Poker Math Quiz #1

[spoonitnow]

With 100bb effective stacks, it is folded to Villain who opens {22+,ATs+,KTs+,QTs+,JTs,AJo+,KQo} for 4x. It's folded to Hero in the big blind with A9s.

Question 1: If Villain continues against a 3-bet with {TT+, AQs+, AKo}, is a 3-bet bluff to 12bb profitable?

Question 2: Assuming Villain calls our 3-bet with the entire range given in question 1, how often does he hit top pair or better on a flop of Q94r, assuming the queen is not the same suit as our A9s?

Question 3: What hand(s) can Hero 3-bet here that will have the most fold equity?

Edit: You can't do this correctly if you have to use PokerStove for any part of it.

[celtic123]

thanks for this, im trying to work out pokerstove.

ill avoid reading replies untill ive worked out my own answers

[wellrounded08]

Ok, I'm going to answer one question at a time. Because, I'm that slow. I might be wrong. But, I'm answering, and I'm learning.

Question 1: If Villain continues against a 3-bet with {TT+, AQs+, AKo}, is a 3-bet bluff to 12bb profitable?

Assuming 147 opening hands, and 42 continuing hands, that gives us 28.5% of his range that will continue. He is therefore folding 71.5% of the time. He needs to fold 68.6% of the time to be profitable.
Needs to fold<Folding Therefore, YES it would be profitable.

[Gefunkt]

Think I got it all.

Villain's raise range is 166 hands. That's 78 for the pocket pairs, 40 for the suited stuff, 48 for the offsuit stuff.

Our hand is blocking some of those combos though. He's losing 3 ways to get 99, 3 ways to get AA, 1 ways from each of 4 A containing suited hands (total of 4), and 4 ways from each of the A containing offsuit hands (total of 9). Total blocked hands is 19, so really what we're looking at is 166-19 = 147 hands that he will raise with.

Villain will continue against a 3-bet with 27 hands worth of pp, 6 hands worth of suited, 9 hands worth of offsuited, total of 42. 42/147 = 28.57% continue, so 71.43% folds.

Now I'm gonna wave my hands. Over in the blockers howto thread there is an example that has the exact same betting sequence as in this quiz, and you said that we need him to fold 68.6% of the time. I don't (yet) know how you came up with that, but there it is.

71.43>68.6, so in answer to Q1, yes, it is profitable to 3-bet bluff to 12bb.

Q1. yes


Question 2:
Flop comes down Q94 rainbow, we want to know how often he hits top pair or better with the range in Q1. I hope I'm reading that right.

Of the hands that he called our 3-bet with, the following hands will give top pair or better: QQ, KK, AA, AQs. That's 3+6+3+3 = 15 hands, out of the 42 he would have seen the flop with, so 15/42 = 35.7%

Q2. 35.7%


Question 3:
To maximize our fold equity, we want hands that block as much as possible out of Villain's 3-bet-call range.

So let's see. Unblocked, he's got:

TT - 6 ways
JJ - 6 ways
QQ - 6 ways
KK - 6 ways
AA - 6 ways
AQs - 4 ways
AKs - 4 ways
AK - 12 ways

We want hands that knock out as much of that as possible.
5 TT - knocks out 5 ways
5 JJ - same as JJ
7 QQ - knocks out 5 ways from the PP and 2 ways from AQs, total of 7
13 KK - 5 ways from the PP, 2 ways from AKs, 6 ways from AK, total of 13
20 AA - 5 ways from PP, 2 ways from AQs, 2 ways from AKs, 6 ways from AK, total of 20

6 JT - 3 ways from TT, 3 ways from JJ, 6 total (hey!)
7 QT - 3 ways from TT, 3 ways from QQ, 1 way from AQs, total of 7
10 KT - 3 ways from TT, 3 ways from KK, 1 way from AKs, 3 ways from AK, total of 10

11 AT - 3 ways from TT, 3 ways from AA, 1 way from AQs, 1 way from AKs, 3 ways from AK, total of 11 ways
11 AJ - same as AT

