The professor has given us hints so I tried doing some myself... but I'm still not sure if I'm approaching this question correctly.
Could someone show me how to approach each of these?
Prove the following statement if it is true, otherwise, state that it is not true:
(a) If E is independent of F and E is independent of G, then E is independent of F∪G.
For this, I wrote P(E) = P(F'∩G') and therefore P(E) = P(FUG)'
(b) If E is independent of F, and E is independent of G, and F∩G =φ, then E is independent of F∪G.
For this, I've done P(E∩F)=P(E)P(F), P(E∩G)=P(E)P(G), so P(E∩F)UP(E∩G) / P(E), P(E)P(F)UP(E)P(G) / P(E) = P(E)UP(F∩G)
(c) If E is independent of F, and F is independent of G, and E is independent of F∩G, then G is independent of E∩F.
Could someone show me how to approach each of these?
Prove the following statement if it is true, otherwise, state that it is not true:
(a) If E is independent of F and E is independent of G, then E is independent of F∪G.
For this, I wrote P(E) = P(F'∩G') and therefore P(E) = P(FUG)'
(b) If E is independent of F, and E is independent of G, and F∩G =φ, then E is independent of F∪G.
For this, I've done P(E∩F)=P(E)P(F), P(E∩G)=P(E)P(G), so P(E∩F)UP(E∩G) / P(E), P(E)P(F)UP(E)P(G) / P(E) = P(E)UP(F∩G)
(c) If E is independent of F, and F is independent of G, and E is independent of F∩G, then G is independent of E∩F.