Two events A and B are independent. Given that P(A) = 0.3 and P(A or B) = 0.65, find P(B).

Write $b$ for P(B). Independence fixes the overlap at P(A and B) = $0.3b$, so the addition rule gives $0.65=0.3+b-0.3b=0.3+0.7b$. Hence $0.7b=0.35$ and $b=0.5$. Check: the overlap is 0.15 and 0.3 + 0.5 − 0.15 = 0.65. The value 0.35 = 0.65 − 0.3 subtracts as though the events were mutually exclusive, which they cannot be — independent events with non-zero probabilities always overlap. The value 0.15 is the overlap P(A and B) itself, a by-product of the working rather than the answer. The value 0.95 = 0.65 + 0.3 adds where the rule asks for subtraction, and it should be rejected on sight for being so close to 1 when neither event is anywhere near certain.