I was pretty sure that looking at all the postflop stuff would be good, but I also thought there wasn’t a whole lot of discussion about why we choose the hands we choose preflop, so this post will cover both aspects of the game. The preflop stuff is mostly mathematical with some reads, and the postflop stuff is mostly reads with some math…any surprises there? There shouldn’t be!! 


Preflop

So…what’s a drawing hand? Basically, it’s a hand that you don’t rate to be the best right now, but has the potential to become something huge later. To be fair, any hand has the potential to become something huge, so let’s take a look at some hand-types and see how likely each of those is! This first set deals with sitting with any two unpaired cards and we’ll consider “huge” to be 2pair plus.

If we have two random, unpaired cards:

flopping EXACTLY two pair by pairing EACH of your hole cards 2.02%
flopping EXACTLY trips by flopping two cards to one hole card 1.347%
flopping EXACTLY a full house, trips of 1 hole card and pairing the other 0.092%
flopping EXACTLY four of a kind, three cards to one of your hole cards 0.01%
flopping four to a flush 2.245%


If we have any 2 suited cards, we figure the first 4 plus:

flopping a flush (including the slight chance of a straight flush in some cases) 0.842%
flopping four to a flush 10.944%

If we have connectors:

flopping a straight (including the slight chance of a straight flush in some cases) 1.306%
flopping an 8 out straight draw* 10.449%

If we have one-gappers:

flopping a straight (including the slight chance of a straight flush in some cases) 0.980%
flopping an 8 out straight draw* 8.08%

If we have a pair:

flopping EXACTLY two pair by pairing the board 16.163%
flopping EXACTLY trips by flopping a set for your pocket pair 10.775%
flopping EXACTLY a full house, a set to your hole pair + pairing the board 0.735%
flopping EXACTLY a full house, by the board tripping up 0.245%
flopping EXACTLY four of a kind, two cards to your hole pair 0.245%

Dealing with SC’s is HIGHLY math intensive…fortunately, this amazing thread has done it. http://archives1.twoplustwo.com/show...0&fpart=1&vc=1

This is a lot of info to take in, but the general idea is that while any two cards can flop a huge hand, the likelihood of flopping something big with 27o is much lower than with 55 or 67s. Think about it. With 78s we can hit all the same types of hands that 27o can hit, but also hit straights (and straight draws), flushes (and flush draws) and occasionally straight flushes. Looking at it from an equity standpoint:

Text results appended to pokerstove.txt

41,095,296 games 0.016 secs 2,568,456,000 games/sec

Board:
Dead:

equity win tie pots won pots tied
Hand 0: 22.475% 22.29% 00.18% 9161616 74394.00 { 76s }
Hand 1: 77.525% 77.34% 00.18% 31784892 74394.00 { AA }


---

123,285,888 games 0.016 secs 7,705,368,000 games/sec

Board:
Dead:

equity win tie pots won pots tied
Hand 0: 11.800% 11.59% 00.21% 14288040 260154.00 { 72o }
Hand 1: 88.200% 87.99% 00.21% 108477540 260154.00 { AA }

It shouldn’t be much of a surprise that a suited connector holds up much more often than 27o, but realize that a HUGE percentage of this difference is from the draws (mainly because they’re both 7-high hands, so the high-card value is nearly the same. For those of you who are interested, 56s has 22.5% equity vs AA and in terms of sole high-card value, 72>56).

Why does any of this matter?

Well, assuming we’re going to play straight-forward (meaning that we’re folding if we totally miss), we’re going to need to make a good deal of money when we do hit (in order to counter the losses we incur when we miss). The amount of money we need to earn can be roughly calculated by:

Money made > preflop investment x (100\percent chance of hitting).

If you’ve heard of the 5 and 10 rule, or the 3 and 6 rule, this equation is where those rules came from. Let’s look at each aspect of this equation.

Money Made

So…what determines how much money we can make?

a) Stack sizes: Obviously, the maximum we can win is the villain’s (or villains’) stacks.
b) Villain range: The stronger the villain’s range, the more likely he is to be willing to put the monies in!
c) Villain tendency: Some villains don’t like to fold…some fold a whole lot. Those that fold less will give us more money (assuming we’re playing straight-forward)
d) Board texture: If the board looks really scary, our villain may find a reason to fold. If it looks safe, he may be more likely to pay us off.
e) Our image: If we look really tight/nitty, an observant player will fold to us more often than if we’re raising and betting all over the place.

