples. You will, for example, often be playing in multiway pots. What effect do additional opponents in the hand have on your play? There are a lot of new possibilities: You might end up having to pay 4 SBs on the turn, but also have the chance to win an even bigger pot.

Another question: When are you going to be getting 24:1 pot odds and have nothing better than a backdoor draw? And how does a backdoor draw increase the strength of an additional draw? Can you simply add up all the outs you have for each draw?

These questions will be addressed in the following sections. We will also take a closer look at various backdoor straight draws (BDSD) and the number of outs they give you.

Multiple turn bets

So far we've only considered two possibilities: Either you get a free card on the turn, or you pay 1 Big Bet. You can easily find yourself confronted with 2+ Big Bets in multiway pots. This requires a few adjustments to the EV formula.

We'll start with the general EV formula:

EV = Payout₁ * Probability₁ + Payout₂ * Probability₂ + ... + Payout_n * Probability_n

Let m be the number of opponents and z the number of Big Bets you must pay on the turn.

This hand can play out as follows:

• You miss the turn.

  Payout₁ = - 1 SB
  Probability₁ = 37/47

• You hit the turn, but miss the river.

  Payout₂ = - (1 SB + 2*z SB)
  Probability₂ = 10/47 * 37/46
  Your costs on the flop are 1 SB plus z Big Bets on the turn (but defined in SBs). If, for example, an opponent 3-bets (z=3), you must pay 2*3 = 6 SBs on the turn.

• You hit on the turn and on the river.

  Payout₃ = Pot + 2*m*z SB
  Probability₃ = 10/47 * 9/46
  You win the pot plus your opponents' turn bets. Against two opponents (m=2), with a 3-bet (z=3): Winnings = Pot + 12 SBs.

This gives us the following formula for taking multiple turn bets into account:

EV = -37/47 * 1 SB - 10/47 * 37/46 * (1 + 2*z) SB + 10/47 * 9/46 * (Pot + 2*m*z SB)

You can now define the EV as > 0 in order to determine how large the pot must be in order to call profitably. The following chart shows how large the pot must be for the EV to be greater than 0.

Chart 2: Necessary pot size to call with a BDFD depending on the number of opponents and the number of turn bets   # of turn bets # of opponents 0 1 2 3 4 1 23.0 29.3       2 23.0 27.3 31.5 35.7 39.9 3 23.0 25.3 27.5 29.7 31.9 4 23.0 23.3 23.5 23.7 23.9

You can now determine how many outs you have by using the transformation formula: Outs = 47 / (Pot+1)

Chart 3: Converted to outs   # of turn bets # of opponents 0 1 2 3 4 1 1.96 1.55       2 1.96 1.66 1.45 1.28 1.15 3 1.96 1.78 1.65 1.53 1.43 4 1.96 1.93 1.92 1.90 1.89

Chart 2 shows that you would need 40:1 pot odds if you got caught in a raising war and the turn is capped. Any additional opponent in the hand hardly has an effect on this, since you have 20% equity. You would even have average equity against 4 opponents and would profit from the higher implied odds. A BDFD gives you almost 2 full outs against more than 4 opponents and against several calling stations, but is only worth 1.3 outs against a maniac (see chart 3).

You will often end up having to pay 1 BB in a 3 handed pot. You may have to pay 2 BBs at times, but you will also get a free card now and then. As to give a concrete value; a needed potsize of 27,3 SB or 1,65 outs is a realistic estimate.  That is how much a BDFD is worth on average.

As a general rule of thumb we can say: the more you are likely to have to pay on the turn and the fewer opponents involved in the hand, the weaker your backdoor draw. In other words: Your call on the turn is nothing other than reverse implied odds and the higher your RIO, the fewer your outs.

Situations in which you can profitably draw with nothing but a backdoor are rare. We will now take a look at a few examples of weak draws + backdoor draws.

Backdoor combodraws

You will often be holding a weak draw and facing a tough call/fold decision. Having a backdoor can make the difference. So far, we have only looked at backdoor draws in and of themselves. Now it's time to take a look at how they can improve a weak (primary) draw and how you can add up your total number of outs.

