I recently wrote an application using a deep learning model designed to classify inputs. There are plenty of examples of this using images of irises, cats, and other objects. If I trained a data model to identify and classify different types of irises and I show it a picture of a cat, is there ...

See this background post for more info: probability and modular arithmetic. So we have the basic fact that $\text{Pr}(\text{a randomly selected } 1 \leq k \leq x \text{ is such that } x \mid k) \equiv p_n(x) = \dfrac{n - n[x]}{xn}$ where we've redefined $n[x] \equiv n_{(x)} \equiv n \pmod x$ or ...

$$ \sum_{x = 2}^a n \pmod x $$ Doesn't seem very interesting until you connect it with expected value of divisors. Since the probability that $x \mid k $ randomly selected from $k \in (m,n]$ is readily seen to be $\dfrac{\lfloor \dfrac{n}{x}\rfloor - \lfloor \dfrac{m}{x}\rfloor}{n-m}$ and sinc...