Question 477724
There are actually two issues arising here, namely, INDEPENDENCE and MUTUAL EXCLUSIVITY.
It seems that you are identifying idependence with mutual exclusivity, which are two different things.
The events B and B' are mutually exclusive (or disjoint), but they are not independent.
Hence P(B and B') = P(empty set) = 0, since B and B' are complementary events.
If two events are mutually exclusive, then the correct rule to use is 
P(B or B') = P(B) + P(B') = 1/3 + 2/3 = 1.