Math Problem: Set of Kings

[DavidC]

Shill's the guy that taught me how to do this...

---

Standard structure (sb = 1/2 bb, SB = 1/2 BB), 10-handed.

Villain in this hand is really really tight, and doesn't like to give too much action without the nuts. PFR 3% or so.

---

Preflop: You're in MP2. Two loose callers and it's folded to you. You raise. Villain re-raises on your immediate left. It's folded to you, you cap, villain calls.

(4 + 4 + 1.5 + 2 = 11.5 SB in the pot)

Flop: K92r

You bet, villain raises, you three bet, villain calls.

(17.5 SB - 1 SB (max rake) = 8.5 BB)

(You put villain on AK or AA.)

Turn Ar.

You bet, villain raises.

11.5 BB in the pot, 1 bet to you.

Villain has either AA or AK. If you hit the case K on the river, villain will go three bets with you with top boat vs quads. If an ace hits the river, you can safely fold ('cause you're awesome).

Folding has an EV of zero. If you think that calling down is the correct option, how much do you expect to make by doing so (this is the math part)?

--Dave.

[DrunkHamster]

I'm going to have a go because I think it will definitely help my game to do these types of analysis, and I need the practice.

We know that villain has either AA or AK. Preflop, he will get AA .45 of the time and AK 1.2 of the time. So, the chances are just under 3 to 1 that you are ahead. Now lets work out the EV of either case.

When he has AA, 1/45 of the time we win 18.5 bets (if the K comes out) and the rest we lose 2 (assuming we call down). So our EV is (1/45*18.5) + (44/45*-2)= 0.41-1.956 = -1.55 bets.

If he has AK on the other hand, we lose 1 bet 2/45 times (when an A falls) and win 14.5 the rest. So the EV is (43/45*14.5) + (2/45*-1)= 13.85-0.04 = 13.8 bets

Overall then, 3/4 of the time we have an EV of 13.8 and 1/4 an EV of -1.55, making this an easy call down.

However, this seems completely wrong to me, and I'm sure I have made a mistake somewhere. I'd appreciate anyone pointing out where I'm wrong.

[topspin]

I think someone with a PFR of 3 is not 3-betting AK. I also think villain caps the flop with AA, so it ought to be less likely he holds that hand. But whatever, I'll go with your turn assumptions that AK is possible and all hands are equally likely.

There are 3 ways he could have AA, and 3 ways he could have AK. That means he has AA half the time. You spike the case K in this case 1/46 of the time. Therefore you hold the winning hand on the river 1/2*1/46 = 0.011 of the time.

You win 15.5BB when your hand is good. You pay 1BB to see the river. Your EV is:

15.5*0.011 - 1 = -0.83BB.

Moral of the story: don't call the turn if you figure you're drawing to 1 out at best.

(PS doesn't this belong in the probability forum?)

EDIT: Forgot about the turn A.

[DrunkHamster]

Oh crap, that's what I did wrong. I didn't take into account the number of ways he could make up AA or AK knowing that we had KK. Dammit!

[DavidC]

[ QUOTE ]
I think someone with a PFR of 3 is not 3-betting AK.

[/ QUOTE ]

I think you're correct on this, but let's assume that I was wrong about my 3%, and that I was right about my AK/AA read. /images/graemlins/smile.gif

[DavidC]

[ QUOTE ]

(PS doesn't this belong in the probability forum?)

[/ QUOTE ]

Meh...

They've probably done this before.

[DavidC]

I'll post my solution tomorrow, after a few guys have taken a kick at it.

I'm not sure that I'm correct (as shill basically pointed out how to work out hand distributions in a few of my earlier posts, not necessarily the EV of it...), but if I get corrected on my errors, that will help me out too, so I can't really lose. /images/graemlins/smile.gif

--Dave.

[mockingbird]

If you assume villain can have only AA or AK then:

On the turn there are 3 aces and 1 king unaccounted for so:

1) There are 3 ways he can have AA. With AA you will win with quads 1/46 times and lose the rest.

2) There are 3 ways he can have AK. With AK he will win with aces full 2/46 times and lose the rest.

We also assume that you fold to a river bet if an ace hits. Otherwise you both pay 3 bets on the river.

So, if our assumptions are correct, calling the turn gives you.



EV= 1/2 [ 14.5*(1/46) - (1/46) - 4*(44/46) ] +
1/2 [ 14.5*(44/46) - 1*(2/46) ] = +5.25

The first term is if he has AA:
1 in 46 times you will hit a king and win 14.5 bets, 1 in 46 times he will hit an ace and you will lose 1 bet since you will fold, and 44 in 46 times neither an ace nor a king will hit and you will lose 4 bets ( one on the turn and three on the river ).

The second term is if he has AK:
You win 14.5 bets the 44 out of 46 times that an ace does not hit and lose 1 bet the two times that an ace does hit.


