If $C$ and $D$ are two events such that $C \subset D$ and $P(D) \neq 0$, then the correct statement among…

If $C$ and $D$ are two events such that $C \subset D$ and $P(D) \neq 0$, then the correct statement among the following is
  1. $P(C \mid D) \geq P(C)$
  2. $P(C \mid D) < P(C)$
  3. $P(C \mid D)=\frac{P(D)}{P(C)}$
  4. $P(C \mid D)=P(C)$

Solution

$\mathrm{C} \cap \mathrm{D}=\mathrm{C} \Rightarrow \mathrm{P}(\mathrm{C} \cap \mathrm{D})=\mathrm{P}(\mathrm{C}) \Rightarrow \mathrm{P}\left(\frac{\mathrm{C}}{\mathrm{D}}\right)=\frac{\mathrm{P}(\mathrm{C} \cap \mathrm{D})}{\mathrm{P}(\mathrm{D})} \geq \mathrm{P}(\mathrm{C})$

Asked in: JEE Main 2011

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