10 KT - 3 ways from TT, 3 ways from KK, 1 way from AKs, 3 ways from AK, total of 10 ways
10 KJ - same as KT

11 KQ - 3 ways from KK, 3 ways from QQ, 1 way from AQs, 1 way from AKs, 3 ways from AK, total of 11 ways

14 AK - 3 ways from AA, 3 ways from KK, 2 ways from AKs, 6 ways from AK, total of 14 ways
12 AQ - 3 ways from AA, 3 ways from QQ, 2 ways from AQs, 1 way from AKs, 3 ways from AK, total of 12 ways

So our possible hands, sorted by fold equity at this point in the hand:

AA
AK
KK
AQ
AJ AT KQ
KJ KT....dunno why I did KT twice, but it's heartening that I got the same answer twice
QJ QT QQ
JT
TT JJ
everything else

[spoonitnow]

I edited and posted a hint above for anyone who missed it in the OP.

[killerkebab]

I am possibly saying something very stupid with this post, but lets hope I'm right...

Villains raise range is 166 combos, 19 of which we block since we have A9s.

Villain will call with 42 of these 147. 42/147 = ~28%.
The pot when it comes to us is 5.5BB (SB+BB, villain raised 4x). We are going to 3bet him to 12BB so we're adding another 11 to the pot.
From villains point of view, this is calling 8BB for a total pot of 24.5BB. 8/24.5 = 0.3265. Villain needs to be in his 3bet calling range 32.7% of the time to make this unprofitable for us.

Since his 3bet calling range is 28%, and 28<32, 3betting with A9s is profitable for us.

[wellrounded08]

Question 2: Assuming Villain calls our 3-bet with the entire range given in question 1, how often does he hit top pair or better on a flop of Q94r, assuming the queen is not the same suit as our A9s?

Bare in mind, Spoon said that the Queen on the flop was not the same suit as the A9 in our hand. (He said that in mIRC, it's your own fault for not knowing that.)

Total Combo's villain could have pre/post flop:
post/pre--TT+ AQs+ AK
pre---------27----6-------9
post--------24----5-------9--Total of 38

Hand combos that could have hit Top Pair or better
QQ+ AQs
12----2----total 14
14/38=36.8% of the time.

It's folded to Hero in the big blind with A9s.

Hero enjoys his free SB moneys. It's a trick question obv.

[wellrounded08]

It says folded to villain who opens for 4xbb. folded to hero. It's a math quiz not a riddle.

[Ragnar4]

My favorite part of this post is that after sppon has an epic conversation about how someone is mentally retarded about blockers... he then posts a pop quiz that blockers must be intrinsically understood in order to profit.

Q1.
2-9 = 9 pps
50 ways to be dealt them
ATs-AJs 6 ways
QTs, KTs, QJs, JTs, KQs 20 ways
KQo, AJo 32 ways
108 ways fold

TT-AA 26 ways
AKo/AKs 12 ways
AQs 3 ways

41 ways to call

around 38% call
let's assume SB is raked off

we're betting 11bb to win 5bb
to be immediately successful we need a fold 45% of the time

so yes, it is profitable

Q2.
Given that the flop contains a q,
TT-AA 22 ways
AQs 2 ways


AKo/AKs still 12 ways

TP or better would be the two AA, two QQ, six KK, two AQs

12 out of 36
or 1/3 of the time

[wellrounded08]

Question 3: What hand(s) can Hero 3-bet here that will have the most fold equity?

Obviously we want hands that block villains continuing range here. I havn't done the math to figure out which hands block the most, but only simple math is need for that, so here is just a list of blocking hands.

TT+AQ+

A VERY basic assumption to be made by this is that the best FE comes from holding cards that villain will continue with. I may be just stating or restating the obvious, but that seems to me, a very simple thing to remember, and build from.

[spoonitnow]

My favorite part of this post is that after sppon has an epic conversation about how someone is mentally retarded about blockers... he then posts a pop quiz that blockers must be intrinsically understood in order to profit.

This also comes right after I wrote a very detailed post about hand combinations and blockers which very few people have really put time into.

[spoonitnow]

Answer 1: The way to do this is to figure up all of the possible combinations Villain opens assuming we were dealt A9s and all of the possible combinations that Villain continues with against our hand A9s. Those two numbers will give us a percentage of the time that Villain folds. Next, we figure how often we need Villain to fold to break even on our bluff. If Villain folds more than that, then it's obviously a +EV bluff.

