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$\quad \mathbf{A}$ and $B$ are two events such that $P(A) \neq 0$. Find $P(B \mid A)$, if
(i) $\mathrm{A}$ is a subset of $\mathrm{B}$ (ii) $\mathrm{A} \cap \mathrm{B}=\phi$
(i) $\mathrm{A}$ is a subset of $\mathrm{B}$ (ii) $\mathrm{A} \cap \mathrm{B}=\phi$
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Verified Answer
(i) $\mathrm{A}$ is a subset of $\mathrm{B} \Rightarrow \mathrm{A} \cap \mathrm{B}=\mathrm{A}$
$$
P(B / A)=\frac{P(B \cap A)}{P(A)}=\frac{P(A)}{P(A)}=1
$$
(ii) $\mathrm{A} \cap \mathrm{B}=\phi \Rightarrow \mathrm{P}(\mathrm{A} \cap \mathrm{B})=0$
$$
P(B / A)=\frac{P(B \cap A)}{P(A)}=\frac{0}{P(A)}=0
$$
$$
P(B / A)=\frac{P(B \cap A)}{P(A)}=\frac{P(A)}{P(A)}=1
$$
(ii) $\mathrm{A} \cap \mathrm{B}=\phi \Rightarrow \mathrm{P}(\mathrm{A} \cap \mathrm{B})=0$
$$
P(B / A)=\frac{P(B \cap A)}{P(A)}=\frac{0}{P(A)}=0
$$
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