I read that the probability that at least one of events A and B will occur in a trial is P(A + B) = P(A) + P(B) - P(AB), where A and B are not mutually exclusive events. That makes sense, because one would count the probabilities of A and B together, and then ignore event AB, because that counts instances of A and B twice, in the same way that the probability that at least one die of two will turn up six is 11/36, not 1/3.
That made sense intuitively, but I couldn't quite follow the proof because it mentioned things I never saw before. I noticed however, that the following relation also appears to be true (AC and BC are complementary events because latex2html is a dick.)
P(ACBC) + P(AB) + P(ACB) + P(ABC) = 1
P(AB) + P(ACB) + P(ABC) = 1 - P(ACBC)
P(A + B) = 1 - P(ACBC)
I did it on some practice examples and it made sense, but what do I know...