I'll go through these backwards, who knows why. Variety I suppose. Also, these are in no way definitive answers, but I feel pretty strongly that I'm "right". I'll quickly review the play up to the decision - to iron out any uncertainty - and then will proceed on the tough yet standard draw decision. Also, for what it's worth I classify a draw as any hand that's unlikely to be best but can improve. Note that some draws have showdown value, others do not, (in the hand examples, #2 has only real showdown win %, #1 is thin, #3 is nonexistant) and also some made hands sometimes have draw potential. As a somewhat odd digression, it's interesting that often an underpair is really just a weak draw in many cases, but since it has so few outs (2!) it's irrelevant of thinking that it has a semibluff aspect. Anycase:
#3. A bad player (140, very loose passive) limps UTG, and we (240) isolate raise 79 on the button to 10 total. BB cold calls, as does the limper. (BB has 383 and is a nit where this cold call is AQ-AT, JJ-22, possibly a little wider range given UTG's presence, but not looser then that). Flop (~30) comes 862, and they check to us. We timebank a little bit contemplating a cbet (validity in checking) but opt to fire away with 23. BB calls, and UTG folds. Ok, lost the bad player, got the good player, terrific. Turn (~76) comes the K and BB bets 44. What do we do?
Preflop I would isolate this player always, in fact, I'd want to isolate anyone with this hand that limps in 6max. Why? 97s is playable, I want to (try) to get a headsup pot, and the best way to do that is raise. A small raise will tend to just build the pot, so I'd just make anything standard. If villain had a short stack (where the effective raises makes a cbet awkward) I'd probably overlimp (since the hand is still worth playing in position but not with short effective stacks) but that is not the case here.
On the flop we have decent equity and a solid shot at picking up the pot; although occasionally we'll face a checkraise, I'd rather give the BB nit a chance to throw away a pair or ace high. I also think despite the draw heavy board, the ability to string the donk in means we dont get checkraised often. Plus, retaining initiative (if both call) in a massive pot (where we can now get a free river) is of the utmost importance. If it's checked through, too many difficult things happen, as in check/donk bets/we need to fold because we dont close action, or we do call and get c/r by the nit, or nit bets a donk minraises etc. When BB calls his range is pretty tight, 4 to 6 or so combos of a flush draw (AQ, AJ, AT, KQ, possibly AK or KJ) but largely an overpair (JJ-99) or a set. I think there is some chance this villain calls with any pair (77, etc.) but the fact you're leading 3way into someone who is probably calling lessens this fact. Anyway, hopefully my range is accurate to the turn, as is why the flop cbet is correct. (Both for equity advantage, retaining initiative, and hell, ability to pickup a pot with 9 high!)
So, on the turn, lets play the game of process of elimination. I think a few people have voted for raise, and I think that's clearly the worst option. Firstoff, any raise commits us to calling an all in, so we may as well deem a raise = push. (Because even if we raise half pot, thats 126 total, we're getting about 5:1 on an all in call, so we pretty much need to call) Secondoff, what is he stop&go-ing? I think sometimes it's a pair he folds (JJ-99, the occasional 77) but most of the time it's a big hand that did slowplay the flop and doesnt want this street checked through. So:
If we push, there is 120 dead money. (76 pot, his 44 bet) If called, 490 in the pot where we have roughly 18% equity - how did I come up with that? When he has a set, we have 8 outs. When he has a flushdraw, we have either 6 outs or 10 outs (KQs vs AJs, for example). Or, we have roughly 88 +EV in the total pot. We're putting in 207 -EV however on the turn, so net on always getting called is -119. Interestingly, if villain calls and folds to a push half the time, our move is 0EV with a ton of variance.
What does our line look like, raising the donk, cbetting 3people, then pushing the turn? I think we can rep a big hand (AK+, esp bc of the king) enough that JJ-99 do fold. But we cant be sure those even bet. So from a pure combinatorics perspective, ~4 combos of flushdraws that do not bet/fold, 9 set combos, and ~18 combos of JJ-99 (6 each) which may not fold. What that means is roughly 22:9 call/fold, which means we cant really expect villain to fold more then 30% of the time. I'd estimate a turn push is net -45$ EV push, or 22.5 BBs. That's a HUGE price to pay for image and a real spew.
