How to prove that pairs of probability are independent?

Question

Given the following probabilities:

p(A|C) = 0.5,
p(A|C') = 0,
p(B|C') = 0.6,
p(B'|C) = 0.4,
p(C) = 0.3

How do I prove which of the pairs containing A,B and A,C are independent, without having any information about P(A) or P(B)?

Answer 1

For independent events $A$ and $B$ we must have$$P(A|B)=P(A)$$We proceed by$$P(A)=P(A|C)P(C)+P(A|C')P(C')=0.15$$$$P(B){=P(B|C)P(C)+P(B|C')P(C')\\=[1-P(B'|C)]P(C)+P(B|C')P(C')\\=0.6P(C)+0.6P(C')\\=0.6}$$Since $$P(A)\ne P(A|C)\\P(B)=P(B|C)$$then $A$ and $C$ are mutually dependent and $B$ and $C$ are mutually independent.