Events $E$ and $F$ are such that $P ( $ not $E$ not $F )=0.25,$ State whether $E$ and $F$ are mutually exclusive.
Events $E$ and $F$ are such that $P ( $ not $E$ not $F )=0.25,$ State whether $E$ and $F$ are mutually exclusive.
It is given that $P$ (not $E$ or not $F$ ) $=0.25$
i.e., $P \left( E ^{\prime} \cap F ^{\prime}\right)=0.25$
$\Rightarrow P ( E \cap F )^{\prime} =0.25$ $[ E^{\prime} \cup F^{\prime} =( E \cap F )^{\prime}]$
Now, $P ( E \cap F )=1- P ( E \cap F )^{\prime}$
$\Rightarrow P ( E \cap F )=1-0.25$
$\Rightarrow P ( E \cap F )=0.75 \neq 0$
$\Rightarrow E \cap F \neq \phi$
Thus, $E$ and $F$ are not mutually exclusive.
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