Question

If $E_1$ and $E_2$ are two events such that $P\left(E_1\right)=\frac{1}{4},P\left(E_2\right)=\frac{1}{2};P\left(\frac{E_1}{E_2}\right)=\frac{1}{4},$ then choose the correct options.

• A. $E_1$ and $E_2$ are mutually exclusive
• B. $E_1$ & $E_2$ are dependent
• C. $E_1$ and $E_2$ are independent
• D. $E_1$ and $E_2$ are exhaustive

Answer 1

Answer: (C)

$E_1$ and $E_2$ are independent

Since $P\left(\frac{E_1}{E_2}\right)=P\left(E_1\right)$
$\Rightarrow$ $E_1$ and $E_2$ are independent of each other
Also since $P(E_1\cup E_2)=P\left(E_1\right)+P\left(E_2\right)-P\left(E_1\right)|P\left(E_2\right)\ne 1$
Hence events are not exhaustive. Independent events can't be mutually exclusive.
Hence only (i) is correct
Further since $E_1$ & $E_2$ are independent; $E_1$ and $\overline{E_2}$ or $\overline{E_1},$ $E_2$ are $\overline{E_1},\overline{E_2}$ are also independent.
Hence $P\left(\frac{\overline{E_1}}{E_2}\right)=P\left(\overline{E_1}\right)=\frac{3}{4}$ and $P\left(\frac{E_2}{\overline{E_1}}\right)=P\left(E_2\right)=\frac{1}{2}$