When faced with a difficult decision in no limit hold ‘em, it helps to remember you’re essentially dealing with a mathematical problem. Decisions which seem difficult often have clear outcomes when considered this way, and the answer may not always be intuitive.

[Quick note, the ace of diamonds card hasn't rendered correctly in my post, so I'll be denoting it with 'AceDia'.]

[Edit: A mistake has been pointed out to me as per the comments. An Ace is not an out if villain has hit a set, since it will give him a full house. I'm not going to edit the article, but my statement at the end is not correct! ]

http://www.boomplayer.com/en/poker-hands/Boom/18420554_D8D7D15ED6

Playing this hand out, I felt hard done by. I sent the clip to a few of my friends who told me how unlucky I was, and how they’d have made the same moves. Until I sent it to one of my higher skilled friends, who was horrified by my actions after the post flop c-bet.

Hero is dealt , UTG+1.
Hero raises 3x the big blind, and is called by the Villain immediately to his left (UTG+2)

All other players fold.

I'm thinking, great! I’m AJo, and I’m heads up against one opponent. I’m out of position, but considering the early position I raised from, this was a good result.

I didn’t give much thought to Villain’s cards, which proved to be my downfall.

The flop comes:

, AceDia,

My thoughts:

I’ve got top pair with a reasonable kicker. It’s a rainbow flop, and whilst there’s a straight draw on the board, it can’t be an open ended straight.

I read in Harrington on Hold’ em that if you’re the pro-flop aggressor, it can be a good option to make a 1/2 pot continuation bet (c-bet). This bet has several purposes:

It’s a big enough bet that Villain is getting 3:1 pot odds. Those odds are generally not good enough to call with drawing hands, as a gut-shot or one sided straight draw have about 11:1 odds, an open ended straight draw has about 5:1 odds and a flush draw has about 4:1 odds.

As a consequence of offering these odds, one of three things can happen:

Villain can fold:     Perhaps he didn’t hit anything on the flop, or identified that the odds weren’t good enough for him to chase his draw. Perhaps he’s put us on a better hand, even if he did hit his flop. If he had a pocket pair and was hoping to flop a set, but didn’t, he might have decided his pair is no good against the over-cards, or that we hold a higher pocket pair.

Villain can call:     Villain might think he has us beat, could have the nuts, could be slow playing, or could be chasing a draw. If Villain IS chasing a draw, it’s probably a mathematical mistake, or he’s relying on the implied odds of us making further bets, which could pay him off in the long run.     Ideally, he’s made a mistake, and has put us on a weaker hand than his own, when we are in-fact stronger, or he’s decided to chase a draw against the odds.

Villain can raise:     I’m not really sure what this could mean, I’ll have to research and play a lot more, but I’d imagine this is a real show of strength most of the time, and sometimes a complete bluff. I wouldn’t imagine anything in between triggering a large raise. If Villain has a strong hand right now, but is worried about us out-drawing him, he could be trying to win the pot right now. He might figure that we’re sufficiently scared of his hand that he can win the pot right here with a bluff.

In this case, Villain called. I wasn’t thrilled with this, as I knew I didn’t have a monster hand, and would have liked to have won the pot right here. Still, I had a hand, and felt like there were worse situations to be in.

The turn came .
A good card, I thought. It’s very unlikely to have helped Villain, and I still feel like it’s likely that my pair of aces with Jack kicker is a decent hand.

Keen not to see another card, I made a move for the pot.

Hero bets 750, just over 3/4 of the pot.

Villain calls.

I was less than ecstatic by this call, but there’s a chance my ace is still good… right?

The river comes .

I check, and Villain shoves.

Before relentlessly mocking me for my call, I’ll tell you why I did call.

My odds were 3:1, and I’m thinking ‘does my Ace Jack win at least one in four times?’ I thought yes. A mistake of course, but at the time, it seemed a fair guess.

If I folded now, I’d be short stack in this tournament. I’d still have plenty of blinds, but I’d go from a decent position to a small fry, and have a hard battle to make it back, I felt fairly stack committed.

Hero calls.

Hero loses all of his chips.

Villain shows a pocket pair of sixes, giving him three of a kind, destroying my pitiful pair of aces.

‘That’s poker’ I thought.

But that’s not poker. That's an almost classic example of a bad player, making bad decisions, and losing a lot of chips because of it.

In an attempt to stop this kind of thing from happening again, my friend teaches me about the importance of ranges and putting Villain on a range. Considering the combinations of those ranges, accounting for cards we know about, and making our plays based not on our cards, but on our approximation of his.

So let’s start the hand again.

Hero is dealt UTG+1. Hero raises 3x BB. Villain calls UTG+2.

Immediately, we know some things about Villain.

I 3xBB bet from an early position, my hand must be pretty strong. I’m probably on a pocket pair, or something in the region of [AJ+, 22+, KQs].

This is a notation my friend taught me.

The AJ+ denotes an Ace and anything from Jack and above. The lack of a letter afterwards indicates we’re considering all combinations. An ’s’ (AJs+) would mean An Ace and everything Jack and above, suited. An ‘o’ (AJo+) means the same, but off-suit.

