Hey asilvester635.

I agree with 1, 2, 3, 4, 5, 6, 7: well done.

For number 8 this just means that P(A and B) != P(A)P(B)

When an event is independent it means that conditioning on that event doesn't change the outcome. Typically we write this as P(A|B) = P(A) for any event B not including the null event which is not a proper event in a probability space even though it is accepted as a special kind of set member in set theory.

If you have a sequence of events in any probability space, then P(A|B) = P(A) for independence (i.e. its memory-less) or if not then P(A|B) != P(A) for dependent and non-memory less events.