Preflop Investment:

What do I mean by “preflop investment”? Basically, I mean the amount of money you put in preflop. (Whether you’re calling a raise, or limping in). It should make sense that the equation is easier to satisfy as our preflop investment goes down. In fact, some hands that are highly profitable to play for a small investment, are worthless when the “price of poker” gets too high. Note that this is also directly related to the effective stack size. Let’s look at a couple of situations.

Pokerstars $1/$2 (assume all other players are 100BB deep)

Villain ($ 120) (UTG)

Hero ($120) (BTN)

Hero has 6h 6d

Villain raises to $xx

Hero…?

If we look at our equation, then money made needs to be > investment x (100/10.8)
Investment x (9.26)

Since the cap on the money made is $120 (assuming everyone folds), then the most we should be willing to invest is $12.96. Of course, this assumes that villain is stacking off on every flop (which isn’t the case) so, it should actually be less than that. (The rule of 5/10 says we should be willing to pay $12 max, given good reads). In essence, if the villain raises to $13 or more, we should be folding now if our plan is to fold the flop if we miss.

However, if the stacks are $250, we can call a much larger raise and still be able to make monies!

Percent Chance of Hitting:

We’ve covered that, pretty much! Feel free to take a look back for some percentages, or…..

CLIFF NOTES (rougly)!
Hitting with jank: 5.714%
Hitting a straight or straight draw with connectors: 11.85%
Hitting a set with a PP: 10.8%
Hitting a flush or flush draw with sooted cards: 11.85%
Flopping sexy with a SC (sd, fd, combo draw, or made hand: roughly 25%

Note: Flopping sexy is not my term, but it’s awesome.

Postflop

Once we’ve gotten to the flop and villain has bet into us, we’ve got three actions at our disposal. I’ll talk a bit about each and the reasons we can choose each, then go into more detail! But FIRST…some basic definitions.

Express Odds:

Basically, the odds that we’re getting to call a bet, based on the current pot size. For example, If the pot is $12 and villain bets 6 into the pot, we need to pay $6 into a pot of $18 to see the next card. The pot is offering us 3-1. We often compare these with our odds to hit (approximated by multiplying the # of outs we have by 2).

Implied Odds:

Much less basic, I’m afraid. Here, we’re interested in the amount of money we stand to make later in the hand. Whereas express odds are mathematic, implied odds are more psychological and depend on villain’s range/tendencies, your perceived range, remaining stack sizes, and board texture.

Fold Equity:

Technically, the amount of money in the pot multiplied by the likelihood that villain folds to a raise. This is used when figuring out whether a raise is going to carry more equity than a call. For our purposes, we will consider the likelihood that villain will fold to a raise (another psychological estimation).

So…what are our choices?


Raise!

Almost every time I’ve brought up the topic of playing draws, the person I’ve been talking with has said “Raise, ldo!!”. I think that in any situation, raising should be the first move we consider because it gives us a second way to win the pot. (Either we can hit our draw later, OR villain can fold his hand to our raise). Aside from that, it lets us balance our range for raising on the flop. Consider an 8/2/1 player cold-calling your preflop raise in position 60BB deep. The flop comes down Kh, 5s, 6s. You make a cbet with AsKs, and the villain raises pot. It’s pretty safe to say that the villain has 55/66 here. How are we holding up?

Text results appended to pokerstove.txt

5,940 games 0.005 secs 1,188,000 games/sec

Board: Kh 6s 5s
Dead:

equity win tie pots won pots tied
Hand 0: 29.949% 29.95% 00.00% 1779 0.00 { AsKs }
Hand 1: 70.051% 70.05% 00.00% 4161 0.00 { 66-55 }

Yeah…poorly. I’d actually advocate a fold here (I’ll explain later). It sounds really nitty, but those are the breaks. However, what if the villain was 14/11/3 and somewhat tricky? His range now includes a whole mess of hands that we’re glad to get AI in with on the flop, including draws and 2pair hands (we’re slightly ahead here, and with the dead money in the pot, we’re certainly not folding…I’d probably just jam here). Why did I bring this up? Because when villain looks at our flop raise, we want him to have a hard time putting us on a hand. It means we’ll be more likely to get paid off when we do have a set, and many villains will be less inclined to cbet with air since we’ll stick around with a wider range.