22+ means any pocket pair (from two to aces, so the whole range).

KQs means any suited King and Queen.

If Villain is calling me here (particularly in such an early position) he must have a hand that can tangle with the hands in that range.

My first thought was that we were looking at a similar range, probably something like

[AQ+, 22+].

I ran this potential range by my much more experienced friend, who suggested that probably wasn’t a fair assessment.

He mentioned if Villain had a hand in the range [KK+, AKs], he’d probably have re-raised.

This was interesting for a couple of reasons - to start with, I’d not considered that I might have to remove stronger hands from Villains range because he called. Secondly, I probably wouldn’t have re-raised with those hands.

Another lesson learned, re-raising the opening 3x BB with a premium hand is recommended when in early position. Probably because it keeps the number of players in the hand small, or wins you the pot out right.

We can actually remove [KK+, AKs] from Villain’s range, assuming he’s a better player than I.

So the range we have Villain on now is [AQ, 22-QQ, KQs, AJs].

But it’s not enough to simply guess which hands Villain might have (unless we lose to, or beat all of them).

Putting Villain on a range should allow us to reason about the probability of us winning the hand, so what is that probability?

Here I was introduced to another important poker concept, hand combinations.

There are 4 ways to deal 2 suited cards.

For instance, AKs can be dealt: , AceDia, , .

There are 6 ways to deal a pocket pair.

For instance, 66 can be dealt: , , , , ,

There are 12 ways to deal 2 unsuited cards.

For instance, AJo can be dealt: , , , , , , AceDia, AceDia, AceDia, , , .

There are 16 ways to deal any 2 cards, the 4 suited combinations and the 12 unsuited combinations.

Why does this matter? Well it relates to the probability of Villain having the various cards in his range. Villain is clearly less likely to have KQs than AK, simply by virtue of there only being 4 ways of dealing KQs, and 16 ways of dealing AK.

The next step in working out Villain’s range is to attach these distributions to their respective hands:

[AQ(16), 22-QQ(66), KQs(4), AJs(4)]

Much more useful, now we have some idea of the likelihood of each of the hand’s we’re up against. We can determine how many we beat, and how many we don’t, and factor those into our equations when making decisions. For instance, there’s a 74:16 chance that Villain has AQ.

But we’re not quite done attaching probabilities to Villain’s range, for instance, is it true that there are 4 ways to deal AJs to Villain?

No, I have AJo, so there’s at least two combinations of AJs that Villain can’t possibly have. If Hero has the , then of AJs possible combinations: , , AceDia, , Villain can’t possibly have or . Thus, the relative distribution of AJs in Villain’s range isn’t 4, it’s 2. Our hand actually disrupts quite a few of Villain’s hands.

After updating Villain’s range to account for the cards in our hand, the distribution of hands look like this:

[22-QQ(63), AQ(12), KQs(4), AJs(2)]

We’ve removed 3 possible hands from 22-QQ, because our J interferes with JJ, removing three possible hands (, , ) , we’ve removed 4 hands from AQ, (, , , ), and we’ve removed two hands from AJs, as previously mentioned.

Immediately, I’ve got much more information than I had before. When I played the hand, I had no idea what Villain’s cards could be. This time, I’ve got a range, and a relative probability for each set of related hands in that range. One obvious use of this range, is to work out my odds of being ahead.

My AJo is superior to KQs, and inferior to every other possible hand in Villain’s range. Wow. To think how confident I was when I went in to the flop.

I’m ahead against 4 of Villain’s possible hands, and I’m behind against 77 possible hands. a 4:77 underdog.

If I don’t hit anything on the flop, I’m probably in trouble.

As it happens, the flop comes , AceDia, .

Brilliant, I’ve hit top pair and I have a decent kicker. Is what I thought last time around. This time, I can actually do some analysis of the situation. This is often called figuring out the texture of the flop.

There’s two things to do here. First, we have to re-calculate the relative probabilities of Villain’s range.

[22-55(24), 77-QQ(33), KQs(3), 66(3), AQ(8 ), AJs(1)]

I’ve separated the pairs into three distinct groupings now. The reason for this is that it’s useful to reason about hands with respect to the hands that beat us and the hands that don’t. We’re not beaten by 22-55, we’re also not beaten by 77-QQ, but we are beaten by 66, so it’s useful for it to be in it’s own range.

The second thing to do is to re-evaluate which hands we beat.

Our AJo has paired with the AceDia, so we now beat:

[22-55(24), 77-QQ(33), KQs(3)]

And we’re beaten by:

[66(3), AQ(8 )]

We potentially split the pot against AJs(1).

Whereas before the flop, we were a 4:77 underdog, we’re now 60:11 to win.

So we’re golden, right?

Not at all, we don’t have any further information on Villain, and this is why the c-bet is so important.

We were the pre-flop aggressor, and our hand isn't hopeless, so we're going to c-bet. This is important, because if we behave similarly with a monster as we do with a marginal hand we’re not giving any information away. That makes it much more difficult for Villain to adjust our range and determine how well he’s doing. After all, if Villain simply folds one in three times, we’re breaking even.

