Hi everyone,
I am trying to build up the intuitive explanation of reasons behind c-bet in NLH - why we c-bet and what's the optimal bet size - by using the following simple example:
we raised preflop from LP with A5s, BB called and the flop has two spades, so we hit the nuts fd.
BB checks.
Should we bet here? and why? and how much?

First of all, let's reduce this game to a very simple one : we bet and either BB folds OR BB cals and both of you check to the showdown.
In the first case we win 100% of times.
In the second case we win ~33% of times - because we have 9 outs and 2 streets ahead.
Now, let's say that BB folds with frequency X : [0, 1], hence he calls with frequency 1-X[0,1]

So, on average we win this game with the following frequency:

Y = X * 1 + (1 - X) * 1/3 = 2/3 * X + 1/3

For example if BB folds 1/2 of times, then Y = 2/3

Now, the game of poker is not that simple :
1) sometimes BB check-raises OTF and we don't see the turn and the river
2) sometimes we don't complete the flush OTT and fold.
etc..

So, we win this hand less than Y of times from the formula above. For example if BB folds 50% of times, we win the hand less than 2/3 of times. If we can estimate actual winning frequency as accurately as possible, we can derive the bet size.

Questions :
- is this explanation correct?
- how we can reduce our win frequency of the simple game to be closer to the actual one?

Looking at a single hand in isolation is already a mistake.

It would be kind of hard to compute the solution for this even if you knew villain's precise hand, because you still have to factor in how he plays that hand on every possible runout.

e.g. If he has A2s (a hand you dominate) with a BDFD, it might be more profitable for you to check back the flop, because on the rare occasions you both bink an ace, you could win more money than by just betting the flop and winning immediately. You could also get value from him if he bluffs the turn. If he flopped a set, then obviously you'd do better by checking back in order to suck out for free. Conversely, if he has AJ or something like that (which is beating you), you'd like him to fold ASAP.

Villain actually has a range, and so do you, so not only do you need to find the best strategy for this combo vs villain's particular range, but you also have to work out the max EV line for this combo in the context of your entire range.
And this is why solvers were invented. They can compute all the different runouts and action sequences to find out which lines are the most profitable for each hand in your range, given the probability that villain calls, folds, or raises the flop, and which cards can come on the turn and river. Poker is not a simple mathematical game. It's incredibly complex.

Switching to a toy game with check-down totally misses the point of flop c-bets.

The flop is the heart of the hand where different decisions set up very different scenarios on the turn and river.

It is so complicated that going the other route and using purely logic and specific definable terms of poker theory, works better.