This is definitely a positive EV call.

Even if you assume he has AA 3/4ths of the time. ( Not sure why you would assume that though.)
The EV is still + 0.8

This is also my first time doing this, hope I'm not way out in left field. Please let me know, gently, if I am.

[DavidC]

[ QUOTE ]
If you assume villain can have only AA or AK then:

On the turn there are 3 aces and 1 king unaccounted for so:

1) There are 3 ways he can have AA. With AA you will win with quads 1/46 times and lose the rest.

2) There are 3 ways he can have AK. With AK he will win with aces full 2/46 times and lose the rest.

We also assume that you fold to a river bet if an ace hits. Otherwise you both pay 3 bets on the river.

So, if our assumptions are correct, calling the turn gives you.



EV= 1/2 [ 14.5*(1/46) - (1/46) - 4*(44/46) ] +
1/2 [ 14.5*(44/46) - 1*(2/46) ] = +5.25

The first term is if he has AA:
1 in 46 times you will hit a king and win 14.5 bets, 1 in 46 times he will hit an ace and you will lose 1 bet since you will fold, and 44 in 46 times neither an ace nor a king will hit and you will lose 4 bets ( one on the turn and three on the river ).

The second term is if he has AK:
You win 14.5 bets the 44 out of 46 times that an ace does not hit and lose 1 bet the two times that an ace does hit.


This is definitely a positive EV call.

Even if you assume he has AA 3/4ths of the time. ( Not sure why you would assume that though.)
The EV is still + 0.8

This is also my first time doing this, hope I'm not way out in left field. Please let me know, gently, if I am.

[/ QUOTE ]

You misread the question just a little, but you've got the right idea, and you added a new idea, which way more than makes up for it. /images/graemlins/smile.gif

Basically, if we hit our K, we're going 3 bets on the river, and if we don't we're calling down. Villain wouldn't raise the river than a full house after all that action, so by betting the river, we'd have a 50% chance of being best, and if we weren't, we'd still have to call the bloody raise. /images/graemlins/smile.gif

But yeah, this seems to me to be the right approach.

The really cool thing here, though, is that you wrote something about us making our calculations based on villain having AA 3/4 of the time, speculatively...

IF you thought that villain would call your turn bet with AK half the time, rather than raise, then you do this:

AK = 1/2 (hand distribution) * 1/2 (behavioral distribution) = 1/4, therefore AA = 3/4.

Neat! /images/graemlins/cool.gif

[topspin]

Ugh, I suck. Thank you.

Fixed (I hope):

[ QUOTE ]
There are 3 ways he could have AA, and 3 ways he could have AK.



That means he has AA half the time. You spike the case K in this case 1/46 of the time. Therefore you hold the winning hand on the river 1/2*1/46 = 0.011 of the time. You win 15.5BB when your hand is good.



He has AK half the time. He spikes an A 2/46 of the time. You hold the winning hand on the river 1/2*44/46 = 0.478 of the time. You win 13.5BB when your hand is good.



You pay 1BB to see the river.



Your EV is:



15.5*0.011 + 13.5*0.478 - 1 = +5.62BB.

[/ QUOTE ]

[mockingbird]

[ QUOTE ]
IF you thought that villain would call your turn bet with AK half the time, rather than raise, then you do this:

AK = 1/2 (hand distribution) * 1/2 (behavioral distribution) = 1/4, therefore AA = 3/4.

Neat!


[/ QUOTE ]

Thanks, that is pretty cool.

[DavidC]

To summarize the problem:
-------------------------

-Villain has AA or AK.
-Board is K92Ar.
-Pot is 11.5 BB.
-Villain will go 3-bets on the river when you hit quads.
-You can fold to an ace on the river.

----

So, first step is to figure out how likely it is that villain has AA rather than AK.

AA hands = 4 aces * 3 remaining aces / 2 ways of getting each hand (since AcAd is the same hand as AdAc). Therefore there's 6 ways of getting AA preflop.

However, we know that one of the aces is removed, so it's 3 * 2 / 2, or 3 ways.

AK = 3 * 1 ways... oh crap. For some reason I think it should be a 2:1 ratio in favour of AA, but the math thus far shows me 1:1 (equally likely). Please correct me if I screwwed up this part.

If he has AA:
1/46 times we hit the A: -1BB.
1/46 times we hit the K: +14.5BB
44/46 times we hit nothing: -2BB
Total: 14.5 - 1 - 88 = -74.5/46 =~ -1.62BB

If he has AK:
1/46 times we hit the K: +14.5BB
2/46 times we hit the A: -2BB
43/46 times we hit nothing: +12.5bb
Total: 14.5 - 4 +537.5 = +548/46 =~ +11.91BB

Since these outcomes are equally likely, you split the difference, and get +5.145BB EV for calling down (and giving action with your lovely quads).

