Let A and B be two non-null events such that

Question:

Let $A$ and $B$ be two non-null events such that $A \subset B$. Then, which of the following statements is always correct?

  1. (1) $\mathrm{P}(\mathrm{A} \mid \mathrm{B})=\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A})$

  2. (2) $\mathrm{P}(\mathrm{A} \mid \mathrm{B}) \geq \mathrm{P}(\mathrm{A})$

  3. (3) $\mathrm{P}(\mathrm{A} \mid \mathrm{B}) \leq \mathrm{P}(\mathrm{A})$

  4. (4) $\mathrm{P}(\mathrm{A} \mid \mathrm{B})=1$

Correct Option: , 2

Solution:

$\because A \subset B ;$ so $A \cap B=A$

Now, $P\left(\frac{A}{B}\right)=\frac{P(A \cap B)}{P(B)}$

$\Rightarrow P\left(\frac{A}{B}\right)=\frac{P(A)}{P(B)}$

$\because P(B) \leq 1$

$\Rightarrow P\left(\frac{A}{B}\right) \geq P(A)$

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