I really want to have a deep understanding of the independent probabilities of two events. That means to me that I just do not want to use and know the definition. I want to fully understand the why.
Definition 3.1. (a) Two events $A$ and $B$ are independent if $\text{P}(A \cap B) = \text{P}(A)P(B)$.
(b) $A$ (possibly infinite) collection of events $(A_i)_{i \in I}$ is an independent collection if for every finite subset $J$ of $I$, one has $$\text{P}\left(\bigcap_{i \in J} A_i\right) = \prod_{i \in J}\text{P}(A_i).$$ The collection $(A_i)_{i\in I}$ is often said to be mutually independent.
Therefore my question:
Let's consider two events of two train crashes $A$ and $B$. $A$ is in London and $B$ is in New York.
If the intersection of the two trains is a multiple of the probabilities of the two events then these two events are independent.(In my opinion this should be $0$ for independent events) If not they are dependent.
If we just know this kind of information, logically these two events should not be dependent, because a train crash in London should not have anything to two with one in New York. (Could I also say shares the same information?) However, if I get that these two events are dependent is my equation wrong? AND is this value the probability of their dependency?
I appreciate your answer!