So Hero makes a half pot c-bet.

Villain calls.

One of the major goals of the c-bet is to gather information. And one of the biggest blunders a new player can make is to ignore the information they’ve just paid for.

In this case, I’ve just bought some information about Villain. Just as I’ve got him on a range, he’s got me on one, and he’s just decided that pot odds of 3:1 are good enough to keep playing.

Villain’s either worked out that his hand is at least better than a 1:3 underdog against us.
Villain is on a drawing hand and figures his pot odds (and possibly implied odds) are good enough to keep playing.
Villain is bluffing.
Villain has made a mistake, which is the ideal situation.

We’ve got no real reason at this stage to assume Villain is bluffing, but WSOP pro Dan Harrington has a rule which states that every player, no matter how tight, will bluff at least 10 percent of the time. It seems high to me, but Dan Harrington is one of the most successful players of all time, so it’s worth keeping in mind.

With this new information, we can reduce Villain’s possible range once again.

I’m sure I could work out which hands in Villain’s range give him better than at least the 3:1 odds to play, but that seems like overkill, and something I’d be really unlikely to be able to manage at the table… Something I could learn to do at a later date, perhaps.

We can clearly reduce Villain’s range a bit without the math.

Villain’s range before the c-bet was:

[22-55(24), 77-QQ(33), KQs(3), 66(3), AQ(8 ), AJs(1)]

Given the likelihood of Hero having entered the pot UTG+1 with either an ace or a king, we can likely discount most of the low pairs, with two overlords on the board that Hero is likely to have hit.

A more interesting question, is do we rule out KQs, and are we prepared to say Villain would stick around after the c-bet with a high pocket pair like QQ?

These are difficult questions to answer, and there may be no concrete correct or incorrect answer, particularly with no info on Villain’s play-style.

We could refer to the maths I mentioned earlier to determine which range of hands gives Villain at least 3:1 odds, but we’re getting into the realm of real approximations now.

I’ve decided to drop QQ, and keep KQs around.

Villain’s new range is [KQs(3), AJs(1), AQ(8 ), 66(3)]

So the information we’ve purchased with our c-bet is that we’re not 60:11 to win. In fact, we’re much more likely to be 3:11 underdogs. (I’m ignoring the AJs, since it’d split the pot).

The turn comes, and is a .

As I mentioned before, this is a great card. It’s unlikely to have helped Villain (in fact, based on our range, it’s certainly not helped him). The problem, is that it didn’t help us at all either. A Jack would have changed the odds from 3:11 to 11:3, putting us in a good position to make a move on the pot.

As it happens, 3:11 clearly aren’t good odds. Approximately one in five games, we’ve lost this hand. The correct play is to check and fold in most cases. Now my 3/4 pot raise is clearly identifiable as the massive blunder it was.

But it happened, so lets continue.

Villain calling this raise reduces his range from [KQs(3), AJs(1), AQ(8 ), 66(3)] to something more along the lines of [AQ(8 ), 66(3)], possibly even just [66(3)]. We’re now down to 0:11 or 0:3. Clearly it doesn’t matter whether or not we remove the AQ(8 ), since we’re completely beaten. The only hope at this stage is that we’re hoping to draw an ace, which has us beating the 66(3), and still losing to the AQ(8 ). We could also be hoping for a Jack, or a higher pair than 66 to arrive on the board.

The river comes .

We were doomed before, and somehow, we now appear even more doomed, although our expected odds of winning have gone from zero to zero, so we’re actually no worse than before, other than the fact we can no longer out-draw Villain.
Hero senses death is looming and checks.

Villain shoves.

For reasons that defy logic, Hero calls, and loses all his chips.

‘I should have checked myself before I recked myself.’

Hero.

This was a clear fold, but what if it hadn’t been? What if this was a potentially difficult decision?

Imagine instead we’re on 22, and the shared cards are , , , ,

Imagine the rest of the hand had played out exactly as before, and Villain has shoved after our check on the river.

The pot is offering us 3:1 odds, and we’ve got Villain’s range as: [66(3), TT(3), KQs(3), AQs(3), QQ(3)]

We beat AQ(3) and KQ(3)

And we lose to 66(3), TT(3) and QQ(3)

We’re 3:6 underdogs.

The pot is offering us 3:1 odds, we need to win one in four times for this to be a break-even bet.

If our range is accurate (which is questionable, but for the sake of the maths we’ll say it is), we’re going to win 1 in 3 times.

Just like using pot odds and outs to calculate whether or not we should chase a draw, we do exactly the same things here, using the probability we have won.

In this case, the pot odds are greater than the odds we have won, and over time, this should be a profitable bet.

Hero should call.
In fact, not only can we work out whether or not Hero should call, we can also determine what this bet is likely to make us in the long run. This is called calculating Expected Value (EV).

EV = (Size of Pot * likelihood of winning) - (chips to call * likelihood of losing)
EV = ((2357 + 1218 ) * 0.33 - (1218 * 0.66)
EV = +375.87
On average, this bet will be profitable to the sum of 375.87 chips. A good call indeed.