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If $E$ and $F$ are independent events such that $0 \lt P(E) \lt 1$ and $0 \lt P(F) \lt 1$,then
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$P(E \cap F)=P(E) \cdot P(F)$
Now, $P\left(E \cap F^c\right)=P(E)-P(E \cap F)=P(E)[1-P(F)]=P(E) \cdot P\left(F^c\right)$
and $P\left(E^c \cap F^c\right)=1-P(E \cup F)=1-[P(E)+P(F)-P(E \cap F)$
$=[1-P(E)][1-P(F)]=P\left(E^c\right) P\left(F^c\right)$
Also $P(E / F)=P(E)$ and $P\left(E^c / F^c\right)=P\left(E^c\right)$
$\Rightarrow P(E / F)+P\left(E^c / F^c\right)=1$
Now, $P\left(E \cap F^c\right)=P(E)-P(E \cap F)=P(E)[1-P(F)]=P(E) \cdot P\left(F^c\right)$
and $P\left(E^c \cap F^c\right)=1-P(E \cup F)=1-[P(E)+P(F)-P(E \cap F)$
$=[1-P(E)][1-P(F)]=P\left(E^c\right) P\left(F^c\right)$
Also $P(E / F)=P(E)$ and $P\left(E^c / F^c\right)=P\left(E^c\right)$
$\Rightarrow P(E / F)+P\left(E^c / F^c\right)=1$
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