"Mutually exclusive" sounds like it should mean "unrelated." Mathematically, it means the exact opposite.
These two terms get confused constantly, and the confusion is understandable — both descriptions involve two events that seem to have nothing to do with each other. But they describe opposite relationships. Independent events are ones where learning that one occurred gives you zero new information about the other — its probability doesn't budge. Mutually exclusive events are ones where learning that one occurred gives you total information about the other — you now know with certainty it did not happen. One relationship carries no information at all. The other carries the maximum possible amount. Treating them as similar, or worse, interchangeable, is a real and common source of wrong probability calculations.