On top of that, raising allows us to get more money into the pot while our equity is the highest. More often than not, we are going to miss our draw on the next turn, and when that happens our equity plummets. Case in point:

Text results appended to pokerstove.txt

990 games 0.005 secs 198,000 games/sec

Board: 9h Th Ad
Dead:

equity win tie pots won pots tied
Hand 0: 55.354% 55.35% 00.00% 548 0.00 { 8h7h }
Hand 1: 44.646% 44.65% 00.00% 442 0.00 { AcKd }


---

We’re ahead!!


Board: 9h Th Ad 3s
Dead:

equity win tie pots won pots tied
Hand 0: 34.091% 34.09% 00.00% 15 0.00 { 8h7h }
Hand 1: 65.909% 65.91% 00.00% 29 0.00 { AcKd }


Not anymore, though. When we miss on the turn, we’re very likely going to be behind, and in poker, we want to get as much money in the pot while we’re ahead. Raising the flop allows us to do just that!

Pressure!

This is especially good when the stacks are deeper and we know villain won’t go to the felt with an OP, but will call a flop raise, and possibly a turn bet. You really want to have a good read here. Villain will NEED to put you on 2pair+ in order to fold, but also, you’ll have to know that he won’t take this line for value against an aggressive player. Sometimes, you’ll also be doing this with strong draws to build the pot (preferably if villain won’t fold an OP) and get the stacks in on the river (something tough to do without a flop raise).

Cliff notes:

Raising is good because it lets us get more money in while we have the most equity, allows us to win by getting our opponent to fold, and balances our raising range. Aside from that, it lets us build the pot when we want to and put pressure on our opponents.

Given all that, why the hell would we do anything else?


Call! (?)

Why call when you can raise? Primarily because there will be a TON of spots where the advantages of raising won’t apply.

But…fold equity! Yeah…what if villain doesn’t fold? If you’ve assessed that villain’s range is very strong, or if villain never folds in general, you’re probably better off calling. Case in point (adapted from a 2p2 HH).

Hero (100BB) 17/14

BTN (100BB) 14/10

SB (28BB) 60/5

Hero opens KQcc from MP1 for 3BB. BTN and SB call, all others have folded.

Flop (10BB pot) Th Jc 7d

SB checks, Hero bets 6BB, BTN raises to 12BB, SB folds…Hero?

Given BTN’s stats and the fact that he minraised with an obvious donk behind him, I’d say his hand is pretty strong (I’d say we’re looking at 2pair+ a great deal of the time). We have an OESD here, but I’m pretty certain that villain’s not folding if we reraise. However, we’re getting 28-6 on our call (almost 5-1) and we’ve got 8 clean outs (to nut draws, no less). Even though we will be OOP, calling is the clear play here and it’s certainly better than reraising. (Not to mention that if we show down, villain will see that our range for calling a flop raise OOP includes draws when the price is right.)

What about balancing our range? Balancing our range is very useful against opponents who are putting us on ranges to being with. Against unknowns or players that aren’t observant, we really don’t benefit from range balance.

We have the most pot equity now…If we call, it’s likely to go down. This is true, yes, but unless our draw is VERY strong, we’re likely to be an underdog vs villain’s range. If we don’t think he’s folding, then all we’re doing is getting the most money while we’re behind.

Okay, okay…so…any other reasons to call? Sure! If we know that folding isn’t going to work that well, it probably means people either like their hand, or hate folding. In this case, the amount of money we stand to win if we hit (implied odds) is going to be nice. Here are some clues that your IO (implied odds) are good.

Wide field: More players= more money to win

Calling stations: People who hate to fold will pay you off!

Safe Board: If you hold 56ss, your chances of getting paid are better on a 4d 7h 8c Qs board than on a 2s 7s Jd As board.

