We have a random variable $X$ uniformly distributed on the set $\{1,\ldots,n\}$. Assume $s<<n$. Can anyone please advise, how to find the conditional probability $P\{X = k | X\le s \}$, where $k\in\{1,\ldots,s\}$?
More specifically, let us construct a random variable $Y$ in the following way. We generate a random number $X$ uniformly distributed on $\{1,\ldots,n\}$ and if $X\le s$, we assign $Y:=X$. Otherwise we discard $X$ and generate again. Is it true, that $Y$ will follow uniform distribution on $\{1,\ldots,s\}$?
Yes of course it would.
As a side note you should be a little careful because you're double using X and Y for your random variable and for the value the random variable takes on. You should use lower case for the latter.