For two given events A and B , P,(A cap B) =

English

Gujarati

Hindi

14.Probability

normal

For two given events $A$ and $B$, $P\,(A \cap B) = $

A

Not less than $P(A) + P\,(B) - 1$

B

Not greater than $P(A) + P(B)$

C

Equal to $P(A) + P(B) - P(A \cup B)$

D

All of the above

(IIT-1988)

Solution

Correct options are $A), B)$ and $C)$

$\because P(A \cup B)=P(A)+P(B)-P(A \cap B)$

on rearranging we get

$P(A \cap B)=P(A)+P(B)-P(A \cup B) \quad$ which is option $C$

Also $\because \quad 0 \leq P ( A \cup B ) \leq 1$

$\Rightarrow P(A)+P(B)-1 \leq P(A \cap B) \leq P(A)+P(B)$

Hence $B$ and $C$ are also correct

Now if $P(A \cap B)=P(A)+P(B)+P(A \cap B)$

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

which is not necessary for the events $A$ and $B$

Hence the correct options are $A, B$ and $C$

Standard 11

Mathematics

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