Quick on the Draw (Part II)

By Tony Bee (a.k.a. ‘Tony’ on the Low Limit Poker Forum).


In my previous article I outlined the fundamental principles of drawing in Limit Hold’em.  We briefly considered outs, pot odds, implied odds and position.  In this article I’ll investigate drawing in more detail.  First of all I’d like to introduce a very useful concept.

Expected Value

Let’s revisit the scenario we looked at in the first part of this article (if you haven’t read it, take a look now.  What follows will make more sense).  To cut a long story short, the pot contains 4 big bets on the turn and it will cost you 1 big bet to see the river.  You have a gutshot to the nuts and there are no flush draws available.

We came to the conclusion that we should fold, since the amount we would win if we should hit our straight would be insufficient to make up for the times we called and didn’t hit.

It would be nice if there was a way we could quantify this, and other decisions like it.  Well there is.  We can calculate the expected value of calling.

Expected value can be applied to many different aspects of life, not just poker.  However, in this context expected value is a term used to describe the value of a poker decision taking into account all possible outcomes.  Effectively it is a number that indicates whether, in the long run, a decision is profitable or not.  If the expected value of our decisions at the poker table are mostly positive (that is, greater than zero), then we will make money in the long run.  If not, we will lose money in the long run.

Sometimes the expected value (from here on in I’ll abbreviate this to EV) can only be estimated, as we must make some assumptions about our opponents playing style.  However, provided we are prepared to do so then any poker decision can have an expected value associated with it.

It is useful to be able to calculate the EV, since many poker concepts are rather counterintuitive.  An EV calculation can often convince someone that a decision is correct, even though their intuition tells them otherwise.

Let’s look at the example I mentioned and calculate the EV of calling.

In this example the pot contained 4 big bets, but we were confident we could win another big bet on the river if we hit our gutshot.  We were faced with calling a single bet, so our implied odds were 5:1.  The EV of calling is calculated as follows:

Outcome if We Win * Probability Of Winning - Outcome If We Lose * Probability of Losing

So in this case,

5 * (4/46) - (42/46) = -0.48, rounded to 2 decimal places.  Since the EV is negative, calling loses money.

The reason I’m mentioning EV in an article about playing draws, is that we can use this idea to provide an alternative approach to drawing which some of you may prefer.  The derivation involves a little algebra but is not too difficult.  See if you can follow it.

I’m going to assume we’re drawing on the turn initially, since this makes matters slightly simpler.  The basic idea is that we require our EV to be greater than zero.

We’ve just seen that the EV is calculated as follows:

Outcome if we win * P(Winning) - Outcome if we lose * P(Losing)

The outcome if we win is the size of the pot we think we can win if we hit our draw, taking implied odds into account.  I’ll call this the expected pot size, or EPS.

The outcome if we lose will usually be one or two big bets, but it could be three or more in rare cases.  I’ll denote the amount we will have to call as the BET from here onwards.

P(Winning) is the probability of hitting our draw and therefore winning.  On the turn there are 46 unseen cards, so this is calculated by simply dividing our OUTS by 46.

P(Losing) is the probability of missing our draw and therefore losing.  We calculate this by dividing our non-outs (that is, cards that will NOT complete our draw) by 46.  I’ll call the non-out NOUTS for short.

Since we require EV > 0, we can now write this inequality in the following form:

EPS * (OUTS/46) + (– BET) * (NOUTS/46) > 0,

Or equivalently:

EPS * (OUTS/46) – BET * (NOUTS/46) > 0,

To make things simpler I’ll multiply both sides of this inequality by 46:


Then we replace NOUTS by the equivalent term (46 – OUTS):

EPS * OUTS - BET * (46 - OUTS) > 0

Expand the brackets:

EPS * OUTS + BET * OUTS - 46 * BET > 0

Add 46 * BET to both sides:

EPS * OUTS + BET * OUTS > 46 * BET

And factorise the LHS:

OUTS * (EPS + BET) > 46 * BET

PHEW!  Did you manage it?  Well done.  If you didn’t, don’t worry.  You don’t need to understand the derivation to use the inequality in your games.

Let’s assume for a moment that the amount we have to call is 1 bet, since this was the scenario we were looking at.

In this case BET = 1, so our inequality looks like this:

OUTS * (EPS + 1) > 46

The clever part is that we can use this inequality to tell us if our draw is profitable as an alternative to memorizing a chart.  All we have to do is to calculate the LHS in our head, and if this value is greater than 46 we should call.  If it is less than 46 we should fold.  If it is exactly 46 then it won’t matter in the long run what we do.  We can call or fold, take your pick.

So in the example we looked at we had the following information:

OUTS = 4 (since we had a gutshot to the nuts)

EPS = 5 (assuming we will win an additional 1 big bet on the river if we hit).

Substituting this into the LHS gives

4 * (5 + 1) = 24 (notice that we add the 1 first, since this is in parenthesis).

Now, since 24 is definitely not greater than 46, we should fold.  Simple, huh?

If we were instead drawing on the flop, the inequality would be identical except that the RHS would be 47, rather than 46.

Also, if we have to call more than one bet, we simply reduce the EPS to a ratio of n:1 then proceed as usual.

