From the note on union bound I found the following
Fact 1.3 (Reverse Union Bound) Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be arbitrary events, not necessarily independent. Suppose that $\operatorname{Pr}[\mathcal{F}_1]\geq 1-p_1$ and $\operatorname{Pr}[\mathcal{F}_2]\geq 1-p_2$. Then $\operatorname{Pr}[\mathcal{F}_1 \cap \mathcal{F}_2]\geq 1-(p_1+p_2)$.
Here if $\mathcal{F}_1$ and $\mathcal{F}_2$ were known to be independent, does that imply the following?
$$\operatorname{Pr}[\mathcal{F}_1 \cap \mathcal{F}_2] \geq (1-p_1)(1-p_2)$$