Nut draws: By playing these passively, we might encourage someone with a worse draw to call behind us and if we hit, we’re getting his stack.

Moves of obvious strength: As per our example above, sometimes a villain will make a very strong move, tipping the strength of his hand. If he really likes his hand, he’s gonna have trouble folding it.

Deep stacks: Bigger stacks= more money to win!

Important Note

In situations with low FE, the difference between raising and calling is going to show in how much you lose when you miss. If villain isn’t folding, you’re getting their stack no matter what (for the most part), but by playing it passively, you can lose less when you miss.

Cliff notes

When fold equity is bad and implied odds are good, calling is usually going to be the better play. Keep aware of both in any drawing situation.

Fold!

If you’re not getting the express odds you need, your implied odds suck, and there’s no good chance of getting villain to fold, you should probably just let it go. This happens to me a decent amount where I’ll call a ½ pot bet vs a strong range IP on the flop, and then villain will pot the turn after I miss. Unless I think he’ll do this with anything other than an OP that he’s not going to fold, I’ll just lay it down. There’s no point in taking any other line, unless it’s for metagame or you just like burning money.

Let’s go back to that first example and look at it from the Hero’s side.

We open for 3BB with AsKs. Folds to the 8/2/1 who is playing 60BB deep (LOL). Blinds fold.

Flop (7.5BB) Kh 5s 6s

Hero bets 5BB, Villain raises to 22.5 BB.

What can we do here?

First, realize that even though you have TPTK, you have a drawing hand. If we have a read that villain only does this with a set…assume he has a set. I can’t tell you how many times I’ve seen incredibly optimistic ranges for villains who are super-tight. No, he doesn’t have KQ or a draw…he has a set. Deal with it.  Now that we’ve gotten that out of the way…we can…

Call? We’re getting 2-1 express odds on our call (risking 17.5 to win 35), and villain has 34.5 left behind. Counting the implied odds, we’re risking 17.5 to win 69.5 (a bit worse than 4-1) and our odds of hitting the turn are a bit worse than 4-1. Calling is slightly –EV even before we factor that sometimes we hit the turn and he boats up on the river. It’s not worth it to call.

How about raising? Well, given the stacks, we’d basically have to shove all-in and risk 52 (our remaining stack) to win 69.5. Our equity is better since we get to see both cards, but if you remember, we’ve only got about 30% equity here. We’d need better than 2-1 odds on our shove to justify it and we’re not getting them (we’re getting roughly 7-5). Therefore, we’ve gotta groan and fold. (This assumes that villain never folds a set here…which I’m more than happy to assume).


In short, it's all about analyzing the situation and knowing your opponents. If you can put your opponent on a range (given the action) and you know how willing he will be to play for stacks with it (given his tendencies), you should have enough info to determine whether fold equity, pot equity, or implied odds are going to be your hero for any particular hand.

1rd. tl;dr. omgwtfbbq

I love the maths you took the time to provide here. All micro players should study this stuff and pass a test on it--misunderstanding the maths behind playing draws is one of the primary weaknesses of micro players imo (and I include myself in there).

Where's the rest of this?

Great Post!

This is sooo important to remember in deciding what we are playing preflop and how we play it. I think a lot of players at the micros look at PFR% and actually have the exact opposite starting strategy then the should. Playing too many speculative hands against loose player that fold a lot post flop, and playing waaay too tight against tight sticky players.

tl/dr, RE is crazy, etc., etc.

Seriously, I've been waiting for this one! I'll read/digest/comment when I'm done work.

Well, seriously it was gonna be longer, but I figured between your CotW post and my Pooh-Bah post, half the forum would be blind right now and ready for something concise.

+1. I recommend WOT's Mathematics of NL Hold 'Em vid and Professor Plotkin's vid..... both on DC.

JoeTall needs to give me a sales commission b/c I spend a ton of time pumping the site......

Sticky this immediately!

play at 25NL or lesser makes me wonder if there is fold equity in this world. ever. not sure any villan fold anything at all.. regs included. So the raise part should be sparingly used i feel.

mostly the way to go is if you have the initiative by all means do bet your draw. But if not, its best to call when the odds are right.

