[5NL] A9s in CO, in depth

This is a hand that we were talking about in the IRC. We determined our two best options were either C/F or B/F 1/3 pot OTR, and I said I'd do an in depth analysis breaking down our opponent's range to figure out which was the better option. So, here it is!

Hero (CO): $5.22 (104.4 bb)
BTN: $4.86 (97.2 bb) {26/5, 0/2 fold to CB, 0 3b, 20 hands}
SB: $5 (100 bb)
BB: $5 (100 bb)
UTG: $5 (100 bb)
MP: $5.12 (102.4 bb) {29/10, 0 3b, 21 hands}

Preflop: Hero is CO with A 9
2 folds, Hero raises to $0.15, BTN calls $0.15, SB folds, BB calls $0.10

Flop: ($0.47) T A T (3 players)
BB checks, Hero bets $0.33, BTN calls $0.33, BB folds

Turn: ($1.13) 7 (2 players)
Hero bets $0.54, BTN calls $0.54

River: ($2.21) 3 (2 players)
Hero ??

I've made the assumption that our opponent will raise AT, AA at some point earlier in the hand, and probably raise 77 OTT. He's a fairly passive player so I've assumed that he won't raise his trips OTF and OTT the vast majority of the time. He'll probably call these two bets with his gutters and JJ-KK sometimes, so I discounted two thirds of his JJ-KK. So his range should look something like this to the river:

Board: AdThTs7c3h
Equity Win Tie
Hero 64.50% 62.72% 1.78% { As9s }
BU 35.50% 33.73% 1.78% { TT, KdKh, KdKc, QdQh, QdQc, JdJh, JdJc, AJs+, A9s-A2s, KTs+, QTs+, JTs, T8s+, AJo+, A9o-A2o, KTo+, QTo+, JTo, T9o }

As for breaking down his range and how he will play it, these are my assumptions:

- When we check he'll bet AJ+, his Tx, his TT, and check back the rest
- When we bet 1/3 pot he'll fold his few JJ-KK and his BW gutters, call all his Ax, raise a few of his Tx (say KT, QT), and of course TT


EV FOR CHECKING

We have 97.32% vs his check back range, and he's checking back 66.3% of his range so

(0.663)(0.9732)(2.21)=$1.43


EV FOR BET/FOLDING 1/3 POT

When we bet he's raising 10%, calling 58% (against which we have 56% equity), and folding 32%.

(0.1)(-0.74)=-0.074
(0.58)(0.56)(2.95)-(0.58)(0.44)(-0.74)=0.769
(0.32)(2.21)=0.707
Total: $1.40

So... based on my assumptions and assuming I didn't make any mistakes it's pretty damn close!


edit: I'm interested in what betting bigger does to our EV so I'm gonna do that real quick, betting 2/3 pot without changing his range breakdown even though it probably does change a bit..

(0.1)(-1.47)=-0.147
(0.58)(0.56)(3.68)-(0.48)(0.44)(-1.47)=0.885
(0.32)(2.21)=0.707
Total: $1.44

Though his calling range probably actually does narrow a bit... I bet if we even took out A2 it'd be an inferior option to 1/3 pot.