A and B are two events such that P (A) ≠ 0. Find P (B|A), if
(i) $A$ is a subset of $B$ (ii) $A \cap B=\Phi$
It is given that, $P(A) \neq 0$
(i) A is a subset of B.
$\Rightarrow \mathrm{A} \cap \mathrm{B}=\mathrm{A}$
$\therefore \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{B} \cap \mathrm{A})=\mathrm{P}(\mathrm{A})$
$\therefore \mathrm{P}(\mathrm{B} \mid \mathrm{A})=\frac{\mathrm{P}(\mathrm{B} \cap \mathrm{A})}{\mathrm{P}(\mathrm{A})}=\frac{\mathrm{P}(\mathrm{A})}{\mathrm{P}(\mathrm{A})}=1$
(ii) $\mathrm{A} \cap \mathrm{B}=\phi$
$\Rightarrow \mathrm{P}(\mathrm{A} \cap \mathrm{B})=0$
$\therefore \mathrm{P}(\mathrm{B} \mid \mathrm{A})=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{A})}=0$
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