So, folding (0EV) is better then pushing. (Or any raise which essentially is a push)
What about calling? From a pure odds perspective, if we think we have ~18%, we're getting effective 44:164, or 27% equity required. Pot will be 164 with effective 153 behind (perfect stacks!). If villain plays perfectly, ie pushes when we miss (or check calls a bluff always, so that us bluffing is just -EV) and check folds when we hit, he makes 14.5$. (Us calling then is a -14.5$ mistake, or 7BBs, ignoring river action) With no implied odds, this would be a fold. Interestingly though, since we have position and a large portion of his hands are sets (where we do have implied odds) the "best case" scenario (we get his stack when we hit) mean we actually earn 490 x (.18) = +88 vs (-44) turn call, or +44$. (A +22BB earn!) Therefore a turn call is anywhere from a +22 BB good play to -7 BB. I often feel people overestimate their implied odds, and here even if we label them only snapping us off 1 in 3 times (roughly 1 in 3 we think they show up with a set) it balances the play to just about a number >7 on a call.
So you could see how I posted some very marginal situations, but even reserved, this is a turn call. In fact I'd label it: Turn Call (~+2 BB) > Fold (~0 BB) > Push (-22.5 BB). If you said fold, that's OK, but you probably didnt think enough on the river. Another note is that since we're in position (and almost guaranteed to get ATLEAST the set to pay off, if not more) vs. out of position (where we're not guaranteed those things) does factor into a call, since out of position I dont think we can call this ~58% pot bet.
#2. We are UTG and pick up AQo, suits irrelevant. We (200, unknown to these players) make it 8, button ("defends his button religiously" but a tag/lag with 245) calls.
Sub Q: If this player 3bets to 25 total, what's our standard play?
Any case, flop (~19) comes T J 4, we bet 12, villain calls. Turn (43) comes a blank, 3. What's our play? If we check, villain bets 25. What's our play then?
So, in this hand we have a few things going on. AQo is a somewhat sub-premium but nice looking hand to have shorthanded, and I would hope even real nitty players open this shorthanded UTG. Full ring some people would fold this UTG (I would not) but that is understandable yet debatable. Anyway, the open is standard, and 3 to 4x bb or whatever you make really is irrelevant, it's stylistic. That said, when you get 3bet on the button, there are a few things to consider:
1. Your own image, if you're nitty you should be more apt to fold. If you're a maniac, you should be more apt to 4bet. If you're in the middle, closer to a call.
2. Their image, the one given means they're probably [censored] us around alot on their button. When a good player has the button, its good to be UTG or the BB, since you'll be affected less by them playing more hands, so we shouldnt worry about "silencing" them by making a strong play. That's the image / metagame considerations.
3. We raised UTG, which tightens up our range somewhat. If villain loves his button, he should atleast respect this somewhat, making his fluff % a little lower.
Anyway, my default would be to muck AJo and worse, but take a flop with AQo. This is somewhat of a trick question, since I wont argue with anyone folding here, but I think AQo plays well enough (and yeah, on TPTK we're getting AI) in a reraised pot even out of position to merit defending your open.
However, villain simply cold called, and we take a somewhat dry board which is good and bad for us; vs 99 or worse (the hands I'd imagine most cold calls to mean from a TAG here) we have roughly 40% equity and alot of ways to win the pot on a bluff. However, vs broadways, our hand is deceptively bad, since we often are no reverse dominated, or if we're against JT / some of the KJ type stuff (where they suck an out away) we have less equity. (Yet more implied odds). That said, it's an easy cbet here, given ace high figures to be best alot, and I think we can knock 99-22 (minus 44) off the pot alot with an easy stab. The cbet # is my optimal amount but I wont go into that here.
On the turn I think a 2nd barrel has merit, but I'd realign the hand range into float (we beat) straight draw (we beat, they reraise AK, so AQ, KQ, Q9, 89, etc.) a marginal pair (this would be an underpair or midpair hand) a bluff catcher (something like TP which is NOT folding) or a lock (44 or JT, or a weird AA type thing that slowplayed or TT which didnt repop). A person who messes with us in position is more apt to have both the slowplayed hand and the float. Anyway, if we think he'll drop floats and marginal hands, and wont pop us with anything (ever) a 2nd barrel is nice since (depending on float % and how often they call down to showdown) that shows profit AND our straight draw isnt so massive that we want to see a river. If we had KQ, for example, and expected top pair+ to shut us out of the pot, a 2nd barrel is horrific. (In fact, I'd normally check push all in, or check call, depending on their bet size)
So, if we check, villain bets 25, leaving effective stack if we call 155 with a pot of 93 for the river. Lets once again review the options, with raising first.
