What is statistical "independence"? How can probabilities verify that two events are statistically independent?

It means, the occurrence or non-occurrence of one event has no effect on the probability of occurrence of the other event.

P(A|B) = P(A|not B) = PA

meaning, the probability of an event A occurring on condition that B has occurred is equal to

the probability of occurrence of A on condition that B has not occurred, which is equal to

the general probability of occurrence of event A.

You can verify independence checking that this equality holds.

It also has an implication which can be used to test independence:

P(A and B) = P(A)*P(B)

meaning, the probability of two events occurring at the same time is equal to the product of the probabilities of each event.

Here is an identity that you need to know:

P(A u B) = P(A) + P(B) - P(A intersect B)

If P(A) = 0.3, P(B) = 0.4 and P(A u B) = 0.5, then

P(A u B) = 0.5 = 0.3 + 0.4 - P(A intersect B)

P(A intersect B) = 0.2, right?

FACT: Independence means P(A intersect B) = 0.0. Always.

So, if you can show that P(A intersect B) = 0.0, then A & B are independent.