For any two events A and B in a sample space - Vidyadip

MCQ

For any two events A and B in a sample space

✓
$P \left(\frac{ A }{ B }\right) \geq \frac{ P ( A )+ P ( B )-1}{ P ( B )}, P ( B ) \neq 0$ is always true
B
$P ( A \cap \overline{ B })= P ( A )- P ( A \cap B )$ does not hold
C
$P ( A \cup B )=1- P (\overline{ A }) P (\overline{ B })$, if A and B are disjoint
D
None of these

✓

Answer

Correct option: A.

$P \left(\frac{ A }{ B }\right) \geq \frac{ P ( A )+ P ( B )-1}{ P ( B )}, P ( B ) \neq 0$ is always true

(A)
We know that $P ( A / B )=\frac{ P ( A \cap B )}{ P ( B )}$
Also we know that $P ( A \cup B ) \leq 1$
$\Rightarrow P ( A )+ P ( B )- P ( A \cap B ) \leq 1$
$\Rightarrow P ( A \cap B ) \geq P ( A )+ P ( B )-1$
$\Rightarrow \frac{ P ( A \cap B )}{ P ( B )} \geq \frac{ P ( A )+ P ( B )-1}{ P ( B )}$
$\Rightarrow P ( A / B ) \geq \frac{ P ( A )+ P ( B )-1}{ P ( B )}$

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