Exercise 1: Preflop, it folds to the small blind who had 15bb before he posted the blind and you have him covered. He open shoves {22+, K2+, A2+, QT+, JT}. What hands can you call with profitably in the big blind?

Step 1 - I'm trying to figure this out in a logical way, but honestly don't know how. Let me see if I can figure something out. I will have to call 14bb to win a pot of 30bb. Thus the pot odds are roughly 2 to 1. If I understand this correctly, I need to win the hand at least 1 out of 3 to make 2bb (win 1 = 30 -lose twice (28)). So to accomplish that I need to be ahead of his range 33% of the time. His range is 22+, K2+, A2+, QT+, JT.... the top 1/3 of his range is probably any pair, AT+, KT+. Thus I think I would feel comfortable calling with 99+, AK, AQ, KQ to give myself a little extra headway...

Step 2 - I don't know if I'm figuring out the range I need to hit odds wise to be ahead enough, but since I'm hampered a bit at the moment, let me download and analyze with pokerstove. According to Pokerstove's estimate his range is about 37%. With me opening the top 10% of hands, the following results occur:

equity win tie pots won pots tied
Hand 0: 38.315% 36.78% 01.53% 35734063096 1491179408.00 { 22+, A2s+, K2s+, QTs+, JTs, A2o+, K2o+, QTo+, JTo }
Hand 1: 61.685% 60.15% 01.53% 58439707048 1491179408.00 { 77+, A9s+, KTs+, QTs+, AJo+, KQo }

Thus top 10% of hands does not give us a high enough win percentage to beat him on average to make the call profitable over time. If I run my range I guessed at above, I get the following results:

This gets us closer to our goal of winning 2/3 of the time, making us a favorite to win... but to be honest, I have gotten to the point that I think I'm doing the whole exercise wrong. It may be more correct to be searching for a point where I am at least a slight favorite, then beat it. If I need to win at least 1/3 of the time to make it a profitable call, I may only need to beat him 33% of the time also... I'm actually quite confused now.


Exercise 2: Preflop, a 20bb stack opens for 5x with {AQ+, 99+} UTG. You call on the button with A K and you have him covered. The flop pot is 11bb after the rake and you have 15bb behind. The flop comes AQTr. Villain open shoves 100% of his range. Can you call profitably?

Step 1 - The pot is now 20 bb from villian plus your 5 bb, for a total of 25bb in the pot. You have to call 9 bb to call. Thus the pot is offering you a little less than 3 to 1. You have top pair. You still have 9 outs to improve. Two cards left to come. Thus roughly you have a 36% chance to improve, though improving may chop the pot or still not win for you without improving to a straight. To cut it a little narrower let's say you have a 30% chance to improve. With pot odds of 3 to 1 you need to win one time in 4 at least, or 25%. You have a 30% chance to improve or are already ahead, call.

Step 2 - Let me try to confirm those numbers in pokerstove... Pokerstove puts you at
Board: As Qd Tc
Dead:

equity win tie pots won pots tied
Hand 0: 44.580% 34.63% 09.95% 11656 3349.50 { 99+, AQs+, AQo+ }
Hand 1: 55.420% 45.47% 09.95% 15305 3349.50 { AcKh }

A much higher percentage than I figured, making it a call.

Exercise 3: Preflop, it folds to you on the button. Both blinds had 20bb stacks before posting the blinds and you have them covered. You open raise to 4x, the SB shoves {AT+, 22+}, then the BB shoves {AQ+, 88}. What is your profitable calling range?

I look forward to your answers.

Step 1 - The SB range is looser than BB. Effectively you will be calling 44 bb with a bet of 16 bb. Thus you are getting a little less than 4 to 1 pot odds. Let's call it 3.75 to 1 (I did that in my head, yeah me!). To make the math easier, I am going to say 3.5 to 1. Thus I need to win more than 1 in 4.5 attempts to make this profitable. The BB range is the one I need to be beating as often as possible. 1 in 4.5 is about 22 % so I need to be ahead of 78% of his range. To do that I think I'm only calling here with AA, KK, AKs, maybe QQ.

Step 2 - Pokerstove confirms that this is close to the truth, IMO, with my QQ+, AKs range winning 54% of the time... I think I'm doing this wrong but that is still a good percentage of time ahead of the others, so even though my explanation in step 1 is not what I figure to be correct, the results might be...