EXTREME novice here. How do people (mathematicians, physicists, engineers, etc.) come up with their own formulas?

The IMDB one is likely a heuristic, essentially trial and error.

The Drake equation they've made a list of the important factors, then multiplied them together. e.g. There are 100 millions stars in a galaxy, 40% have at least one planet, the average number of planets in a system is 5, 1% of planets are habitable, Life starts on 0.1% of habitable planets. You multiply them together and it gives the approximate number of civilisations in the galaxy.

In physics and engineering in general you start with dimensional analysis. This reduces the number of variables to a minimum, then you experiment to find formulae that match experimental results.
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You're not gonna like the solution:

Experience and years of throwing shit at the wall and seeing what sticks are the two most tried and true ways for coming up with your own formulas

:)
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You can read up on the history of the heat equation and Fourier series. I'd say it's pretty well documented how Fourier came up with the idea of Fourier series (and Fourier's law of heat conduction) to solve the heat equation. 3Blue1Brown also has a video talking about this.
Trial and error, but the rule of thumb persists: if you want the rating to be proporcional to a certain value, it should be in the numerator. If you want it to be inverse proporcional to a certain value, it should be in the denominator.
This is a great question that touches on math before it's math.

A general simple guideline is:

1. Is the thing you want a formula for "measurable"?  Here I mean 'measurable' in a very vague sense, as in you don't actually have to have the capability of measuring a correct value (like the number of planets with intelligent life in the universe, you know that it's "a" number).
2. Do you know what properties affect the thing you want a formula for?  By this I mean if you change the property, the outcome would change.
3. can you isolate the properties that affect the outcome into just the independent ones?  There are many properties that are entwined causally or correlational-y, figuring out the underlying independent properties can be difficult, but very important.
4. Are those independent properties "measurable"?  Do they have a minimum/maximum interval they work on?

Once you know the independent properties and the interval they work on, and if you're lucky enough that you can get scientific measurements, a formula should just drop into your lap.  Whether it's the 100% correct one?  Ask Newton about that.
The imdb formula is the overall average rating if you imagine that in addition to the v voters who rated it R, there are also m voters who rated it C. This is a standard sort of way to dial back extreme results; you pretend that you have some more data that is pretty average. If v is high, then W will be pretty close to R; if v is low, then W will be pretty close to C.

The Drake equation is just keeping track of what everything means. If you're not familiar yet with keeping track of units, and how they cancel, then it's essential to master it.

* R\_\* is stars created per unit time
* R\_\* × f\_p is stars with planets created per unit time
* R\_\* × f\_p × n\_e is possibly-life-supporting planets created per unit time
* R\_\* × f\_p × n\_e × f\_1 is life-having planets created per unit time

etc.
you are essentially asking "how do you do mathematics". the answer is through understanding, practise, and experimentation.
Sometimes you can numerically prove the relation between multiple variables, sometimes its experimentally proven, but most of the time it’s just trial and error and running with whatever sticks.
As others said, its trial and error, though there is a power tool used by physicist to have pointers to deduce formulas, its dimensional analysis, look it up there are some neat stuff you can deduce, for example you can deduce (kinda) the period of a pendulum based on the assumed variables it should depend on (weight, length of the string, gravitationnal constant, initial angle, etc. See what you can ditch and try to make homegeneity work)
You do a sanity check:

When R=1, C=1, W=1
When R=5, v=1, m=0, W=R

And others.

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