Considering independent events/multiplictaion theorem:
In probability theory, to say that two events are independent means that the occurrence of one does not affect the probability of the other
Two events A and B are independent if and only if their joint probability equals the product of their probabilities:
Suppose this: $A=\xi$ and $B\subseteq A$ Then, $P(AB)=P(A)P(B)$, since $P(A)=1$, thus making them independent, but I can't see how actually they are independent. Suppose A doesn't happen then B can't happen so I see them dependent.
Another Case, let $S=\xi$ and $A,B\subseteq S$, consider $n(S)=10,n(A)=4,n(B)=5,n(AB)=2$, how can these be independent, because if one event doesn't happens or happens, the probability of happening of other is affected.
Why intutively, this seems wrong?