1. Poker Stars, $1/$2 NL Hold'em Cash Game, 6 Players
LeggoPoker.com - Hand History Converter

SB: $134
Hero (BB): $304.70
UTG: $136
MP: $202.70
CO: $200
BTN: $239

Pre-Flop: 6 4 dealt to Hero (BB)
3 folds, BTN raises to $5, SB calls $4, Hero calls $3

Flop: ($15) 8 3 7 (3 Players)
SB checks, Hero bets $12, BTN folds, SB calls $12

Turn: ($39) Q (2 Players)
SB checks, Hero checks

River: ($39) A (2 Players)
SB bets $30, Hero raises to $168, SB folds

Results: $99 Pot
Hero showed 6 4 and WON $234 (+$189 NET)

My opponent here is a huge fish, 44/8. I opted not to bet the turn because I felt that he would call again with any pair, any open-ender, and potentially J9/JT, all of which beat me. It would also suck to get jammed on, but I doubt that happens very often versus this player. Given how wide I think his calling range is on the turn, the correct play here is probably to bet turn and bet a lot of rivers - hell, maybe any river? does he check-call A8 all-in on a jack river? I don't know.

Anyway, I wanted to go over the math of my river shove. I'm risking $87 to win $69, so my bet has to work roughly 56% of the time. His range probably looks something like this:

Bluffs (91 combos) -
1. missed straight draws: {9T, JT, J9, 56, 45, 46, 96s} (84 combos)
2. missed heart draws: {KJ, KT, K9, K6, K5, K4, K2} (7 combos) - I am unsure what else to include here as I don't know if J2s/94s/25s stuff are in his range preflop. They probably are some of the time but I'll leave it where it's at for a conservative estimate. It probably won't make much of a difference anyway since that's only a handful of combos.

Value hands (72combos) -
1. Axhh {AJ, AT, A9, A6, A5, A4, A2} (7 combos)
2. slowplayed flop hands {87, 88, 77, 33} - let's say he slowplays half the time, leaving 18/2 = 9 combos.
3. turned/rivered two pair {Q8, Q7, Q3s, AQ, A8, A7, A3} (56 combos)

Now, I went into this assuming that I would be doing math to justify my shove since I have this horrible thing where I always assume I'm right, but already it's pretty clear my shove is -EV. If he bluffs the river 100% of the time my shove is basically break-even:

91 / (91+72) ~= 56%

And obviously we can't assume he's going to bluff the river 100% of the time or anything close to it. I think the moral of the story here is that when you're talking about a fish who plays tons of unpaired cards, while those missed straight draws comprise a ton of combos, so do two pair combos (assuming they are feasible given flop/turn actions). Intuitively I would've expected his value range to be something like half the # of combos it actually is because I rarely do calcs like this (obviously much to my detriment).

2. Poker Stars, $1/$2 NL Hold'em Cash Game, 6 Players
LeggoPoker.com - Hand History Converter

BTN: $200
SB: $286.65
Hero (BB): $203
UTG: $208
MP: $75.65
CO: $251.30

Pre-Flop: 8 T dealt to Hero (BB)
3 folds, BTN raises to $5, SB folds, Hero raises to $20, BTN calls $15

Flop: ($41) 7 2 7 (2 Players)
Hero bets $20, BTN raises to $45, Hero raises to $77, BTN raises to $180 and is All-In, Hero folds

Results: $195 Pot
BTN showed and WON $295 (+$198 NET)

I find a spot like this is super hard to analyze. Obviously quite a bit of leveling goes on here and no conclusive answer can be reached w/o very precise reads on his frequencies, but at the very least I'd like to do a little math to at least get a feel for what % he would have to be bluffraising here to make my 3bet profitable.

