I couldn't decide on a topic for my first post on Settling the Tab, so I went through my recent emails to see what interesting nuggets could be found. Also, I should note that what follows is an example of something that I consider to be "interesting." However, you should proceed only if prepared to be bored by poker minutia...
We're six handed, so my 88 is pretty good against 1 player. It's normally the kind of hand you call the BB with and see a flop, hoping for a set or a particularly favorable texture--typically, a larger pocket pair will have raised. 1010, for instance, has as lot of pre-flop equity, but if you're going against, say, 2 players and don't raise, 1010 is looking for a 2-outer most of the time (probability that there's a card >10 on the flop = 69.5%. So you certainly raise with 1010. 88 is a similar hand but, while dominated by 1010, it plays a called raise much the same way as does 1010 (it's just slightly weaker, becasue 88 is losing to 99 and 1010). The hands that are calling a rasise 6-handed are likely a pocket pair up to 1010 or JJ. JJ or QQ-AA are raising (as, if you're Sean, is AQ [and by extension AK]). AJ is probably calling. Pocket pairs maybe down to 66 or 55 are probably calling (or maybe even raising if the opponent is trying to get tricky).
We both obviously know that you have to use a mixed strategy--if every time you have a mid-pair you limp, your opponents have a really easy time of putting you on a hand and when you do win, you win less. So let's assume, for the sake of argument, that raising with 88 is okay some of the time.
So when you raise with 88, your callers are probably something like 55-JJ, AJ, AQ, and AK. Your raisers are probably something like JJ-AA, AK, AQ, and I'll add in AJ because hey--if Sean thinks AQ is that good, he probably doesn't think a whole lot less of AJ. Maybe (let's say 25% of the time) some other pair (to simplify, let's split the difference between the pairs bigger than 88 and the pairs lower than 88--meaning that for 25% of raises, you have a 50% chance of having the better pocket pair.
So we have:
6 ways to make JJ
6 ways to make QQ
6 ways to make KK
6 ways to make AA
8 ways to make AJ
8 ways to make AQ
8 ways to make AK
...these account for (we're assuming) 75% of hands that have raised.
So we'll assume that there are 12 ways to make a split pot
The only face card that came up on the flop was Q, so suddenly, there are only 3 ways to make QQ with hole cards.
We need to update our preflop-raise breakdown...
6 ways to make JJ (I'm losing to this)
3 ways to make QQ (losing)
6 ways to make KK (losing)
6 ways to make AA (losing)
8 ways to make AJ (winning)
6 ways to make AQ (losing)
8 ways to make AK (winning)
These hands can be made in 43 ways.
We're still using 25% for the probability that the opponent has some other pair, so we'll say that there are 14 ways to make his other hands
I didn't set up my eights on the flop. However, we don't know whether the opponent did (but we do know that the Q has come up, and that case is already covered above. So we need to think about the possibility of a random pair not mentioned above setting up on one of the first two cards of the flop--this comes out to about 8% of the time.
Right now we're dealing with a total of 57 hands:
6/57 opponent has JJ (I'm losing, with about a 8% chance of winning)
3/57 opponent has QQ (drawing dead)
6/57 opponent has KK (losing with 8% chance of winning)
6/57 opponent has AA (losing with 8% chance of winning)
8/57 opponent has AJ (winning with 24% chance of losing)
6/57 opponent has AQ (losing with 8% chance of winning)
8/57 opponent has AK (winning with 24% chance of losing)
14/57 opponent has an unknown pair below JJ
So my odds of winning are determined as follows:
= (24/57)(0.08) + (16/57)(0.76) + (7/57)
= 0.034 + 0.213 + 0.123 = 0.370
So once that flop comes out, I've got a 37% chance of winning the hand.
If my preflop raise is just called, my calculations can include more low pocket pairs, which means that I'm probably (currently) ahead.
When I'm raised, I call, and the Q is out there, I'm probably going to win 35-40% of the time.
So there are a couple of things here:
1) Knowing now that I'm winning 35-40% of the time, does hindsight tell me that I should have I bet on the flop or just check and give up when Sean bets?
2) Did I make a mistake by calling the raise to 2x my bet (my initial bet was somewhere between 3.5 and 4.5 BB)
I'm inclined to think that, once at the flop with one scary overcard that isn't an ace, I should bet.
I'm also inclined to think that I shouldn't call the re-raise (even though it ended up being the case that Sean was just holding AQ) in the first place.