Two Plus Two Older Archives - math quiz "answers"

i put answers in quotes because everyone seems to have a different answer from me for number 3. it looks like it's because people underestimated the equity of K2 vs a hand like K6 (as i did until i pokerstoved) but i'm not ready to present this as the definitive correct answer.

check out aplus's post here for a nice answer to number 1.

also, the original quiz is here .

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question 1

hero has 8 [img]/images/graemlins/heart.gif[/img]6 [img]/images/graemlins/heart.gif[/img]. pot is 2k. villain has 4k, hero covers easily.

board is 7 [img]/images/graemlins/spade.gif[/img] 9 [img]/images/graemlins/heart.gif[/img] 2 [img]/images/graemlins/club.gif[/img] 3 [img]/images/graemlins/diamond.gif[/img]

villain bets 2k. hero can only push or fold. what percentage of the time must villain fold to make pushing better than folding? villain only calls with 9x or better.

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approach 1

if villain calls, pot is 10k, hero has 1/6 equity = 1.7k. hero paid 4k for 1.7k, so being called is -2.3k. stealing the pot is +4k, so hero has to steal a little over 1/3, so call it around 36%. (lmn calculated it exactly; it's 36.5%)

approach 2

i think this way is worse, but i'm not sure. villain has 1/6 equity, so he could call a 1/4-pot bet with 0 EV. here that is 500. so hero is putting in 3500 at 1/6 equity. to figure out the (negative) value of doing that, double 1/6 to 1/3, subtract from one and multiply by 3500 to get 2/3*3500 = 2.3k.

-2.3k when called, +4k when not, so around 36%.

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question 2

blinds are 150/300. it's folded to the SB (2k, hero covers) who pushes. what is hero's calling range if SB's pushing range is

a) KJ+ A8+ 66+
b) JT+ Q9+ K7+ A4+ 22+ (take 45 sec's each)

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600 in dead money, 1400 to call. to call a 2x pot, hero needs 40%, so call it 42%.

using pokerstove these hands are 42% vs the range in a) KJ+ A8+ 66+:

KQs, A9s, AT, 33

these hands are 42% against the hands in the range for b) JT+ Q9+ K7+ A4+ 22+

22, QJ, QTs, KT, K8s, A2o,
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question 3

you hold K [img]/images/graemlins/spade.gif[/img] 2 [img]/images/graemlins/heart.gif[/img]. villain has 4k, you cover. pot is 3k. villain pushes. the board is:

K [img]/images/graemlins/diamond.gif[/img] 7 [img]/images/graemlins/club.gif[/img] 3 [img]/images/graemlins/heart.gif[/img]

a) villain has 1 pair. 67% of the time he has Kx, and 33% he has 7x. should you call?

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i totally blew this one when i did it in my head. i said "ok, K2 has 3 outs against Kx, multiply by 4 and K2 has about 12% equity." this works find if Kx is KQ or AK, but when it's K6, it's not close because of the chops.

K2 v Kx (where Kx is one pair) is 27%.

K2 v 7x is 81%. (this is more straight-forward: 7x has 5 outs, 5*4 = 20, so 7x has around 20%).

next we need to do (27*2+81)/3. i don't really have a good way to do that in my head cause we need to be fairly accurate. i'm going to start keeping a calculator by computer. for now i'll cheat, it's 45.

to call a 4/3 pot bet, we need to be between 33% (full pot) and 40% (2x pot), so call it 35%. (it's actually 4/11 = 36.4%)

45 is bigger than 36, so it's a call.

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b) what % of the time must villain have 7x to make the call barely profitable for hero?

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again, not sure how to do this quickly with no calculator. we only need to get from 27% to 36%, so we don't need a lot of 81%'s thrown in. i'll guess 20%. then by calculator, if villain has 7x 20% then we are 37.8%. so he probably needs to have 7x around 18%.