So this problem I made up based on this video. I've made three parts to it just to keep things fun
A bunch of $n$ students take an exam. Out of all the students, $x$ of them are cheaters. A teacher, Bob, has too many students to grade and just checks a random bunch of $y$ papers, where $y \leq n$. If it is guaranteed that a student will be caught cheating if Bob checks their paper, what is the probability that Bob will catch a cheater while grading the exam?
Second follow-up:
Now let's say Bob is tired and is no longer perfect. He can catch a cheater with a $p$ probability. What is the probability then that he catches a cheater while grading the exam?
Third part:
Let us say that the $n$ students take one exam each day that is graded instantly. What is the expected number of days until perfect Bob catches his first cheater, and until imperfect Bob catches his first cheater?