Scenario: A company that makes cartons finds the probability of producing a carton with a puncture is 0.05. The probability that a carton has a smashed corner is 0.08. The probability that a carton has a puncture and a smashed corner is 0.004.
Question: If a quality inspector randomly selects a carton, find the probability that the carton has a puncture or has a smashed corner.
Let $P$ and $S$ respectively be the events "the carton has a puncture" and "the carton has a smashed corner":
$$P(P \cup S) = P(P) + P(S) - P(P \cap S)$$
Thus, what you are looking for is 0.126
The property invoked is true for all non mutually exclusive events.