Question

Let $E^C$ denote the complement of an event $E$. Let$E_1,E_2$ and $E_3$ be any pair wise independent events with $P\left(E_1\right)>0$ and $P\left(E_1\cap E_2\cap E_3\right)=0$. Then $P\left(E_2^C\cap E_3^C/E_1\right)$ is equal to:

• A. $P\left(E_3^C\right)-P\left(E_2^C\right)$
• B. $P\left(E_3\right)-P\left(E_2^C\right)$
• C. $P\left(E_3^C\right)-P\left(E_2\right)$
• D. $P\left(E_2^C\right)+P\left(E_3\right)$

Answer 1

The correct option is C

$P\left(E_3^C\right)-P\left(E_2\right)$

Explanation for correct option:

Finding the value of $P\left(E_2^C\cap E_3^C/E_1\right)$:

Given the equation is,

$\begin{array}{rcl}P\left(\frac{E_2^C\cap E_3^C}{E_1}\right)& =& \frac{P\left(E_2^C\cap E_3^C\cap E_1\right)}{P\left(E_1\right)}\\ & =& \frac{P\left(E_1\right)-P\left(E_1\cap E_2\right)-P\left(E_1\cap E_3\right)+P\left(E_1\cap E_2\cap E_3\right)}{P\left(E_1\right)}\\ & =& \frac{P\left(E_1\right)-P\left(E_1\right)·P\left(E_2\right)-P\left(E_1\right)·P\left(E_3\right)+0}{P\left(E_1\right)}\\ & =& 1-P\left(E_2\right)-P\left(E_3\right)\\ & =& P\left(E_3^C\right)-P\left(E_2\right)\left[\because P\left(E_3\right)+P\left(E_3^C\right)=1\right]\end{array}$

Since,

$P\left(\frac{E_2^C\cap E_3^C}{E_1}\right)=P\left(E_3^C\right)-P\left(E_2\right)$

Hence, the correct answer is option (C).