A and B are two events such that

Question:

A and B are two events such that P (A) ≠ 0. Find P (B|A), if

(i) $A$ is a subset of $B$ (ii) $A \cap B=\Phi$

Solution:

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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