Two ordinary fair dice (one red and one blue) are thrown.
Event A: The red die will show a 5 or a 6.
Event B: The sum of the two dice will be 7.
Event C: The sum of the two dice will be 8.
Using the test for independence: $$P(A\cap B) = P(A).P(B) $$ It can be seen that $A$ and $B$ are independent events whilst events $A$ and $C$ are not independent.
I can't understand why this is the case. I understand the mathematics but I can't understand the logic behind it. I drew out the sample space but I am none the wiser. Is there some intrinsic difference between events $B$ and $C$ that results in one being independent and the other not?