Here we could technically raise/fold to an all in, since our draw is so weak once pushed on we cannot assume we have 10 outs, in fact it's closer to 4. (or not quite 10% equity) So if we do that, say we make it 75 total to pick up 93. It's very, very difficult to see if villain ever simply calls, but from a combinatorics, if we say he's "always" floating with a suited connector 56, 67, 78, that makes 12 combos. We'll say this is the type to check a KQ, 89 type draw as to not get checked through, and will also bet/fold an underpair. (99-55, 33/22, or 42 combos, wow!). So they're bet/folding roughly 56 floats, this is somewhat overly optimistic but against this type not really all that surprising. We dont think of a type like this to bet/call an AT type deal, and it's kinda hard to put them on hands like that, but we'll say there are JJ/TT/44/JT, (we grouped JJ in as the slowplayed AA as well) which constitute 9 set and if they do not call w/ suited JT, only 2 twopair. If they truly are TAG/LAG in defending their button, we'll lump JTo, as well, so 6 more combos. If we also limp AJ, KJ, and QJ, which is tough to see someone playing those like that always, but in this case we will if they are applying that much pressure, then that is an additional 27 combos offsuit and 9 combos suited. Since we gave an overly optimistic range for float, we'll give an overly pessimistic range for "defending the float," or 56 vs 9+6+27+9, or 51 hands that dont fold. If we think that we also stack them when we hit, and give up on the hand/bluff (even when we improve to the best hand) then we're winning the pot ~52% on the turn, so that's -75 48% of the time, 52% +78, +40.5 vs -36, or in isolation +4.5$ (2BB). We also push when call and hit the nuts (10%) of the time, or 400$ pot (48% x 10%) = 19.2$. (Check folding when we miss) Since sometimes villain doesnt give us that opportunity to hit the nuts (simply pushing the turn over a checkraise) I'd say a c/r is anywhere from a +2BB to +12BB "earn." I think alot of people just call to let the pfr bluff, but alot of people simply push as to not let river cards (another broadway) mess things up, so we'll say this parlay only happens ~1/3, to make this roughly +5BB. (Also, very very rarely we're rebluffed all in, but this is less then 2% occurrence, or 1 combo switching teams so to speak; even if we allow for this, it swings the move from +5 BB to roughly +3 BB.)
Sorry for the long mindedness, it was not my intention. So right away we can eliminate folding, since we think check raising shows more profit. What about calling? Here, since villain is floating, (or floating with a better hand, ala 66) sometimes ace high is the best hand. However, out of position, if we think we're JUST drawing, it's much harder to extract. For what it's worth, I'd check every river, and consider calling raising and folding on all rivers, depending on bet. I think leading out as both a blocker (since any pair hits 2pair for them or straight) is a waste as is a bluff, since any low card possibly hits a set/float and may make them payoff since then they "beat AK". We're getting 25:93, or we require 27% equity to move on. If we think we have ~10 outs to ~4 outs, (safe to assume 7, or roughly 18% equity) once again we're making in isolation a -EV call. (a 9$ mistake, in fact, or -4 BB). However, occasionally ace high shows down as the best hand, occasionally we pair up and show down the best hand, and occasionally we hit the nuts and check raise to bust a set. It's very, very difficult to do the math here, but: This is somewhat of a "secret" of mine, but I never call one street and fold the next with a made hand. I make a decision on the turn and carry through on the river, although every now and then I make a bet size read and make a weird laydown. However, I do call with marginal draws in order to fold to blanks, as that naturally balances out my calling frequencies; it makes less exploitable in the fact that I believe im drawing, but when I hit the fact they also will bluff into my hit hand makes the original call +EV. An odd concept and not worth getting too much into, but the concept applies here; AQ high could be good, but we're not check calling twice with it. But, this is a case an opponent may fire one shell with a float (like 7 high) then give up. Or fire a shell w/ 77 and then give up when we pair, so we showdown a sucked out hand. Unfortunately I cant "math this out" (yes, I'm using math as a verb) but I feel a call here seems like inherent -EV, but since we are able to showdown the best hand ~40% of the time in a 93$ pot (or 37$, 18 BBs which helps balance the loss of 4 BBs on the turn call) but my gut reaction is check calling here is better then checkraising. If we ourselves had complete air, and felt villain was truly floating as much as he was, since the value in the c/r came from them folding and not us improving, I may bluff c/r with air. (Obviously makes a fold to a push very easy, too). Generally if I have air though I'll check fold until a frequent floater has been established, and then I stack a donk with anything good and a stack a donk jr (c/r air and fold) with anything bad. Anything marginal - yes, AQ high is marginal here - we're seeing a river, baby!