Yeah, most of these calculations are under the assumption that we are operationwithout initiative (mostly in spots where we've cold-called preflop). In spots where we open preflop and flop a good draw (usually, overs and a flush draw or some kind of combo draw if we're stealing a ton), we can usually tack on some fold equity. (Depending on the villain, of course)

Down at 5NL/2NL and with some opponents at 25NL, calling their flop and turn bets in position when I flop a good draw is the best thing ever. They always pay off when you hit your draw, and you can just fold the river if you don't. There's no need at all to raise to balance your range or disguise your hand. They don't care what you have and if they has a top pair they're not going to fold.

Poker Stars $0.01/$0.02 No Limit Hold'em - 9 players
The Official 2+2 Hand Converter Powered By DeucesCracked.com

BTN: $0.96
SB: $3.14
BB: $1.00
UTG: $3.48 29/0/1.0
UTG+1: $0.94
UTG+2: $4.60
Hero (MP1): $7.68
MP2: $0.58
CO: $5.00

Pre Flop: ($0.03) Hero is MP1 with A 8
UTG calls $0.02, 2 folds, Hero calls $0.02, MP2 calls $0.02, 2 folds, SB calls $0.01, BB checks

Flop: ($0.10) 3 7 K (5 players)
SB checks, BB checks, UTG bets $0.10, Hero calls $0.10, MP2 folds, SB folds, BB folds

Turn: ($0.30) 9 (2 players)
UTG bets $0.36, Hero calls $0.36

River: ($1.02) 5 (2 players)
UTG bets $0.50, Hero raises to $2.25, UTG calls $1.75

Final Pot: $5.52
UTG mucks K T
Hero shows A 8 (a flush, Ace high)

The lesson: Playing draws in position is very, very sexy.

The funny thing is that he was betting big on flop and turn to deny me odds to draw... but because he's a fish and will never ever let go of TOP PEAR ZOMFG, he's still ATM'ing me the monies.

At 50NL against a reg who apparently is in love with his TPTK after we had some sick history and I was basically very very tight.

His hand is transparent (thanks to his sick bet sizes and c-bet sizes) and my equity is pretty high on the flop. But unless he was gonna fold, call call and value bet is far more profitable than playing this big draw aggressively. its only unnecessary variance IMO.

Hands where particularly you can detect equity changes better than your opponent are better played passively at 25NL or below.

Full Tilt Poker $0.25/$0.50 No Limit Hold'em - 9 players
The Official 2+2 Hand Converter Powered By DeucesCracked.com

UTG+1: $48.35
UTG+2: $93.10
MP1: $100.30
MP2: $76.45
Hero (CO): $52.80
BTN: $25.50
SB: $66.05
BB: $33.60
UTG: $48.50

Pre Flop: ($0.75) Hero is CO with T A
3 folds, MP1 raises to $1.75, 1 fold, Hero calls $1.75, BTN calls $1.75, 2 folds

Flop: ($6.00) J A 5 (3 players)
MP1 bets $4, Hero raises to $12, BTN folds, MP1 raises to $71.50, Hero calls $39.05 all in

Turn: ($108.10) 5 (2 players - 1 is all in)

River: ($108.10) 3 (2 players - 1 is all in)

Final Pot: $108.10
MP1 shows A K (two pair, Aces and Fives)
Hero shows T A (two pair, Aces and Fives)
MP1 wins $105.10
(Rake: $3.00)

Text results appended to pokerstove.txt

329,378,119 games 134.206 secs 2,454,272 games/sec

Board: Ah Jc 5c
Dead:

equity win tie pots won pots tied
Hand 0: 45.151% 44.54% 00.61% 146720852 1996335.50 { AcTc }
Hand 1: 54.849% 54.24% 00.61% 178664596 1996335.50 { AsKs }

+1

I always struggle with how to play draws postflop. Thank you for the villian dependent advice. I always wonder what is the best way to play my draws because I hate getting to them and not getting paid off. I realized a few weeks ago that I have lost a lot when I flop a flush draw, villian leads with 3/4 and I am always like?????

This will help me a lot. I am going to run through some hands this evening I think.