By definition of conditional probability,
[tex]\mathbb P(E\mid F)=\dfrac{\mathbb P(E\cap F)}{\mathbb P(F)}=\dfrac{\mathbb P(F\mid E)\cdot\mathbb P(E)}{\mathbb P(F)}[/tex]
We're told that [tex]\mathbb P(E\mid F)=\mathbb P(E)=0.03[/tex], so
[tex]0.03=\dfrac{\mathbb P(F\mid E)\cdot0.03}{\mathbb P(F)}[/tex]
[tex]1=\dfrac{\mathbb P(F\mid E)}{\mathbb P(F)}[/tex]
[tex]\mathbb P(F)=\mathbb P(F\mid E)[/tex]
which means [tex]F[/tex] has the same chance of occurring regardless of whether or not [tex]E[/tex] had occurred. In other words, [tex]F[/tex] is independent of [tex]E[/tex].
