Question

For two events $A$and $B$,if $P\left(A\right)=P(A/B)=\frac{1}{4}$ and $P(B/A)=\frac{1}{2}$, then

• A. $A$and $B$ are independent
• B. $P(A\text{'}/B)=\frac{3}{4}$
• C. $P(B\text{'}/A\text{'})=\frac{1}{2}$
• D. All of the above

Answer 1

The correct option is D

All of the above

The explanation for the correct options:

Step1. Check $A$ and $B$ are independent:

We know that

$P\left(\frac{B}{A}\right)=\frac{P(B\cap A)}{P\left(A\right)}=\frac{1}{2}$

$\Rightarrow P(B\cap A)=\frac{1}{8}$

$P\left(\frac{A}{B}\right)=\frac{P(A\cap B)}{P\left(B\right)}=\frac{1}{4}$

$P\left(B\right)=\frac{1}{2}$

$\Rightarrow P(A\cap B)=P\left(A\right).P\left(B\right)=\frac{1}{4}\times \frac{1}{2}=\frac{1}{8}$

So, $A$ and $B$are independent.

Step2. Find the value of$P\left(\frac{A\text{'}}{B}\right)$:

Now,$P\left(\frac{A\text{'}}{B}\right)=\frac{P(A\text{'}\cap B)}{P\left(B\right)}=\frac{P(A\text{'})P\left(B\right)}{P\left(B\right)}$

$=\frac{{\displaystyle \frac{3}{4}}\times {\displaystyle \frac{1}{2}}}{{\displaystyle \frac{1}{2}}}$

$\Rightarrow P\left(\frac{A\text{'}}{B}\right)=\frac{3}{4}$

Step3. Find the value of$P\left(\frac{B\text{'}}{A\text{'}}\right)$:

Now,$P\left(\frac{B\text{'}}{A\text{'}}\right)=\frac{P(B\text{'}\cap A\text{'})}{P(A\text{'})}$

$$\begin{gathered} =\frac{\left[P\right(B\text{'}).P(A\text{'}\left)\right]}{P(A\text{'})} \\ =\frac{1}{2} \end{gathered}$$

Hence, Option (D) is the correct answer.