“Independent” does not mean “unrelated in spirit.” It means: knowing one outcome tells you nothing about the other. Dice rolls? Independent. Midterms and sleep? Probably not. Don’t confuse mutually exclusive (can’t both happen) with independent (don’t inform each other). If events are mutually exclusive and both have positive probability, they are not independent. Start with the English, then do the math: P(A∩B) = P(A)P(B) only when independence is actually true, not merely convenient.
