By independence, \(P(A \cap B) = P(A) \cdot P(B)\text{.}\) However, by mutually exclusivity, \(A \cap B = \emptyset \Rightarrow P(A \cap B) = 0\) gives
\begin{equation*}
P(A) \cdot P(B) = 0.
\end{equation*}
Hence, one or the other (or both) must be zero.