TITLE: Why are events $A$ and $B$ independent whilst events $A$ and $C$ are not?
QUESTION [5 upvotes]: 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?

REPLY [6 votes]: If the red die is 5 or 6 it doesn't reduce or increase the chance of the sum being 7 because every value the red die could be could lead to a sum of 7 with equal chance.  
This can't be said of a sum of 8 though. Because if the red die would have been a 1, a sum of 8 is now impossible. By saying that the red die is a 5 or 6, we are increasing the chance of rolling an 8.