Concept guide
Event independence: deciding by the product rule, with the conditional form, complements and a zero-probability event
Fair-die model: Ω={1,2,3,4,5,6}, each outcome with mass 1/6. Let A={2,4,6}, B={1,2,3,4}, D={1,3,5} and Z=∅. (1) Decide by the product definition whether A and B are independent, and with P(B)>0 check P(A|B)=P(A). (2) Prove that independence of A and B implies independence of A and Bᶜ, and check it numerically. (3) Decide whether the mutually exclusive events A and D are independent. (4) Decide whether A and the zero-probability event Z are independent, and explain why the conditional form cannot be used.