How could one calculate probability for how many questions you need to learn for a test. These are the numbers:
I have 63 test questions written on a sheet of paper from which 10 are going to be on an exam and 5 are needed to pass the exam. What are the odds in percentage that random 5 questions that I learned are going to be on the exam? Please help, not a great mathematician.
If you have $N$ questions from which a $10$ question exam can be built, and $5$ of those questions are needed to pass, and you completely understand and can answer perfectly $k$ of the $N$ questions, what is the probability that an exam chosen uniformly at random from the set of all possible exams contains at least $5$ of the $k$ problems you have mastered?
Assuming we do not care about the ordering of the questions on the exam, there are a total of $N \choose 10$ exams, and among those there are $$\sum\limits_{i=5}^{10} {k \choose i}\cdot {N-k \choose 10-i}$$ exams which you can pass based on mastered material. Thus, the probability that any given exam can be passed based on your mastered material is the ratio of these two sums.
For $N=63$ and $k=30$ you get about a 57% chance of passing. Not all that great.
For $N=63$ and $k=40$ you get about a 91% chance of passing. Not all that bad.
For $N=63$ and $k=50$ you get about a 99.7% chance of passing. Basically a sure thing.