Question 441203
a.  A, B disjoint means P(A and B) = 0.  Since P(A), P(B) > 0, {{{P(A)*P(B) > 0}}}, hence {{{P(A)*P(B) <> P(A and B)}}}, and so are not independent.

b.  A, B independent means {{{P(A)*P(B) = P(A and B)}}}
==> {{{P(A) + P(B) = P(A) + P(B) - P(A and B) = P(A) + P(B) - P(A)*P(B)}}}
Unless P(A)*P(B)= 0, A and B can never be disjoint.  But from the given, P(A)*P(B) > 0, hence A, B are never disjoint.

c. If A &#8834; B, then P(A and B) = P(A).  If we assume independence, then 
P(A) = P(A)*P(B), which means 
P(B) = 1, a contradiction of the hypothesis that 0 < P(B) < 1.  Hence A, B are not independent.

d. P(A and (A &#8746; B)) = P(A), since A &#8834; A U B.  Hence the argument is reduced to one like in part (c) above, the result being the same.