Let A and E be any two events with positive probabilities: Statem… - Vidyadip

MCQ

Let $A$ and $E$ be any two events with positive probabilities:
Statement $- 1$: $P\left( {E/A} \right) \geq P\left( {A/E} \right)P\left( E \right)$
Statement $-2$ : $P\left( {A/E} \right) \geq P\left( {A \cap E} \right)$

✓
Both the statements are true
B
Both the statements are false
C
Statement $-1$ is true, Statement $- 2$ is false
D
Statement $-1$ is false, Statement $-2$ is true

✓

Answer

Correct option: A.

Both the statements are true

a
Let $A$ and $E$ be any two events with positive probabilities.

Consider statement-$1$:

$\mathrm{P}(\mathrm{E} / \mathrm{A}) \geq \mathrm{P}(\mathrm{A} / \mathrm{E}) \mathrm{P}(\mathrm{E})$

$\mathrm{LHS}: \mathrm{P}(\mathrm{E} / \mathrm{A})=\begin{array}{c}{\mathrm{P}(\mathrm{E} \cap \mathrm{A})} \\ {\mathrm{P}(\mathrm{A})}\end{array}$ ......$(1)$

$\mathrm{RHS}: \mathrm{P}(\mathrm{A} / \mathrm{E}) \cdot \mathrm{P}(\mathrm{E})=\frac{\mathrm{P}(\mathrm{E} \cap \mathrm{A})}{\mathrm{P}(\mathrm{E})}-\mathrm{P}(\mathrm{E})$

$=\mathrm{P}(\mathrm{A} \cap \mathrm{E})$ .......$(2)$

Clearly, from $( 1)$ and $( 2)$

we have

$\mathrm{P}(\mathrm{E} / \mathrm{A}) \geq \mathrm{P}(\mathrm{A} \cap \mathrm{E})$

Thus, statement - $1$ is true.

Similarly, statement-$2$ is also true.

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