# Solve the following

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 $\mathrm{P}\left(\mathrm{E}_{1} \cap \mathrm{E}_{2} \cap \mathrm{E}_{3}\right)=0$. Then $\mathrm{P}\left(\mathrm{E}_{2}^{\mathrm{C}} \cap \mathrm{E}_{3}^{\mathrm{C}} / \mathrm{E}_{1}\right)$ is equal to :

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

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

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

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

Correct Option: , 4

Solution:

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

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

$\left[\because P\left(A \cap B^{C}\right)=P(A)-P(A \cap B)\right]$

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

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

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

$=1-P\left(E_{2}\right)-P\left(E_{3}\right)$ $[\because P(A \cap B)=P(A) \cdot P(B)]$

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