Question

For any two independent events $E_1$ and $E_2$ in a space $S,P\left[\right(E_1\cup E_2)\cap (E_1\cap E_2\left)\right]$ is equal to

• A. $\le \frac{1}{4}$
• B. $>\frac{1}{4}$
• C. $\ge \frac{1}{2}$
• D. $>\frac{1}{2}$

Answer 1

The correct option is A $\le \frac{1}{4}$

$P\left(E\right)=P[(E_1\cup E_2)\cap (E_1\cap E_2)]$

Using De Morgans law

$(E_1\cap E_2)=(E_1^{\prime }\cup E_2^{\prime })={(E_1\cup E_2)}^{\prime }$

$\Rightarrow P\left(E\right)=P[(E_1\cup E_2)\cap {(E_1\cup E_2)}^{\prime }]$

Now $(A\cap A\prime )$ is always $\varphi$ set

$\Rightarrow P\left(E\right)=\varphi =0$

Out of the options

$\Rightarrow P\left(E\right)\le \frac{1}{4}$

So option $A$ is correct.