I have a confusion in knowing when to use $P(A∩B$) and $P(A\backslash B)$

Question

I often stumble upon problems which requires me to take a hard decision on what should I apply in problems. I really want to know the diffrence between $p(A\cap B)$ and $p(A\mid B)$ in a specific way(actual mathematical meaning is clear) but.. let me explain by example.

Consider an exam where $P(E_k)$ is the prob that exactly $k$ people pass the exam so consider a situation in which you defien an event $A$ such that $P(A)=$ prob that you select one student and that student has passed the exam.

So if I ask you what's the prob that only one student has passed the exam. What will you say $P(E_1\cap A)=p(E_1)P(A\mid E_1)$ or $P(E_1\mid A)$?

Kindly help me with this thank you in advance!!!!

Answer 1

Use $P(A \mid B)$ when $B$ is something given, i.e., read $P(A\mid B)$ as $\textit{probability of A given B}$.

Use $P(A \cap B)$ when both $A$ and $B$ have to happen but nothing is given.

Let $F_k$ be the probability that at least $k$ students pass the exam

1. What is the probability of $10$ students passing the exam if we know that $9$ students passed the exam? $P(F_{10} \mid F_9) = P(F_{10}\cap F_9) / P(F_9) = P(F_{10}) / P(F_9)$.
2. What is the probability of $10$ students passing the exam and $9$ students passing the exam? $P(F_{10} \cap F_9) = P(F_{10})$

The second probability is always lower since the event that is already given in the first case might not happen in the second case.