What is the probability of choosing the same number?

Question

Combinatorics

Good afternoon. I read the article www.vokrugsveta.ru/article/215452 and in the section about "Favorite number", the probability of getting the same number on roulette is calculated.
Who can explain why the probability of falling out on the second spin is 1-2/37, because numbers can also fall out with repetition, i.e. the probability of any number falling out, it seems, should always be 1/37, and its non-falling out, respectively. 1-1/37.
And I also can’t understand why then these probabilities are multiplied and how the final formula is obtained as a result.
I've read a lot of articles, but I can't figure out this case...

Answer 1

There it is somehow awkwardly explained - in this calculation (where 1-2/37) the probability of any number falling out again is calculated. If before that there was not a single repetition for N laps, then N rooms were visited, and the probability of getting into them is not the next. circle N/37.
The probabilities are multiplied according to the probabilities multiplication theorem.
The final formula is also sloppy - at first it says about 36 spins, and the formula again pops up a hell of a thing.
37! - the number of possible series of 37 sectors falling out, provided that none of them falls out twice.
37^n - the number of possible series of 37-long dropout sectors without this condition.
37!/(37-n)! - the number of possible series of long (37-n) falling out sectors, provided that none of them falls out twice. For example, for 2 spins this number is 37*36.