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In each of the following choose the correct answer:
If $A$ and $B$ are events such that $\mathbf{P}(\mathbf{A} \mid \mathbf{B})=\mathbf{P}(\mathbf{B} \mid \mathbf{A})$ then
(a) $A \subset B$ but $A \neq B$ (b) $\quad \mathbf{A}=\mathrm{B}$
(c) $A \cap B=\phi$
(d) $\mathbf{P}(\mathrm{A})=\mathbf{P}(\mathrm{B})$
If $A$ and $B$ are events such that $\mathbf{P}(\mathbf{A} \mid \mathbf{B})=\mathbf{P}(\mathbf{B} \mid \mathbf{A})$ then
(a) $A \subset B$ but $A \neq B$ (b) $\quad \mathbf{A}=\mathrm{B}$
(c) $A \cap B=\phi$
(d) $\mathbf{P}(\mathrm{A})=\mathbf{P}(\mathrm{B})$
Solution:
1412 Upvotes
Verified Answer
$P(A \mid B)=P(B \mid A)$
$$
\Rightarrow \frac{P(A \cap B)}{P(B)}=\frac{P(\mathrm{~B} \cap A)}{P(A)} \Rightarrow \mathrm{P}(\mathrm{A})=\mathrm{P}(\mathrm{B}) .
$$
Thus option (d) is correct.
$$
\Rightarrow \frac{P(A \cap B)}{P(B)}=\frac{P(\mathrm{~B} \cap A)}{P(A)} \Rightarrow \mathrm{P}(\mathrm{A})=\mathrm{P}(\mathrm{B}) .
$$
Thus option (d) is correct.
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