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Question:
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Choose the correct answer is each of the following:
If $A$ and $B$ are two events such that $P(A) \neq 0$ and $P(B \mid A)$ $=1$, then
(a) $\mathbf{A} \subset \mathbf{B}$
(b) $\mathbf{B} \subset \mathbf{A}$
(c) $\mathbf{B}=\phi$
(d) $\mathbf{A}=\phi$
If $A$ and $B$ are two events such that $P(A) \neq 0$ and $P(B \mid A)$ $=1$, then
(a) $\mathbf{A} \subset \mathbf{B}$
(b) $\mathbf{B} \subset \mathbf{A}$
(c) $\mathbf{B}=\phi$
(d) $\mathbf{A}=\phi$
Solution:
1220 Upvotes
Verified Answer
$P(B / A)=1 \Rightarrow \frac{P(A \cap B)}{P(A)}=1$
$$
\therefore \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{A})
$$
Where $\mathrm{A} \subset \mathrm{B}, \mathrm{A} \cap \mathrm{B}=\mathrm{A} \quad \therefore \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{A})$
$\therefore$ Option $(a)$ is correct.
$$
\therefore \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{A})
$$
Where $\mathrm{A} \subset \mathrm{B}, \mathrm{A} \cap \mathrm{B}=\mathrm{A} \quad \therefore \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{A})$
$\therefore$ Option $(a)$ is correct.
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