Caclulate Probablity that student will pass the test? There is a test of Algorithms. Teacher provides a question bank consisting of N questions and guarantees all the questions in the test will be from this question bank.Due to lack of time Ram can practice only M questions. There are T questions in a question paper selected randomly. Passing criteria is solving at least 1 of the T problems.Ram can’t solve the question he didn’t practice. What is the probability that Ram will pass the test?
 A: We will first find the probability that Ram fails the test. To fail the test, Ram must have not been able to answer any question. This means that there is no intersection between the $M$ questions he practiced and the $T$ questions on the test.
We assume $N \geqslant M+T$ (otherwise, there must be atleast one question which is both in the $M$ questions he practiced and in the $T$ questions on the test, which would give him a guarantee of passing).
Normally, there would be $N \choose T$ ways to choose problems for the test. To ensure that Ram fails, we have to choose from the $N-M$ questions he didn't study, giving $N-M \choose T$ ways to choose for the test. Thus:
$$P(\text{fail})=\frac{N-M \choose T}{N \choose T}$$
$$P(\text{pass})=1-\frac{N-M \choose T}{N \choose T}$$
I leave it to you to simplify the expression.
A: In order to fail the test, all the questions cannot be studied by Ram. Notice that the probability that the first question is not studied by Ram is $\frac{N-M}{N}$. Then, for the next question to not be on the studied ones, the probability is $\frac{N-M-1}{N}$, and so on. This can be expressed as a product:
$$P(\text{failing})=\Pi_{n=0}^{T-1}{\frac{N-M-n}{N}}$$
Hence the probability of passing
$$P(\text{passing})=1-\Pi_{n=0}^{T-1}{\frac{N-M-n}{N}}$$
