Question

If $A$ and $B$ are two events of a random experiment such that $P\left(A/B\right)=P\left(B/A\right)\ne 0$, then which of the following is/are correct:

• A. $A\subset B$ but $A\ne B$
• B. $n\left(A\right)=n\left(B\right)$
• C. $A\cap B=\varphi$
• D. $P\left(A\right)=P\left(B\right)$

Answer 1

Answer: (B), (D)

$P\left(A\right)=P\left(B\right)$

Given : $P\left(A/B\right)=P\left(B/A\right)$
Now, we know that by definition of conditional probability,
$P\left(A/B\right)=\frac{P(A\cap B)}{P\left(B\right)}$
$P\left(B/A\right)=\frac{P(A\cap B)}{P\left(A\right)}$
We have
$P\left(A/B\right)=P\left(B/A\right)$
$\Rightarrow \frac{P(A\cap B)}{P\left(B\right)}=\frac{P(A\cap B)}{P\left(A\right)}$
$\Rightarrow P\left(A\right)=P\left(B\right)$
$\Rightarrow n\left(A\right)=n\left(B\right)$