Let E^C denote the complement of an event E.
Question:

Let $E^{C}$ denote the complement of an event $E$. Let $E_{1}, E_{2}$ and $E_{3}$ be any pairwise independent events with $P\left(E_{1}\right)>0$ and $P\left(E_{1} \cap E_{2} \cap E_{3}\right)=0$.

Then $\mathrm{P}\left(\mathrm{E}_{2}^{\mathrm{C}} \cap \mathrm{E}_{3}^{\mathrm{C}} / \mathrm{E}_{1}\right)$ is equal to :

1. $\mathrm{P}\left(\mathrm{E}_{3}^{\mathrm{C}}\right)-\mathrm{P}\left(\mathrm{E}_{2}\right)$

2. $\mathrm{P}\left(\mathrm{E}_{2}^{\mathrm{C}}\right)+\mathrm{P}\left(\mathrm{E}_{3}\right)$

3. $P\left(E_{3}^{C}\right)-P\left(E_{2}^{C}\right)$

4. $\mathrm{P}\left(\mathrm{E}_{3}\right)-\mathrm{P}\left(\mathrm{E}_{2}^{\mathrm{C}}\right)$

Correct Option: 1

Solution:

Given $\mathrm{E}_{1}, \mathrm{E}_{2}, \mathrm{E}_{3}$ are pairwise indepedent events

so $\mathrm{P}\left(\mathrm{E}_{1} \cap \mathrm{E}_{2}\right)=\mathrm{P}\left(\mathrm{E}_{1}\right) \cdot \mathrm{P}\left(\mathrm{E}_{2}\right)$

and $P\left(E_{2} \cap E_{3}\right)=P\left(E_{2}\right) \cdot P\left(E_{3}\right)$

and $\mathrm{P}\left(\mathrm{E}_{3} \cap \mathrm{E}_{1}\right)=\mathrm{P}\left(\mathrm{E}_{3}\right) \cdot \mathrm{P}\left(\mathrm{E}_{1}\right)$

$\& \mathrm{P}\left(\mathrm{E}_{1} \cap \mathrm{E}_{2} \cap \mathrm{E}_{3}\right)=0$

Now $\mathrm{P}\left(\frac{\overline{\mathrm{E}}_{2} \cap \overline{\mathrm{E}}_{3}}{\mathrm{E}_{1}}\right)=\frac{\mathrm{P}\left[\mathrm{E}_{1} \cap\left(\overline{\mathrm{E}}_{2} \cap \overline{\mathrm{E}}_{3}\right)\right]}{\mathrm{P}\left(\mathrm{E}_{1}\right)}$

$=\frac{P\left(E_{1}\right)-\left[P\left(E_{1} \cap E_{2}\right)+P\left(E_{1} \cap E_{3}\right)-P\left(E_{1} \cap E_{2} \cap E_{3}\right)\right]}{P\left(E_{1}\right)}$

$=\frac{P\left(E_{1}\right)-P\left(E_{1}\right) \cdot P\left(E_{2}\right)-P\left(E_{1}\right) P\left(E_{3}\right)-0}{P\left(E_{1}\right)}$

$=1-\mathrm{P}\left(\mathrm{E}_{2}\right)-\mathrm{P}\left(\mathrm{E}_{3}\right)$

$=\left[1-\mathrm{P}\left(\mathrm{E}_{3}\right)\right]-\mathrm{P}\left(\mathrm{E}_{2}\right)$

$=P\left(E_{3}^{\mathrm{c}}\right)-P\left(E_{2}\right)$