When to use intersection and union?

Question

In a probablity problem how do one come to know that one have to use intersection between two events or union between two events to arrive at the solution?

Answer 1

$P(A \cap B)$ means the probability that both $A$ and $B$ happen.

$P( A \cup B)$ means the probability that $A$ or $B$ happens. It is ok to have both happening at the same time, and it include the case where exactly one happens.

Answer 2

If you have n number of events and you are asked to find when at least one of them occurs you have to use union. The keyword is "or".

If you are asked to find probability of all of them occurring together then you have to use intersection. The keyword will be "and".

Venn diagrams will help you understand this better.