# If C and D are two events such that $\dpi{100} C\subset D$ and $\dpi{100} P(D)\neq 0,$ then the correct statement among the following is : Option 1) $P(C/D)< P(C)\;$ Option 2) $\; P(C/D)=\frac{P(D)}{P(C)}\;$ Option 3) $\; P(C/D)=P(C)\;$ Option 4) $\; P(C/D)\geqslant P(C)$

As we learnt in

Conditional Probability -

$\dpi{100} P\left ( \frac{A}{B} \right )= \frac{P\left ( A\cap B \right )}{P\left ( B \right )}$

and

$\dpi{100} P\left ( \frac{B}{A} \right )= \frac{P\left ( A\cap B \right )}{P\left ( A \right )}$

- wherein

where $\dpi{100} P\left ( \frac{A}{B} \right )$ probability of A when B already happened.

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

If C and d are independent events

$P\left ( C\cap D \right ) =P\left ( C \right ).P\left ( D \right )$

other wise if $C\subseteq D$

$P\left ( C\cap D \right ) = P\left ( C \right )$

$\therefore \: P\left ( \frac{C}{D} \right )=\frac{P\left ( C \right )}{P\left ( D \right )}\geqslant P\left ( C \right )$

Option 1)

$P(C/D)< P(C)\;$

Incorrect option

Option 2)

$\; P(C/D)=\frac{P(D)}{P(C)}\;$

Incorrect option

Option 3)

$\; P(C/D)=P(C)\;$

Incorrect option

Option 4)

$\; P(C/D)\geqslant P(C)$

Correct option

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