So first we figure the number of Villain's opening combinations

Hand   Combos  Blockers   Total
-------------------------------
22+      78       6        72
ATs+     16       4        12
KTs+     12       0        12
QTs+      8       0         8
JTs       4       0         4
AJo+     36       9        27
KQo      12       0        12
-------------------------------
                          147

Now we figure the number of Villain's continuing combinations

Hand   Combos  Blockers   Total
-------------------------------
TT+      30       3        27      
AQs+      8       2         6
AKo      12       3         9
-------------------------------
                           42

The percent of the time that Villain folds is 1 - (42/147) = 0.7143 = 71.43%.

Since it's folded to us in the big blind and we are raising to 12x, we are putting another 11bb into a pot of 5.5bb. The percent of the time that Villain has to fold for us to break even is 11/(5.5+11) = 0.6667 = 66.67%. Villain folds more than that, so our 3-bet bluff with A9s is forced to be +EV regardless of any value we get post-flop.


Answer 2: The way to do this is to figure up all of the possible combinations that give Villain top pair or better on the Q94r flop. We already have the total combinations that make up {TT+, AQs+, AKo} from answering question 1 and we can just subtract 3 from that to compensate for the blocker Queen on the flop for QQ and subtract 1 from that to compensate for the same with AQs. Then the answer to question 2 is a simple percentage of those two numbers.

Villain's range is {TT+, AQs+, AKo} on a board of Q94r and the Queen on the flop is never the same suit as Villain's AQs to simplify the calculation a little. Villain hits top pair or better with QQ+, AQs. There are 3 combinations of QQ left (because there is a Queen on the flop), 6 combinations of KK left, and 3 combinations of AA left (because we hold an Ace).

However, the combinations of AQs are a little trickier -- there are only 2 combinations of AQs left. Remember: the Queen on the flop is NOT the same suit as our Ace. So if our Ace was Spades and the Queen on the flop was Clubs, that leaves only two possible combinations of AQs (Hearts and Diamonds).

So this leaves us with 3+6+3+2 = 14 combinations that hit top pair or better on the flop, and 14/38 = 36.8%, so Villain hits the flop 36.8% of the time.


Answer 3: This was by far the easiest question. Another way of asking this question would be 'Which starting hands in Hold'em hold the most effective blockers against the range {TT+, AQs+, AKo}?' There are a few different ways to do this, but I'll outline a way that's easy to learn.

First, consider each "part" of Villain's range. That is, consider TT+, AQs+, and AKo seperately. Now, ask yourself which hand(s) provide the most blockers for each?

a. For TT+, any non-pair hand where both cards are T or higher provide 6 combinations worth of blockers.

b. For AQs+, any *offsuit* AQ or AK provides 3 combinations worth of blockers. For example, if we hold AsQd, then this blocks the combinations AsQs, AdQd, and AsKs. (Note that a suited AQ or AK only blocks 2 combinations).

c. For AKo, any AK provides 5 combinations worth of blockers.

Now we can see that AKo provides the most effective blockers for each part of the range, so it will provide the most effective blockers against the entire range as well.


Now it's time for you to compare your answers to my own and ask questions.

[spoonitnow]

You actually seemed pretty interested in this so I went through and marked in red the parts you need to reconsider.

Think I got it all.

Villain's raise range is 166 hands. That's 78 for the pocket pairs, 40 for the suited stuff, 48 for the offsuit stuff.

Our hand is blocking some of those combos though. He's losing 3 ways to get 99, 3 ways to get AA, 1 ways from each of 4 A containing suited hands (total of 4), and 4 ways from each of the A containing offsuit hands (total of 9). Total blocked hands is 19, so really what we're looking at is 166-19 = 147 hands that he will raise with.

Villain will continue against a 3-bet with 27 hands worth of pp, 6 hands worth of suited, 9 hands worth of offsuited, total of 42. 42/147 = 28.57% continue, so 71.43% folds.

Now I'm gonna wave my hands. Over in the blockers howto thread there is an example that has the exact same betting sequence as in this quiz, and you said that we need him to fold 68.6% of the time. I don't (yet) know how you came up with that, but there it is.

71.43>68.6, so in answer to Q1, yes, it is profitable to 3-bet bluff to 12bb.