Example: You have two overcards and give yourself n outs: you also have a backdoor draw. If you completely miss on the turn, you will have to fold. If you hit on of your n outs, you will have the best hand. If you hit your backdoor, you will have 9+n outs to the river.

Let's return our attention to the EV formula:

EV = Payout₁ * Probability₁ + Payout₂ * Probability₂ + ... + Payout_n * Probability_n

The hand can play out as follows:

• You completely miss the turn.

  Payout₁ = - 1 SB
  Probability₁ = (37 - n)/47
  If you miss the turn, you will have to fold. Your investment will be exactly 1 SB.

• You hit your backdoor on the turn, but miss on the river.

  Payout₂ = - 3 SBs
  Probability₂ = 10/47 * (37 - n)/46
  There are 9+n good and 37-n bad cards that could show up on the turn. You will miss (37 - n)/46 and lose 3 SBs.

• You hit your backdoor on the turn and complete for a flush on the river.

  Payout₃ = Pot + 2 SBs
  Probability₃ = 10/47 * (9 + n)/46
  You have 9 outs for a flush and n outs for your overcards on the turn. You can win the pot plus all opponents' bets.

• You hit one of n outs on the turn.

  Payout₄ = Pot
  Probability₄ = n/47

Now that we have defined all the variables we can plug them into the formula:

EV = - (37-n)/47 * 1 SB - 10/47 * (37-n)/46 * 3 SB + 10/47 * (9+n)/46 * (Pot + 2 SBs) + n/47 * Pot

Define the EV of calling as > 0 and you arrive at:

Pot > (1316 - 48n) / (45 + 28n)

You can check this formula by defining n as 0. Our previous result of pot > 29.3 should then be the result. Once you have this answer, you can then use Formula II to determine how many outs you can give yourself.

This gives us the total number of outs. Since, however, we want to know how many outs the backdoor flush draw gives us, we simply remove the outs (n) from our primary draw:

Outs = (2115 +1316n) / (1361 - 20n) - n

The graph below illustrates this formula and shows you how many outs a BDFD gives you in relation to the number of outs your primary draw gives you.

Value of a BDFD combined with a n-outer y-Axis: Outs for the BDFD x- Axis: n outs of your primary draw

The more outs in your primary draw, the more valuable your backdoor. This is due to the fact that you will usually have to fold your primary draw if you don't hit on the turn, meaning you give up pot equity. Any time your backdoor hits, you have enough outs to stay in the hand and don't have to forfeit the equity you would otherwise have to give up. This applies to weak draws, since you would call with a strong draw (OESD) even if you didn't have a backdoor open. This is why the chart does not go beyond 6 outs in the x axis.

Backdoor straight draws

Backdoor straight draws (BDSDs) vary in strength. You will almost always win when you complete a BDFD, since it is unlikely that someone else will have a higher flush and a full house is unlikely (unless the board is very inviting). The problem with BDSDs is that they can end up losing to flushes, which is why they are only particularly strong on a rainbow board. BDSDs must be discounted on 2 suited boards and are completely worthless on single suited boards.

There are also three kinds of BDSDs: 1, 2 and 3-gappers.

Chart 4: Backdoor Straight Draws Your Hand: J9 Flop Outs to turn 0-Gap T32 8 outs to OESD (Q, 8) 8 outs to gutshot (K, 7) 1-Gap 832 4 outs to OESD (T) 8 outs to gutshot (Q, 7) 2-Gap 732 8 outs to gutshot (T, 8)

But be careful: KQ on a J32 flop is a 1-gapper, since the T alone would give you an OESD. An A or J would merely give you a gutshot. Another example is: AK on a Q64 flop. This BDSD is ultimately a 2-gapper, since it only gives you 8 outs to a gutshot.

Another less important factor that influences the strength of a BDSD is the strength of your hole cards. There is a big difference between QT and 86 on a 922 flop. Aside from pair outs, QT lets you draw for two nut straights (QJT98, KQJT9), whereas 86 does not (98765 = nuts, T9876 = 2nd nuts).