So in this case I think call (more then 4BB, closer to 10BB) > raise (+4ish BB) > fold (0 BB)
#1. We're in the CO with KT, we (200, unknown image) raise to 8 pf. BB (245, you have a few notes/stats but just a typical player with somewhat loose TAG stats) defends his blind. Flop (~17 minus rake) comes A 6 3, checked to us, we bet 12, BB check raises to 32 total. We call. Turn (80) 2 villain bets 40. What do we do?
This is the most common occurrence of the draws mentioned. Villain simply isn't betting enough to price us out; there is some debate here, because if we look from a pure effective odds standpoint (and we will) we should fold. But there is more to the hands then that!
First, though, I hope preflop is standard for everyone. KTs in LP should be an auto raise, even for the nittier crowd. (Or full ring crowd) Flop is very nice, since we can get alot of pairs to fold given the ace, and we have a very nice 2nd nut flush draw. The 2/3 bet is more or less a standard cbet, with air I think you pickup the pot alot, and the fact we have 35 to 45% equity vs any hand that calls means we're freerolling on it anyway. When checkraised, though, we should evaluate: Does anyone 3bet here? If I had a stronger draw, I may (45c, 75c, etc). If I had a weaker draw (where I cant call getting 3:1, like 75d) I may. However, the general plan with a strong ace or better made hand is to call and shove over the turn. With a marginal draw, like ours, we will call and take a turn. Alot of turns could be good; Q or J gives us a nut gutterball, a K or T gives us a pair, a club gives us the 2nd nuts. I dont think the flop call is too debatable, but if villain raised more it would be. That said:
We're calling 40:160, or we require 25% equity. Villain himself could have a set or 2 clubs (lowering our equity from 9 outs, or 20%, to 7 outs, or 16%) or even AXc which has us in alot of trouble. I alluded in a hint that because we're drawing dead sometimes this SHOULD be a fold. I'd mentally draw that line around the Jack high flush draw. Here though we could be dominating a 87c, a QJc, etc.
Anycase, once again, effectively, if we call and they play perfectly on the river (even with us having just 16% equity) we're making a 14$ mistake by calling (or -7 BB). If we play perfectly, as in us having 20% equity and stacking AX or whatever they have, we're stacking them (160 in the pot, effective 120 behind, or 400 total$ pot) this is a +40$ call. (Or +20 BB) Needless to say the turn call is anywhere from a -7 BB mistake to +20 BB earn; this is perhaps why people would view this as marginal. In position I'd up the fact we could get more of their stack in there if we improve (hell, if we dont, and they give up with a lower flush draw or air, we COULD win unimproved, but I'd put this as 10% - the Harrington rule tops) I'd average this call to roughly a +14 BB call. Meaning, it's a pretty big mistake to let it go here. If villain bet more - like 60$, 3/4 pot, you're now more likely to make a mistake. (or rather, you're forced into folding, since even optimally it's barely +EV).
What about raising? Many people argued for that, and it makes me cringe. I think villain will have to fold similar club draws, but AXc wont fold w/ top pair as well, which has us locked up. If villain is semibluffing alot - the problem with the turn is 45 got there as well, which we only have 20% equity, as well. I'd expect (it's tough spewing combinatorics for them folding) but 10% them folding air, half/half calling us dead and folding a draw (perhaps 20%?) and if they weirdly misplayed top pair (like AJo, etc.) they MAY call or they MAY fold. This is pretty gray area ish. Clearly 2pair + will not fold; when called roughly ~18% equity. I'd estimate get called 60% of the time, and this is pretty generous. That in mind, We pick up 120 when they fold (40% of the time) and play a 400$ pot with an equity of +72, requiring us to put in -160, for a net of -88. Interestingly, a push is close; (+120 x .4, -88 x .6) = +48 -52.8, or net -4$ or 2 BB mistake. (With lots of variance) This isnt a horrible image play, and if we had more escape outs (like K4c, the additional gut shot) this would be +EV. (Although, again, not likely as MUCH +EV as calling!)
So in this hand, I think it's calling (+10BB) > folding (0 BB) > pushing (-2 BB).
Hope ya'll enjoyed this, I got a TINY bit lazy on the end, but only because I was typing so much and it's time to watch NBA playoffs. If anyone disagrees, please enumerate, and I did rush the math on the end so it may be fuzzy. (Although, again, I dont believe it is)