# Let $E^{c}$ denote the complement of an event $E, E_{1},E_{2},E_{3}$ be any pair wise independent events with $P(E) > 0$ and  $P\left ( E_{1}\cap E_{2}\cap E_{3} \right )=0$ then $P\left ( \frac{E_{2}\cap E_{3}}{E_{1}} \right )$ equal to: Option: 1 Option: 2 Option: 3 Option: 4

$P(\bar{E_2} \cap \bar{E_3} / E_1)=\frac{P(\bar{E_2} \cap \bar{E_3} \cap E_1)}{P(E_1)}=\frac{P(\bar{E_2}) P(\bar{E_2}) P(E_1)}{P(E_1)}=P(\bar{E_2}) P(\bar{E_3})$

$=P(\bar{E_2})\{1-P(E_3)\}=P(\bar{E_2})-P(\bar{E_2}) P(E_3)$

$=P(\bar{E_2})-\{1-P(E_2)\} P(E_3)=P(\bar{E_2})-P(E_3)+P(E_2) P(E_3)$

now given that $E, E_{1},E_{2},E_{3}$ are independent and $P(E_1 \cap E_2 \cap E_3)=0$

$\Rightarrow P(E_1) P(E_2) P(E_3)=0$ but $P(E_1) \neq 0$

$\text {hence,} \quad P(\bar{E_2} \cap \bar{E_3} / E_1)=P(\bar{E_2})-P(E_3)$

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