Note: The more gaps you have, the more important for your hole cards to be on the higher end of the possible straight.

Let's get down to business and calculate the number of outs this draw gives us. We start by calculating the EV.

Let k represent the number of gaps in your BDSD. Your hand can play out as follows:

• You improve to an OESD.

  Payout₁ = 8/46 * (Pot + 2 SB) - 38/46 * 3 SBs
  Your odds of completing an OESD are then 8/46; the odds of calling and missing (and losing 3 SBs) are 38/46.
  Probability₁ = (8 - 4*k)/47
  A 0-gapper has 8 OESD-outs, a 1-gapper 4, and a 2-gapper 0.

• You improve to a gutshot.

  Payout₂ = 4/46 * (Pot + 2 SBs) - 42/46 * 3 SBs
  Your odds of completing are then 4/46; the odds of calling and missing (and losing 3 SBs) are 42/46.
  Probability₂ = 8/47

• You miss completely.

  Payout₃ = - 1 SB
  Probability₃ = (31 + 4 * k)/47
  The odds of missing completely are simply: Probability = 1 - (8 - 4k)/47 - 8/47 = (31 + 4 * k)/47

Now we can plug these values into the EV formula:

EV = [8/46 * (Pot + 2 SBs) - 38/46 * 3 SBs] * (8 - 4 * k)/47 + [4/46 * (Pot + 2 SBs) - 42/46 * 3 SBs] * 8/47 - 1 SB * (31 + 4 * k)/4

By defining the EV as greater than 0 we see:

Pot > [(1673 - 136 * k) / (48 - 16 * k)] - 2

That after translating the pot size into outs using Formula 2, we arrive at the values found in the following chart.

Chart 5: Necessary pot odds and number of outs given by BDSD under good conditions Pot odds Outs 0-gap 33:1 1.39 1-gap 46:1 1.00 2-gap 86:1 0.54

Good conditions means: You are drawing for the nuts (rainbow board), you aren't facing very high reverse implied odds on the turn, and the pot is so large that you can call on the turn if you pick up a gutshot.

And now to the third and most important point: When you improve a BDFD on the turn, you will almost always be getting the right pot odds to draw to the river. The same is true when a BDSD turns into an OESD. The only time it isn't true is when you only pick up a gutshot.

You will almost never be getting 46:1 pot odds on the flop, meaning you will never draw on nothing but a 1-gapper. You are much more likely to be holding a primary draw (like overcards) + a BDSD. Take a look at the following example to see why this then becomes much more problematic.

0.5/1 Fixed Limit Hold'em (5 handed)

Pre-flop: Hero is MP3 with J, T
Hero raises, CO folds, BU 3-bets, SB folds, BB calls, Hero calls.

A TAG 3-bets you and the BB cold calls. If the TAG is liberal with his re-raises and throws in a few deceptive 3-bets, his range will be around 11%. The BB will usually have a wider range, but not too large. You can put him on a range of app. 25% excluding QQ+, with which he would have capped.

Realistic ranges are:

BU: 77+, A9s+, KTs+, QJs, ATo+, KQo
BB: 22-JJ, A2s+, K7s+, Q8s+, J8s+, T8s+, 97s+, 87s, 76s, 65s, 54s, A8o+, KTo+, QTo+, JTo

Flop: (9.50 SB) 8, 3, 5 (3 players)
SB checks, Hero checks, BU bets, BB calls, Hero calls?

In order to get a better picture of your outs, take a look at how they effect your equity. A J or T on the turn gives you app. 49% equity. The overcard outs must therefore be discounted by 50%. You are also oop against the PFA, who has a rather strong range, which means your overcards give you RIO. All in all, you can give yourself 2.5 outs for your overcards.

With 11.5:1 pot odds, you need app. 1 additional out in order to call with your backdoor straight draw. Your 3.5 outs mean you need 12.4:1 to call. You may, however, get a free card on the turn.

