I was wondering about this problem: say I have to take a test made of $31$ questions chosen among a database of $140$ questions total. Those questions are open questions (that is, not multiple choice questions).
Say that I know all the database of questions, because it's public, but I have studied poorly and I only know $100$ of the $140$ questions.
Is there a way to find a sort of "average" probability for which al least $25$ of the $31$ questions are among the $100$ I know? I said $25$ but we could do $n$ to generalise.
I was thinking that something I should (?) take into account is the number of ways the professor can choose $31$ questions among the $140$ that is
$$\binom{140}{31} = 11338699879051838313792998934400$$
Now from here I don't know how to proceed. I maybe think that I should take into account the fact that I know $100/140$ of the questions, but I don't know how to put the $40$ questions I don't know into account, in the total ways to choose the questions.
Also my "at least" makes the problem harder...
This is a hypergeometric probability. In the question pool, there are $100$ questions you have prepared for, and $40$ questions you have not. If $31$ questions are selected without replacement and $X$ is the random number of prepared questions on the test, then the desired probability is
$$\Pr[X \ge 25] = \sum_{x=25}^{31} \frac{\binom{100}{x} \binom{40}{31-x}}{\binom{140}{31}}.$$