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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
