If A and B are two events such that P,(A cup B) = P,(A cap B), then true relation is

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14.Probability

medium

If $A$ and $B$ are two events such that $P\,(A \cup B) = P\,(A \cap B),$ then the true relation is

A

$P\,(A) + P\,(B) = 0$

B

$P\,(A) + P\,(B) = P\,(A)\,P\,\left( {\frac{B}{A}} \right)$

C

$P\,(A) + P\,(B) = 2\,P\,(A)\,P\,\left( {\frac{B}{A}} \right)$

D

None of these

(IIT-1998)

Solution

(c) $P(A \cup B) = P(A) + P(B) – P(A \cap B)$

$ \Rightarrow P(A \cap B) = P(A) + P(B) – P(A \cap B)$

{$\because (P \cap B ) = P( \cup B) $}

$ \Rightarrow 2P(A \cap B) = P(A) + P(B)$

$ \Rightarrow 2P(A)\,.\,\frac{{P(A \cap B)}}{{P(A)}} = P(A) + P(B)$

$ \Rightarrow 2P(A).P\left( {\frac{B}{A}} \right) = P(A) + P(B)$.

Standard 11

Mathematics

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