Find probability function of random walk, stochastic processes

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My question is how can I create my own probability functions ?

For example , take a look at this article at medium. At some point it produces

$$ P1 = (1 - q) + (qP1^2) $$

Which is the probability if a drunk man will fall from a cliff.

How the writer reaches to the conclusion that has to multiply q with $P1^2$ ?

Are there any resources(books, papers) that teaches (or have examples) someone how to produce this functions-this kind of solutions ?

I want to understand this problems and solutions but I cant find something relevant on the web. Thanks

Answer to my first question

On "Random Walk: Drunk Man in a Cliff" Problem, Why $qP$ in the equation $P = p + qP^2$?