Q1. yes


Question 2:
Flop comes down Q94 rainbow, we want to know how often he hits top pair or better with the range in Q1. I hope I'm reading that right.

Of the hands that he called our 3-bet with, the following hands will give top pair or better: QQ, KK, AA, AQs. That's 3+6+3+3 = 15 hands, out of the 42 he would have seen the flop with, so 15/42 = 35.7%

Q2. 35.7%


Question 3:
To maximize our fold equity, we want hands that block as much as possible out of Villain's 3-bet-call range.

So let's see. Unblocked, he's got:

TT - 6 ways
JJ - 6 ways
QQ - 6 ways
KK - 6 ways
AA - 6 ways
AQs - 4 ways
AKs - 4 ways
AK - 12 ways

We want hands that knock out as much of that as possible.
5 TT - knocks out 5 ways
5 JJ - same as JJ
7 QQ - knocks out 5 ways from the PP and 2 ways from AQs, total of 7
13 KK - 5 ways from the PP, 2 ways from AKs, 6 ways from AK, total of 13
20 AA - 5 ways from PP, 2 ways from AQs, 2 ways from AKs, 6 ways from AK, total of 20

6 JT - 3 ways from TT, 3 ways from JJ, 6 total (hey!)
7 QT - 3 ways from TT, 3 ways from QQ, 1 way from AQs, total of 7
10 KT - 3 ways from TT, 3 ways from KK, 1 way from AKs, 3 ways from AK, total of 10

11 AT - 3 ways from TT, 3 ways from AA, 1 way from AQs, 1 way from AKs, 3 ways from AK, total of 11 ways
11 AJ - same as AT

10 KT - 3 ways from TT, 3 ways from KK, 1 way from AKs, 3 ways from AK, total of 10 ways
10 KJ - same as KT

11 KQ - 3 ways from KK, 3 ways from QQ, 1 way from AQs, 1 way from AKs, 3 ways from AK, total of 11 ways

14 AK - 3 ways from AA, 3 ways from KK, 2 ways from AKs, 6 ways from AK, total of 14 ways
12 AQ - 3 ways from AA, 3 ways from QQ, 2 ways from AQs, 1 way from AKs, 3 ways from AK, total of 12 ways

So our possible hands, sorted by fold equity at this point in the hand:

We don't care so much about our equity in the hand, just our fold equity.

AA
AK
KK
AQ
AJ AT KQ
KJ KT....dunno why I did KT twice, but it's heartening that I got the same answer twice
QJ QT QQ
JT
TT JJ
everything else

[spoonitnow]

iopq, you're a bit off on a lot of the calculation of hand combinations. I would suggest that you go study http://www.flopturnriver.com/phpBB2/...tc-t75711.html and then come back and try this again.

Q1.
2-9 = 9 pps
50 ways to be dealt them
ATs-AJs 6 ways
QTs, KTs, QJs, JTs, KQs 20 ways
KQo, AJo 32 ways
108 ways fold

TT-AA 26 ways
AKo/AKs 12 ways
AQs 3 ways

41 ways to call

around 38% call
let's assume SB is raked off

we're betting 11bb to win 5bb
to be immediately successful we need a fold 45% of the time

so yes, it is profitable

[spoonitnow]

Ok, I'm going to answer one question at a time. Because, I'm that slow. I might be wrong. But, I'm answering, and I'm learning.

Question 1: If Villain continues against a 3-bet with {TT+, AQs+, AKo}, is a 3-bet bluff to 12bb profitable?

Assuming 147 opening hands, and 42 continuing hands, that gives us 28.5% of his range that will continue. He is therefore folding 71.5% of the time. He needs to fold 68.6% of the time to be profitable.
Needs to fold<Folding Therefore, YES it would be profitable.

You're really close, but the part in bold is a little off. Remember, we are in the big blind and raising *to* 12x.

Your answer to the 2nd part was correct.

[wellrounded08]

ohh ok. (I Know you already posted the answers, so it's kinda silly, but I'd like to correct it anyway.)

Villain folding x% of the time. We figure by:
Amount rasing/(original raise+SB+BB+Amount raising)=% needed to fold

11/(4+.5+1+11)
11/16.5)=.66666...
SOOOO: villain needs to fold 66.6% of the time.

