QUESTION
For two given events A and B, $P(A\cap B)=$ [IIT 1988]
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

Hint: If A and B are two events, then $A\cap B$is the set of those elements which are common in both A and B.
$A\cap B=\left\{ \omega :\omega \in Aand\omega \in B \right\}$
You can also use the set Venn diagram for solving this question.




Formula Used:$P(A\cap B)=P(A)+P(B)-P(A\cup B)$



Complete step by step solution:Since we know that
$P(A\cap B)=P(A)+P(B)-P(A\cup B)$
And since, $0\le P(A\cup B)\le 1$
$\Rightarrow P(A)+P(B)-1\le P(A\cap B)\le P(A)+P(B)$
$\Rightarrow P(A\cap B)\ge P(A)+P(B)-1$and $P(A\cap B)\le P(A)+P(B)$
Therefore, option A. and option B. is true.
And also,
$P(A\cap B)=P(A)+P(B)-P(A\cup B)$
Therefore, option C. is also true.
Hence,



Option ‘D’ is correct

Note: Please note that when you see ‘all the above’ in the last question then you have to definitely check for each and every option for this correctness. Working with the equations please follow the intervals so that you should not make silly mistakes while getting the answers.