As the title says I'd like to know how to distribute the range to bets/raises/checks/folds.

I am only interested in the theory department so let's assume the villain in each situation is a total stranger and I have no reads of him. I just want to know the theory and approach. I'll just give a couple of examples to make it clarify this a little bit further.

Let's say my preflop range from CO is 25%. I raise to 2.5bb and the BB calls. For the sake of example the BB defends 40%. The flop comes A-9-2r. What is the theory behind how large percentage of my hands should I cbet here, how many hands should I check-call, check-fold or check-raise with etc.?

How about if we swap the positions? The villain in CO raises to 2.5bb with the same 25% and now I am in my big blind. To keep it simple, let's assume I defend 40% of my range and never 3-bet. The flop again comes A-9-2r. How do I distribute my range here? For instance if the opponent bets, how often should I fold in theory? How large percentage am I seeking to defend here?

One more thing regarding bet sizing. Same example as before, I am the BB. Let's say there is 5.5bb in the pot on the A-9-2r flop. I check, villain in CO bets 3bb into 5.5bb. How often should I continue here? How about if he bet 5.5bb instead?

Thanks in advance

I think your question is too broad for this format and you would likely benefit more from coaching, training sites, or full publications on the subject.

Working with software including solvers, advanced equity calculator, training tools like snowie is also an option if you have the resources available to purchase and run them. Solvers would require the most resources for running.

first, forget the never 3bet from the big blind idea. 3betting in no limit holdem is ridiculously profitable and you need a very spewy opponent in order for such passivity to be correct.

co 25% range vs bb 40% range(im gonna assume bb 3bets top 10% including some A5s-A2s and suited connectors down to QJs, KJs):

i assume bb checks 100% on the flop A92r.

I like 1/2 pot bet here with full stacks. my range looks like this: any 2 pair or better at high frequency(i slowplay sometimes on this board with monsters), AT+, A8s-A3s with backdoor flushdraws, A8o-A3o at low frequency(i check these often), JJ, TT, 88, 77, 66, some royal draws at medium frequency with KTs+, QTs+, JTs, some T8s T7s, T6s, 87s with backdoor straight flushdraws at low frequency, 54s with backdoor flushdraw at medium frequency, 54s no backdoor flushdraw at high frequency, 43s at high frequency.

now think about the price for the bb:

if strict mdf was applicable, then bb would fold 33%, but since co has positional advantage(checking is profitable, thus ev bet > ev check must be maintained in order to bet) and range advantage, the big blind should probably fold more than that.

In order to continue, the bb must have a profitable call(investing 25% of the pot requires capturing 25% of the pot in the long run just to break even), most hands are clear folds or calls. this is where i think the margin is:

43s, 53s, 54s(near breakeven as check raise or call at medium frequency without backdoor flush draw in equilibrium vs 1/2 pot bet, slightly more profitable with backdoor flushdaw at slightly higher frequency), T8s, J8s, JTs with backdoor flushdraws as near 0ev low equity check raise draws at low frequency(note that fold = 0ev), non royal broadway draws are likely all marginally unprofitable(imo) no matter how you play them, but royal draws JTs, KTs, QTs are slightly more profitabe(or less unprofitable if i happen to be wrong) and i check raise these 100%.

marginal bluffcatchers in the big blind depend on free showdowns, which according to my co betting range (note weak draws and pocket pairs and weak aces) should happen quite often. thus i think any 9x that i called preflop is a profitable flop call(after all i included it in my preflop range for a reason and if i fold here it wouldn't be consistent with preflop strategy). I would fold K2s and Q2s but I think vs very passive turn and river strategies these are a flop call here. pocket pairs? take em or leave em i think its really close either way near 0ev. In practice i call pocket pairs more vs passive turn and river strategies and fold vs overaggressive turn and river strategies.

I haven't solved this one but I think this is one of the rare flops where we have an overbetting range too at eq, because the bb is heavily capped

OP, the calculation you're describing in nearly impossible. You're essentially asking how to solve the game of holdem.

The truth is that the math is far too complicated to do manually. You either have to use very abstract models, or use a solver.

For resources on learning game theory, I would recommend this short introduction to game theory by Zener, or the more advanced Applications of No Limit Holdem by Janda. Mathematics of poker is also a great option.

If you want an example shown to you on 2+2, start with a very simple river scenario. Solving rivers is much simpler (and more informative for a beginner) than solving flops.

Alternatively, you could simply buy a sovler like Piosolver, Simple, GTO+, Monker, etc. Let the computer do the hard work!

https://www.youtube.com/watch?v=K6P9xIoObOM

I think you can range-bet this flop for 1/3 pot and split your range on the turn.

Start as the PFA, and start BU v BB. Run a decent size database and export results to see a list of bet/check frequencies. You want to understand why the solver is range betting/checking in this scenario and then see the bet frequencies down to about 70% and check frequencies down to 70%. These can be simplified down to range bets/checks (you'll want to memorize all the boards you want to range check on first as it's normally better to bet range than a badly implemented mixed strategy. On the 50% boards you want to take the gto solution OTF & round frequencies to either pure bet/check, 75/25 mix or pure mix (50/50). Then nodelock and re-run the solver toying around to get the maximum EV, it's worth nodelocking for more realistic raise frequencies and in general reducing villains x/r as seldom will an opponent reach optimal raise frequency (usually below). When you have a mixed strat that looses the least EV, you need to memorize it and understand why it's choosing what line be it it prefers checking 2nd pair with a gs but barrels second pair with a weak kicker etc. Rince and repeat for all the middling boards (50% check at NE) AND when you've done extensive work for all positions and scenarios and have millions of real play experience also done extensive work on how your range adjusts vs certain player/pool tendencies and also what percentage of your range wants to defend against what size and when to polarize your cbet/range you should be beating hs. GLGL