For any two events A and B in a sample space

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14.Probability

hard

For any two events $A$ and $B$ in a sample space

A

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

B

$P\,(A \cap \bar B) = P(A) - P(A \cap B)$ does not hold

C

$P\,(A \cup B) = 1 - P(\bar A)\,P(\bar B),$ if $A$ and $B$ are disjoint

D

None of these

(IIT-1991)

Solution

(a) We know that $P(A/B) = \frac{{P(A \cap B)}}{{P(B)}}$

Also we know that $P(A \cup B) \le 1$

$ \Rightarrow P(A) + P(B) – P(A \cap B) \le 1$

$ \Rightarrow P(A \cap B) \ge P(A) + P(B) – 1$

$ \Rightarrow \frac{{P(A \cap B)}}{{P(B)}} \ge \frac{{P(A) + P(B) – 1}}{{P(B)}}$

$ \Rightarrow P(A/B) \ge \frac{{P(A) + P(B) – 1}}{{P(B)}}$

Standard 11

Mathematics

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