Question

If $E_1$ and $E_2$ are two events such that $P\left(E_1\right)=\frac{1}{4},P\left(\frac{E_2}{E_1}\right)=\frac{1}{2}$ and $P\left(\frac{E_1}{E_2}\right)=\frac{1}{4},$ then which of the following is/are correct?

• A. $E_1$ and $E_2$ are mutually independent events
• B. $E_1$ and $E_2$ are exhaustive events
• C. $E_2$ is twice as likely to occur as $E_1$
• D. $P(E_1\cap E_2),P\left(E_1\right),P\left(E_2\right)$ are in G.P

Answer 1

Answer: (A), (C), (D)

The correct options are
A $E_1$ and $E_2$ are mutually independent events
C $E_2$ is twice as likely to occur as $E_1$
D $P(E_1\cap E_2),P\left(E_1\right),P\left(E_2\right)$ are in G.P

$P\left(\frac{E_2}{E_1}\right)=\frac{P(E_1\cap E_2)}{P\left(E_1\right)}\Rightarrow P(E_1\cap E_2)=\frac{1}{8}\Rightarrow P\left(E_2\right)=\frac{\frac{1}{8}}{P\left(\frac{E_1}{E_2}\right)}=\frac{1}{2}P\left(E_1\right)P\left(E_2\right)=\frac{1}{4}\times \frac{1}{2}=P(E_1\cap E_2)\&P(E_1\cup E_2)=\frac{5}{8}$
So $E_1$ & $E_2$ are independent but not exhaustive set of events