Complement vs Mutually Exclusive Events

The complement of an event $A$, written $A^c$ (read A complement), collects all outcomes in the sample space where $A$ does not happen. Its probability is $P(A^c) = 1 - P(A)$, since an event and its complement together cover everything. For example, if the probability of rain is 0.3, the probability of no rain is $1 - 0.3 = 0.7$. The complement rule is handy when at least one is easier to compute as 1 minus none.

Two events are mutually exclusive, or disjoint, when they have no outcomes in common, so if one happens the other cannot. For example, on a single die roll, rolling a 2 and rolling a 5 are mutually exclusive. For such events the addition rule simplifies to $P(A \text{ or } B) = P(A) + P(B)$. Being mutually exclusive is about overlap, which is different from being independent.