We determined that a 1-gapper is worth 1 out. You could argue for a call. But keep in mind, the values in the chart only apply under good conditions. We assumed you would see the river if you pick up a straight draw on the turn. But what can change on the turn?

• a) Turn: (6.25 BB) Q (3 players)
  BB checks, Hero checks, BU bets, BB calls, Hero calls.

  You improve to a gutshot and are being offered 8.25:1. You can give yourself 2 BBs in implied odds on this board, and your pair outs may be worth something, as well. Calling is correct.

• b) Turn: (6.25 BB) Q (3 players)
  BB bets, Hero folds, ...

  You improve to a gutshot, but the BB donks on the Q, which makes your overcard outs relatively worthless. The aggressive BU could raise behind you, too. AA, KK, AQ, KQ are all in his range. You have no choice but folding with just 7.25:1 pot odds.

• c) Turn: (6.25 BB) 7 (3 players)
  BB checks, Hero checks, BU bets, BB raises, Hero folds, ...

  Another similar situation. You could have given yourself outs on your overcards if the BB hadn't raised. He could either be slowplaying a strong hand or have hit a set of sevens. Your outs aren't clean even if he only has T9 or a flush draw. The BU could also 3-bet with an overpair. You could then make an easy lay down with only 9.25:2 pot odds.

Sometimes you have to fold even after you improve on the turn, since you only have a gutshot and the pot odds are too low. There was no need to apply the formulas in this situation, since we assumed that you would always be getting the correct odds to call on the turn. As a result, the gutshot outs must be discounted and the OESD outs are the only ones that really help.

Calling on the flop was therefore a mistake. Take a look at an example in which calling could be correct:

0.5/1 Fixed-Limit Hold'em (5 handed)

Pre-flop: Hero is MP3 with J, T
Hero raises, CO folds, BU 3-bets, SB folds, BB calls, Hero calls.

Flop: (9.50 SB) 9, 3, 5 (3 players)
SB checks, Hero checks, BU bets, BB calls, Hero calls!

You can give yourself at least 1 out for your backdoor in this situation. Any Q or 8 would give you an OESD, which would mean you can stay on your draw. You will rarely have to fold an OESD.

BDSDs, on the other hand, must be discounted for these three reasons. Multiway pots (higher implied odds) and likely free cards increase the strength of a BDSD. The following chart provides a good guideline for determining how many outs you can give yourself for a BDSD.

Chart 6: Outs for BDSDs under realistic conditions Rainbow flop 2-suited flop 0-gap 1.5 1 1-gap 0.75 0.5 2-gap 0.25 0

Realistic conditions: The pot is not very large and you can definitely call with an OESD. You may have to fold a gutshot. You may also be confronted by several bets on the turn (higher RIO). Facing several passive opponents gives you higher implied odds and better chances at getting a free card.

You can simply add your BDSD outs to your primary draw outs.

We saw that BDFDs' strength increases with the strength of the primary draw. This was because you can stay in the hand if you pick up a flush draw on the turn and do not have to forfeit the equity provided by your primary draw. This is not the case with BDSDs.

Your primary draw with a BDSD will almost always be overcards, for example: J9 on a T53 flop. A king or queen would give you a straight draw, but you then have to discount your outs on the J and 9. And even if you do hit a pair, the board will become very connected by the river. J9 on a T5389 board could even lose to QJ, J7 or 76. The outs for your primary draw lose value when you improve to a straight on the turn, which means you lose pot equity on your primary draw. This is not the case with BDFDs.

BDFDs can also combine with gutshot draws. A gutshot is a strong primary draw; if you hit you will usually have the nuts, no matter what other card shows up. There is no such thing as a BDSD + gutshot.

You can simply add the outs from a BDSD to the outs of a primary draw, since your primary draw does not profit by being accompanied by a BDSD.

Put to practice

The implied odds basically play the same role as with a regular draw. If you can give yourself 1 BB in implied odds and there are 8 SBs in the pot after your opponent's bet, you can give yourself 10:1 effective pot odds. The main difference between backdoors and regular draws: you only have the chance to cash in on implied odds on the river with a backdoor.

