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Independent events do not influence each other, so P(A and B) = P(A)·P(B). The union then follows from the addition rule with that product subtracted to avoid double counting.
Independence
P(A ∩ B) = P(A)·P(B)
Union
P(A ∪ B) = P(A) + P(B) − P(A)P(B)
Both occur with probability 0.1, either with 0.55, neither with 0.45 and exactly one with 0.45.
Independent events can both happen; mutually exclusive ones cannot. In fact exclusive events with positive probability are never independent.