Two events A and B are independent, with P(A) = 0.4 and P(B) = 0.25. Find P(A or B).

Independence fixes the overlap: $P$(A and B) = $P(A)\times P(B)=0.4\times 0.25=0.10$. The addition rule then gives $P$(A or B) = 0.4 + 0.25 − 0.10 = 0.55. Adding without subtracting gives 0.65, which would be right only for MUTUALLY EXCLUSIVE events — and independent events with non-zero probabilities can never be mutually exclusive, since their overlap has probability 0.10, not 0. The value $0.45=0.6\times 0.75$ is the probability that neither event happens, and 0.10 is the overlap itself. As a check, 0.55 + 0.45 = 1.