AJo is also blocked so I'm off there
also blockers are 3 each from pps, not 4

so actually it's

Q1.
2-9 = 9 pps
51 ways to be dealt them
ATs-AJs 6 ways
QTs, KTs, QJs, JTs, KQs 20 ways
KQo 16 ways
AJo 12 ways
105 ways fold

TT-AA 27 ways
AKo/AKs 12 ways
AQs 3 ways

42 ways to call

around 37% call
let's assume SB is raked off

we're betting 11bb to win 5bb
to be immediately successful we need a fold 45% of the time

so yes, it is profitable

[spoonitnow]

AJo is also blocked so I'm off there
also blockers are 3 each from pps, not 4

so actually it's

Q1.
2-9 = 9 pps
51 ways to be dealt them 22-99 is 8 hands, times 6 each is 48 minus 3 for the 99 blocker is 45
ATs-AJs 6 ways
QTs, KTs, QJs, JTs, KQs 20 ways
KQo 16 ways
AJo 12 ways
105 ways fold

TT-AA 27 ways
AKo/AKs 12 ways
AQs 3 ways

42 ways to call

around 37% call
let's assume SB is raked off

we're betting 11bb to win 5bb
to be immediately successful we need a fold 45% of the time

so yes, it is profitable

You're still a bit off in a few places, but one is that if we were in fact betting 11bb to win 5bb then we would need a fold 68.75% of the time. The calculation for this is 11/(11+5). I put an easy counting mistake in bold above.

Just note that you can't rake the SB because there is no rake when you don't see a flop.

oh yeah I didn't fix that part, thanks for looking over that

[Gefunkt]

Now we can see that AKo provides the most effective blockers for each part of the range, so it will provide the most effective blockers against the entire range as well.

I'd like to make the case that AA provides more fold-equity than AKo. I made some counting errors in my original post, so for the sake of clarity I'm going to identify cards by suit and enumerate everything.

Case1: We have AsKh, We block:
AsAd AsAh AsAc KhKd KhKs KhKc
AsKs AhKh AsQs
AsKh AsKd AsKc AcKh AdKh
14 hands

Case2: We have AsAc, We block:
AsAd AsAh AsAc AcAh AcAd
AsKs AcKh AsQs AcQc
AsKh AsKd AsKc AcKh AcKd AcKs
15 hands

So AA blocks one more hand than AKo.

[killerkebab]

Now we can see that AKo provides the most effective blockers for each part of the range, so it will provide the most effective blockers against the entire range as well.

I'd like to make the case that AA provides more fold-equity than AKo. I made some counting errors in my original post, so for the sake of clarity I'm going to identify cards by suit and enumerate everything.

Case1: We have AsKh, We block:
AsAd AsAh AsAc KhKd KhKs KhKc
AsKs AhKh AsQs
AsKh AsKd AsKc AcKh AdKh
14 hands

Case2: We have AsAc, We block:
AsAd AsAh AsAc AcAh AcAd
AsKs AcKh AsQs AcQc
AsKh AsKd AsKc AcKh AcKd AcKs
15 hands

So AA blocks one more hand than AKo.

We don't need to block if we have AA, surely we beat his entire range by default...

[Gefunkt]

We don't need to block if we have AA, surely we beat his entire range by default...

It's not a matter of "need". Granted, playing texas hold'em, preflop, if you're holding AA then the concept of fold equity isn't too horribly useful. That doesn't change the underlying math.

[spoonitnow]

Now we can see that AKo provides the most effective blockers for each part of the range, so it will provide the most effective blockers against the entire range as well.

I'd like to make the case that AA provides more fold-equity than AKo. I made some counting errors in my original post, so for the sake of clarity I'm going to identify cards by suit and enumerate everything.

Case1: We have AsKh, We block:
AsAd AsAh AsAc KhKd KhKs KhKc
AsKs AhKh AsQs
AsKh AsKd AsKc AcKh AdKh
14 hands

Case2: We have AsAc, We block:
AsAd AsAh AsAc AcAh AcAd
AsKs AcKh AsQs AcQc
AsKh AsKd AsKc AcKh AcKd AcKs
15 hands

So AA blocks one more hand than AKo.

You are correct ldo.