KQo A Big Loser. Hand 1.

[Sasnak]

I like it and it looks like a number of my KQo hands. Yeah I get whipped on a few hands when they hit a set or 2 pr, but more times than not I'm good.

KQ/KQs are way +EV while ATo is my "Oh cr@p" hand.

~

I just tried to work this out mathematically and the result was quite surprising to me.

First lets look at how many outs we have (when we are not drawing dead):

Hand / # of possible combinations / outs
84 / 2 / 8
K4 / 9 / 8
K8 / 9 / 3

This makes an average of 5.7 outs. Thus our draw will come in about once in 7 times (exactly: 1 to 7.07).

Now lets look at the different EV values for different lines.

1 Call turn, call river

This is the line that was suggested the most here.
It's EV (assuming we never have the best hand on the turn, say SB never overplays a weak Kx) is:
0.124*9*20/32 – 0.876*2*20/32 – 2*12/32 = 0,6975–1,095–0,75 = -1,1475.
Explanation:

You have a 0.124 (12.4%) chance of outdrawing 20 of his 32 possible holdings (the 84, K4 and K8) and winning 9BB (the 7 that are in the pot on the turn + 2 more from the river). Thus 0.124*9*20/32 is your expected earning when you outdraw on him.

Now you have to subtract the money you lose for paying him off (call the turn, call the river) when you don't improve:
0.876*-2*20/32. 87.6% of the time your draw will fail and you'll lose 2BB.

And you have to subtract the money you lose when he has a set, leaving you drawing dead:
-2*12/32. You'll always lose 2BB when he has a set.

Now we need to consider the chance he's overplaying a weak Kx or something else (like A8), say the chance that you have the best hand on the turn (and on the river).
So we'll search the break-even point (=bp). That's the point where the chance that your opponent overplays his hand turns your EV from negative to zero (or above).

(-1.1475 * (1 - bp)) + (8 * bp) = 0
1.1475 * (1 - bp) = 8 * bp
1.1475 - 1.1475bp = 8bp
1.1475 = 9.1475bp
bp = 0.125

Explanation (using line 2):
Say there is a 20% chance your opponent has a hand you beat (without improving). 8 * bp = 8 * 0.2, 20% of the time you'll win 8BB for an average of 1.6 per hand.
And you'll lose 1.1475 * (1 - bp) = 1.1475 * (1 - 0.2), say 80% of the time you lose 1.1475BB for an average of 0.918. Your total EV is now: 1.6 - 0.918 = 0.682.

So calling would be correct whenever you estimate the chance that your opponent holds a hand you beat on the turn is > 12.5% (the break-even point).


2 Raise the turn

Now let's do the same for the case we raise the turn (expecting him to 3-bet with his stronger hands and only call with an overplayed Kx etc.):

0.124*11*20/32 – 0.876*4*20/32 – 4*12/32 = 0,8525–2.19–1,5 = -2,8375

The bp is now:

(-2.8375*(1-bp)) + (9*bp) = 0
2.8375 - 2.8375bp = 9bp
2.8375 = 11.8375bp
bp = 0,239

For making raising the superior play, you need a decent read that your opponent is extremely aggressive here (if he overplays 30% of the time thats enough to make raising +EV but it's still worse than just calling).


3 Calling the turn, folding the river ui

This is the line that's considered to be the worst. Let's look at it:

0.124*9*20/32 – 0.876*1*20/32 – 1*12/32 = 0,6975–0,5475–0,375 = -0,225
(Note: you lose now less when he has you dominated or you don't outdraw on him, but you also win less when you have him beaten).

Lets look at the bp now:

-0,225*(1-bp)) + (9*bp*0.124) = 0
0.225 - 0.225bp = 1.116bp
0.225 = 1.341bp
bp = 16.8%
(Note the difference to the other 2 calculations: you win now only if he has his hand overplayed AND you improve because when you don't improve you don't call a river bet).

So what's the best line? If you have a good read that says he will not make this play with a worse hand more than 12.5% of the time - fold the turn. If you have a strong read that he overplays that often - you should call/call.
If you are not sure about your read you should call the turn and fold the river ui. This is because this line is less read dependant.
If you think your opponent overplays 15% of the time but you misjudged him (say he'll only do this 8% of the time) your misestimate will cost you a lot (about 0.8BB when you use the call/call line).
On the other hand if the same happens with the call/fold line you'll lose a lot less (about 0.1BB).

[bottomset]

^^ uh we have a K so there is only 6combos of K4, and 6combos of K8, not the 9 that you have in your calcs

Bah, hate it when such things happen. I'm sorry for the mistake but when I made this post I was quite tired (it was 3 AM here in Germany).
But this mistake doesn't change the results significantly (5.8 outs instead of 5.7). It's much more serious that I obviously missed one out for the case he has K4 (it's 3xQ, 3x8, 3x7) making 9 outs.
I'll recalculate it.

I just recalculated the values - they remained almost the same:

Call/call's EV is now: -1,159 (instead of -1.1475) and the bp is 12.6%.
Call/fold is -0, 234 (instead of -0,225) with a bp of 17.2%.
Raise is now -2,853 (instead of -2.8375) and the bp is 24%.

One more thing I'd like to mention here: many ppl will slowplay something like 88 on this drawless flop against 2 opponents (including 1 pre-flopraiser). If we assume a typical opponent will slowplay 88 50% of the time and 44 only 40% of the time our EV would jump up significantly:

call/call: EV = -0.756; bp = 8.5%
call/fold: EV = -0.066; bp = 5.4%
raise: EV = -2.053; bp = 18.8%

Thus if our opponent could be slowplaying strong hands on drawless boards all lines would become more profitable (but raising still requires a very strong read).
Calling the turn and folding the river ui seems to be the most profitable line, esp. since it's not sensitive to misjudgements on your opponent.

This is really surprising to me since I considered call/fold to be the worst line. I'd like to hear other's comments on this.

[TomBrooks]

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I'd like to hear other's comments on this.

[/ QUOTE ]
I'm having trouble following the math and the rather technical nature of this breakdown.