---

Now, since folding has an EV of zero and calling has an EV of +5.145, we determine that folding is definitely wrong.

There's still the option of raising, though. We can expect that AK will call and AA will raise vs our raise. We will also assume that villain will go into call-down mode with AK if we do this, unless they hit their ace.

Now, if he has AA:
-2BB because we can't call his cap.

Now, if he has AK:
1/46 times we hit the K: +13.5BB
2/46 times we hit the A: -3BB
43/46 times we hit nothing: +13.5bb
Total: 13.5 - 6 + 580.5 = +588/46 =~ +12.78BB

Hmm... interesting... +5.39 as a result of this line...

So... if you can put 100% faith in your reads, then "three-betting to find out where you stand" is +EV (specifically, +0.245 BB per hand).

However, villain would only have to cap with AK (and you would therefore be incorrect in your read) once out of 46 times, and you would have been better off just calling down (excepting the ace of course) lose money by folding. My logic for "46" is that since AK is worth about 11.91 BB, if they push you out of the hand and you lose the 11.91 BB once every 46 times, that comes to -0.258, negating entirely the +EV of your "raising for information".

If you were to call down after AK capped, then you'd also have to call down after AA capped, and you're not getting action on the last king, so other than 1/46 times you hit the K, you lose 4BB that time, and that totally demolishes the times you make an extra bb from the AK hand (pretty sure it's worse than folding to the raise, but that just depends on how often they're going to be capping with AK).

--Dave.

[Mathieu]

3 combos for AK and 3 combos for AA,

AK: He improves by catching an A 2/46 times.
And we assume he'll bet the river anyways.

So EV = 44/46 * 12.5 BB + 44/46 1BB (river) - 1 BB (call on the turn)

EV (vs AK) = 11.9 BB

EV (vs AA) = 1/46 * 12.5 BB + 1/46 3BB (river action) - 44/46 * 1 BB (river calls, we don't call if A falls)- 1BB (turn call)
= -1.68 BB

Total EV = 1/2 * EV (vs AK) + 1/2 * EV (vs AA)
= 5.11 BB(assuming rake had already maxed out)

Is that right?

[bottomset]

how many hands against villian to lock in his hand range this tight?

[DavidC]

[ QUOTE ]
how many hands against villian to lock in his hand range this tight?

[/ QUOTE ]

42 million.

[bottomset]

[ QUOTE ]
[ QUOTE ]
how many hands against villian to lock in his hand range this tight?

[/ QUOTE ]

42 million.

[/ QUOTE ]

/images/graemlins/smile.gif

one note
if he has AK there isn't a K to hit

i think i calldown save a river K

[ClaytonN]

Folding this is NOT an EV of zero.

If you don't get past this, you are useless past micros.

[DavidC]

[ QUOTE ]
Folding this is NOT an EV of zero.

If you don't get past this, you are useless past micros.

[/ QUOTE ]

Folding has an absolute EV of zero, but you're right that it does have an "opportunity cost", which is expressed as the EV of calling or raising.

If you don't get past this difference, you won't get past my (BOOT TO THE HEAD!).

[caggin]

Nit-picky corrections (I think):

1) On the river, there are only 44 unknown cards, not 46:
52 cards - 4 on the board - your 2 hole cards - his 2 hole cards = 44

2) If he has AK and we hit an A on the river, we lose 1 bet, not 2 (from the turn, since we fold the river).

EV=~+5.318 BB (not including any behavioral factors)

[DavidC]

[ QUOTE ]
Nit-picky corrections (I think):

1) On the river, there are only 44 unknown cards, not 46:
52 cards - 4 on the board - your 2 hole cards - his 2 hole cards = 44

2) If he has AK and we hit an A on the river, we lose 1 bet, not 2 (from the turn, since we fold the river).

EV=~+5.318 BB (not including any behavioral factors)

[/ QUOTE ]

1) Really interesting. Very very cool. Thank you!

2) This solution was written at like 8am after being up for nearly 24 hours, because I couldn't go to sleep. I'm pretty sure that I MEANT to say -1bb when the A hits, but thanks for pointing it out anyways.

Again, really cool observation with #1. /images/graemlins/smile.gif

Edit: And hey, check out my location (and response to the guy who said that folding is not zero EV)... I'm a nit, too; don't worry about it. /images/graemlins/smile.gif

[caggin]

Nit-picky correction to my corrections:

Oops, forgot if he has AK, we can't hit another K.

EV=~5.295 BB

[DavidC]

[ QUOTE ]
Nit-picky correction to my corrections:

Oops, forgot if he has AK, we can't hit another K.

EV=~5.295 BB

[/ QUOTE ]

Shite! That too. /images/graemlins/smile.gif