You get two chances to extract value when you complete a normal draw on the turn; when you have a backdoor, you can't bet for value until you've reached the river. Be careful when estimating your implied odds.

The implied odds of a primary draw + backdoor draw can be calculated as follows: Assume you have 4 outs for a gutshot for which you give yourself 3 BBs in implied odds, and a BDFD for which you give yourself 1 BB in implied odds. You have a total of 6 outs and an average in implied odds of (4 * 3 BBs + 2 * 1 BB) / 6 = 2.33 BBs.

As you can see, your overall implied odds are dominated by your primary draw. You can give yourself higher implied odds with a strong primary draw, and lower implied odds with a weaker draw. The backdoor draw doesn't change much.

Any time you are only using one hole card, your draw is significantly weaker. The odds of splitting the pot with a backdoor straight draw increase greatly. The implied odds decrease greatly on very draw heavy boards, since the danger of facing a flush or straight is much higher.

When you have a 1 card flush draw, you could already be drawing dead (unless you are drawing to the nut flush). The most important factor is then the number of opponents in the hand. In general, you should discount 1 card BDFDs as you would 1 card flush draws on a single suited flop.

A BDFD weaker than Q high is worthless in multiway pots. Any flush is worth more heads up, since the ranges are much wider, but you should still discount weak BDFDs by 50%. After all, a weak 1 card flush only has 2/3 equity against a random hand.

Of course, every situation is unique and must be evaluated individually. The following chart provides a good basis:

Chart 7: Modified outs for 1 card backdoor flush draws BDFD Multiway Heads-up and 3 handed A high 1.5 1.5 K high 1 1.25 Q high 0.5 1 J-9 high 0 0.75 8 high or weaker 0 0.5

You can add outs for BDFDs and BDSDs together.

Sometimes you will end up with both. These two draws can overlap, but this combination can also turn into a monster draw on the turn. We explained with backdoor outs are sometimes very weak, because they can only improve to a gutshot. The chance of improving to a flush draw as well means you can easily stay in the hand with 9 + 3 outs. The overlapping has no negative effect on your hand.

Multiple backdoor straights can be added in a somewhat weaker form. Take A5 on a KT2 flop, for example. This gives you two 2-gappers. These two 2-gappers are weaker than one 1-gapper, however, since you cannot improve to an OESD, but rather only to a gutshot.

Another example: 98 on a J62 flop. This gives you two 1-gappers. A ten or seven would give you an OESD (8 outs), a Q or 5 a gutshot (8 outs). Two 1-gappers theoretically give you 8 outs for an OESD and 16 outs for a gutshot. The missing gutshot outs are a result of overlapping draws. These 8 missing outs are so weak on the other hand, that you hardly need to discount.

Conclusion

Note: The higher the reverse implied odds on the turn, the weaker your backdoor draw.

The best situation with 0 reverse implied odds is an all-in situation on the flop. In this case, a backdoor flush draw is worth 2 outs. You can also give yourself 2 outs against loose-passive opponents, especially in multiway pots and when you have other outs. Be careful against one or more loose-aggressive opponents. The RIO on the turn only allow you 1.5 outs for your BDFD.

Pay attention to the gaps when you have a BDSD. A single gap reduces the value of a BDSD by 50%. The strongest BDSD is a 0-gapper on a rainbow board. Discount your outs as heavily as max. 0.5 outs on two suited boards.

The disadvantage of backdoor straight draws is the fact that you may have to give up your hand even if you improve (when you only pick up a gutshot). This is why a 1-gapper is only worth 0.5 outs and why 2-gappers are almost always worthless.

Despite everything we've said, make sure you don't forget what it's really all about: a backdoor draw is usually just enough to tip the scales in a close call situation. It's not really worth anything in and of itself.

There are also other aspects that have a strong effect on your decision on the flop: Can you generate fold equity? Do you have showdown value? Can you buy any outs? How strong is your primary draw? Could you possibly raise for value? How high are the (reverse) implied odds? Be sure to address these questions first.

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