First, His UO PFR% on the button is 50%, and his fold to 3bet is 75%, meaning he continues with a range of 12.5% of hands. Since this is normalized across all opponents I expect this range to be wider against an aggressive 3bettor like myself, but I think his primary adjustment here is going to be 4bet bluffing more and perhaps calling slightly wider. I'm unsure what his 4bet value range is, but I know he doesn't 4bet super light for value, so let's just say AA-JJ and AK for the time being (even though in reality TT and AQ will be in there some % of the time as well). So the range of hands he calls with PF is probably something like {TT-77,AQs-A9s,KTs+,QTs+,JTs,T9s,AQo-AJo,KJo+}, which is 8.7% of hands, or roughly .087*1326 = 115 combos of hands. However, I have blockers to TT and 88, so he'll have 105 combos total. Of these, I believe he is going to raise/get in 88-TT (12 combos) and all his flush draws (10 combos), for 22 combos total. I don't think any of the bare ace highs would ever raise the flop since that doesn't make any sense, so that takes out 35 combos of hands, which leaves us with 48 combos out of 105 that he could choose to bluff with, compared to 22 combos of value hands.

Now, I'm risking 57 to win 106, so my raise has to work about 35% of the time. Therefore if he chooses to bluffraise 8 combos of hands (or, raises his air 8/48 = 17% of the time) my raise will show a profit. This isn't very often at all, so I feel like my 3bet will generally show a profit here, specifically because he's not going to have that many hands he's comfortable raise/getting it in and I get such a good price on my bluff. T8 is one of the better hands to bluff with here since it blocks some of his pocket pair combos. Th8x would be the absolute nut hand to 3bet bluff since it also blocks some FD combos as well. Though, I mean, it isn't going to affect those percentages that much.

3. Poker Stars, $1/$2 NL Hold'em Cash Game, 6 Players
LeggoPoker.com - Hand History Converter

BB: $200
UTG: $234.70
MP: $365
Hero (CO): $410.15
BTN: $210.50
SB: $206.15

Pre-Flop: A 2 dealt to Hero (CO)
2 folds, Hero raises to $6, BTN calls $6, SB calls $5, BB folds

Flop: ($20) 2 3 A (3 Players)
SB checks, Hero bets $14, BTN raises to $42, SB folds, Hero raises to $404.15 and is All-In, BTN folds

Results: $104 Pot
Hero showed A 2 and WON $463.15 (+$415.15 NET)

I don't have any great analysis for this one. I strongly considered folding the flop but he had a raise cbet of 18%, which is quite high in my experience. I wasn't sure at the time if he was capable of bluffing in this exact spot, however he seemed to not be afraid to bluff so I stuck it in.

The main point I wanted to make here is that the parameters that make this a flop fold are pretty common amongst players in my games:
1. low raise cbet (I consider 10% to be relatively low and mostly weighted towards value hands w/ the occasional bluff)
2. always 3bet AK PF (very common obviously)
3. wouldn't raise flush draws on the flop. I think this one is actually very common on this flop texture because they know that they're representing a pretty thin value range by raising so should expect to get jammed on by an ace a lot, and since it's impossible to have a combo draw here a lot of people just flat 100% of their FDs. the one exception might be if they have like KQcc or something and expect me to just jam in any flush draw and get it in very good, but I still think most players flat that on the flop. raising QJcc is unthinkable from most players.

Even if we assume he raises all his king-high flush draws our equity when all-in is:

Board: As 3s 2d
Dead:

equity win tie pots won pots tied
Hand 0: 29.792% 28.42% 01.37% 2532 122.50 { Ah2h }
Hand 1: 70.208% 68.83% 01.37% 6133 122.50 { 33-22, A3s, KsQs, KsJs, KsTs }

Which makes this a snapfold against what's a surprising % of the 1/2 population. Intuitively I would say that heads-up this would never be the case but that the third player in the pot makes many people play straightforward enough in his spot to where you'd have to muck the flop. I'd be interested in hearing people's opinions on my analysis here.

That was really helpful and actually kinda fun (math geek I know). I'm going to start doing more of this in the future and actually put this blog to use.