Question

Events $E$ and $F$ are such that P(not E or not F) $=0.25$ State whether $E$ and $F$ are mutually exclusive

Answer 1

We have P (not E or not F) $=0.25$
$\Rightarrow P(E^{\prime }\cup F^{\prime })=0.25$
$\Rightarrow P{(E\cap F)}^{\prime }=0.25$
$\Rightarrow P(E\cap F)=1-P{(E\cap F)}^{\prime }$
$\Rightarrow P(E\cap F)=1-0.25$
$\Rightarrow P(E\cap F)=0.75\ne 0$
$\Rightarrow E\cap F\ne \varphi \Rightarrow$ Thus $E$ and $F$ are not mutually exclusive.