Links to Part I and Part II, for those who need to catch up.

Brief summary: $1/$2 NLHE, with stacks over $200. MP opened to 3xBB with AQ preflop. Only Button made the (arguable) call with A9s. The flop was Q-7-3, two spades, giving MP top pair, top kicker and Button flopped the nut flush draw. MP opted to bet the pot ($12) and Button called.

This part will look at what will happen if the turn card is a brick, i.e. doesn’t help either player. Let’s say it’s the trey of hearts, for a board of

Q♠ 7♠ 3♣ 3♥

The thought process of MP

What did Button call with? Whatever it is, it’s not likely to have contained a trey, unless he was playing specifically A-3 suited or he has been slowplaying 3-3. The fact that another trey came off makes the risk of 3-3 extremely low. More likely candidates for hands that Button is playing are AQ, KQ, QJ, 77-TT and flush draws (let’s say A8s-AJs and KJs). Some small portion of the time, Button is being tricky with a monster (like AA/KK or a flopped set) but the large majority of the time, we must believe our hand is good.

How should we size our bet to maximize earnings? Calculating EV with multiple outcomes and factoring in bluffing frequencies etc. is more than I can chew in this blog. But let’s at least consider the hand distributions:

Hands we beat with 3 or fewer outs

AQ: 4
KQ: 6
QJ: 6 <– A very loose preflop call unless it’s suited, though.
77-TT: 6*4 = 24

Hands we beat with 9 or more outs

As8s-AsJs: 4
KsJs: 1

Hands that have us beat

AA: 3
KK: 6
QQ: 1
77: 3
33: 1

(It’s also possible that Button made a very loose call on the flop without any hand whatsoever. He could also be “floating.” For the sake of my sanity - and my spare time - I’m going to exclude these possibilities from the calculations this time.)

Sums:

Hands we beat with 3 or fewer outs: 40 (~66%)
Hands we beat with 9 or more outs: 5 (~8.5%)
Hands that have us beat: 14 (~24%)

Note that I have made no weighting of these numbers based on his flop call. For instance, if this is a player who is very straightforward, we might discount the possibility of the hands that beat us, since he didn’t raise. We may also discount the chance of him having AQ and KQ since he will often raise those as well. What I’m hoping you’ll find interesting about these numbers is how overwhelming the number of hands he can have that aren’t flush draws are. A lot of people spend their flops worrying about shutting out flush draws and I think this is a mistake. Perhaps this is an unfortunate time to preach about this particular subject given that we know that Button happens to have specifically the dreaded flush draw.

Anyway.

Based on these numbers, it’s probably best to fold if we’re raised, if our opponent is somewhat unlikely to bluff or raise us with a mediocre hand. It’s also clear that protecting against the flush draw is much less important than getting value from the weaker hands, if we can.

I think the sweet spot is around 1/2 to 2/3rds of the pot. Many of the hands that we beat will call, and it’s a big enough bet to discourage weaker hands to bluff. It’s also enough that the potential flush draw is getting not-quite-sufficient odds to make the call profitably, as long as we don’t pay him off too much on the river.

The pot is $36. MP bets $20.

From Button’s point of view, this bet makes yet another call slightly questionable. The pot is laying him odds of 56:20, or just below 3:1. In order to profitably make the call, he needs to win another $25 or so on the river on average when he hits his flush.

This is the point in the hand where too many other things come into play to make accurate estimates as to what Button should do, because it depends too much on how he perceives MP. Does MP often fire two barrels with weak hands? Or does the turn bet signify strength, such as AA, KK, AQ, KQ, or QQ? If he knew his opponent’s cards, I believe it would be a mistake for Button not to call, because it would be very strange for MP to fold AQ for a $25 bet on the river (in a $76 pot) with TPTK, even if the flush gets there. But of course, the holecards are not known.

Against a player who will mostly fire second barrels with real hands, calling a half-pot bet here with the draw to the nuts is probably good.

What do you think?