Probability of successful sampling fron a population

Question

Say I have a list of 100 questions from which to study for an examination. In the examination I will be asked 10 random questions from the 100 and I pass the exam if I answer 6 questions correctly. What is the way to calculate the number of questions to study (out of the 100) for a given probability? As an example if I wish to pass the test with 100% probability I need to study 96 questions and then I can never fail the test. I ran a computer simulation and I noticed that if I study 72-73 questions the probability of getting 6 out of 10 random questions right is about 90%. How can I calculate this mathematically?

Answer 1

Suppose you studies $n$ many problems, then we try to calculate the probability you will get a pass.

The number of total 10-questions exams is $$C^{100}_{10},$$ the number of the exams you could pass are $$ C^n_6C^{100-n}_4+ C^n_7C^{100-n}_3 + C^n_8C^{100-n}_2+C^n_9C^{100-n}_1+C^{n}_{10}C^{100-n}_0. $$

So you chance of passing the exam is $$P_n=\frac{C^n_6C^{100-n}_4+ C^n_7C^{100-n}_3 + C^n_8C^{100-n}_2+C^n_9C^{100-n}_1+C^{n}_{10}C^{100-n}_0}{C^{100}_{10}}.$$

Since the probability that you pass the exam is increasing when you study more problems, for a given probability $p$, you can always find $n$ s.t. $$P_n\leq p\leq P_{n+1}.$$

Then if want to know what $n$ is given $p$, you just solve the above equation(you may not get exactly $P_n=p$).

I guess it would be hard to have a close form of $n$ in terms of $p$. But if you just want a approximate solution of $n$ given a known $p$(say $p=0.5$), you at most try 100 numbers.