Let $S\subseteq\Bbb N $ and $X,Y\sim\text{Uniform}([0,1])$. What is $\text{Pr}(\lfloor X/Y\rfloor\in S)$?

Question

Let $S$ be a subset of the natural numbers, and sample $x$ and $y$ uniformly from $[0,1]$.

What is the probability that the integer part of $x/y$ is an element of $S$?

For example, if $S$ is the set of even numbers, then the probability is $1-\ln(2)/2$.

Can you generalize this calculation to sets of the form $\{mn+b:n\in\Bbb N\}$?

What about sets of the form $\{p(n):n\in\Bbb N\}$ where $p$ is a polynomial?

Answer 1

The event that $\lfloor X/Y \rfloor=k$ is the event that $kY\le X<(k+1)Y$, which if $k>0$, has probability $E[ P( X\in(Y/(k+1),Y/k)|Y)] = E[Y](1/(k+1)-1/k)$, and probability $1/2$ if $k=0$, from which $$P(\lfloor X/Y \rfloor\in S) = \sum_{k\in S\setminus\{0\}}\frac 1 {2k(k+1)}+\frac 1 2 \mathbb 1_{0\in S}.$$