For example if our BET is 2 and our EPS is 10, we simply reduce this to 5:1 and use the number 5 in our calculation.  So we add 1 (to give 6 in this case), then multiply by the number of outs as usual.  If this number is greater than 46 (on the turn) we call.  Otherwise we fold.

So it’s simple.  We find our implied pot odds, add 1 and multiply by our outs.

I first saw this method explained in an article by Abdul Jalib called “The Theory of Sucking Out”.  Also, in his excellent book “Weighing the Odds in Hold’em Poker” King Yao also utilises a similar method.

With only a little practice you’ll find this method easy to apply as you play.  You may prefer it to memorizing an ‘outs to odds’ chart, particularly if you play live.  Have a go and see what you think.  Just remember that on the turn you use the number 46, but on the flop you use the number 47.

A Closer Look at Outs

So far we’ve looked at an example where it was pretty clear we had exactly 4 outs.  (Remember I’m defining an ‘out’ to be a card that will give us the probable winning hand).

However, often things will not be so simple.  Sometimes one or more of the cards you thought were outs for you will be ‘tainted’.  In other words although they will make your hand, they will make an opponent a better hand.  Therefore they are not outs at all, and you should ignore them.  This process is often called discounting your outs, and is very important.

For example, let’s say you are drawing to a straight in a multiway pot but there is a possible flush draw on the board.  Given the number of opponents and the betting so far you think it is likely that one of your opponents is on a flush draw.  In this case you would discount your eight outs to six, since two of the cards that will make your straight will make that opponent a flush to beat you.

Before we look at an example, it’s important to state that discounting outs is not an exact science.  The amount you should discount your outs will depend upon a number of factors, such as the number of opponents, the board, the betting action and so on.  It takes practice to become proficient at it, and there will always be a certain amount of error involved.

However, ANY discounting you attempt is FAR better than not discounting at all.  Remember that to be a winning player you don’t need to make perfect decisions every time.  You just need to make better decisions than your opponents most of the time.  I can tell you from experience that most players (particularly at the lower limits) definitely don’t bother to discount their outs.  In fact, many don’t consider whether their draw is profitable at all.  So even if you find discounting outs difficult at first, you’ll still have an edge over many (most?) of your opponents.

Let’s look at an example.

You hold QJ of hearts in the big blind.  An early player limps, a middle player raises and the button calls. You also call, and four players see a flop of 9s 8s 8d.  The middle player bets and the button raises.  What should you do?

Well, clearly there’s no chance your queen-high is winning at this stage.  Many players would reason that they have a gutshot and two overcards, giving them 10 outs.  They’re getting about 5:1 pot odds, so can easily call.  However, they’re forgetting to discount their outs for the possibility that things won’t go as planned.  In fact you should probably fold.

Firstly, the board is paired and contains a flush draw.  This means that if someone already holds an eight then hitting a queen or jack will be of no use.  Also the Qs and Js may give an opponent a flush.  Your overcards are quite weak in a raised pot, and even if a queen or jack comes, someone could hit an ace or king to beat you on a later street.  The raiser may even have AA or KK.  If someone has Q9 or J9 then hitting a queen or jack will again give you a second best hand.

In multiway pots overcards lose a lot of value, particularly if the board is coordinated.  A common rule of thumb is to count each overcard as 0.5 outs in these situations, rather than a full out.  If the overcards are also weak you may wish to discount even more.  In this example the pot is multiway and was raised preflop and on the flop, your overcards are weak, the board is paired and there’s a possible flush draw.  It couldn’t really get any worse as far as your overcards are concerned.  You might want to discount them to two outs in this situation, or even less.

The same logic applies to your gutshot draw.  One of your four outs is a spade, which could give someone a flush, and if someone already has an eight they could beat you with a full house if the board pairs.

So in this example you may estimate that you have two overcard outs and three for your gutshot, giving about 5 outs altogether.  You’re getting 5:1 from the pot, and you think you can probably win another big bet on the turn if you hit, so your implied odds are 7:1.

Recalling out ‘outs to odds’ chart (or using Abdul’s method if you prefer) we can see that these are not good enough odds to call, so we should fold.  Even if we were slightly more generous with our outs and the decision was closer, we should probably still fold since we are not closing the betting and the pot could be reraised behind us.

This example illustrates the importance of discounting your outs.  If you don’t do this you will call too often, and these mistakes will cost you money.  This procedure may seem tedious and intimidating at first, but with practice you’ll quickly be able to make a reasonable estimate in your allotted time.

Backdoor Draws

The addition of a backdoor draw (such as a backdoor flush or straight draw) gives some extra value to your hand (usually something between half an out and 1.5 outs), and can sometimes turn a fold into a call if the decision is otherwise close.  So if you have a close decision and you have a backdoor draw or two, go ahead and call.  But don’t overdo this.  It’s rarely correct to continue with only a backdoor draw, and you should usually only consider them in addition to other aspects of your hand.

And finally…

There is a lot of information in this and the previous article, and if you are a new player it will take you some time to assimilate these ideas into your game.  However, don’t be tempted to ignore these principles and just ‘go with your instincts’.  Although it will be challenging at first, you can be sure your hard work will be rewarded at the end of each month when you count your winnings.