\(\mathbb P(A\cup B)=\mathbb P(A)+\mathbb P(B)-\mathbb P(A\cap B)\)
If \(A\) and \(B\) are independent, then \(\mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B)\), so under this assumption
\(\mathbb P(A)\mathbb P(B)=\mathbb P(A)+\mathbb P(B)-\mathbb P(A\cup B)\)
The only way for \(\mathbb P(A\cup B)\) to be equal to \(\mathbb P(A)\mathbb P(B)\) is if both \(\mathbb P(A)\) and \(\mathbb P(B)\) are zero.
