If A and B are two given events, then P(A ∩ B) is

not less than P(A) + P(B) -1 not greater than $P(A) +P(B) - P( A ∪ B)$ equal to  $P(A) +P(B) + P( A ∩ B)$ equal to  $P(A) +P(B) + P( A ∪ B)$

not less than P(A) + P(B) -1

We have, $P(A ∪ B) ≤ 1$ $∴ -P(A ∪ B) ≥ -1$ $⇒ P(A) +P(B) - P(A ∪ B)  ≥ P(A) + P(B) - 1$ $⇒ P(A ∪ B) > P(A) +P(B) -1$ $⇒ P(A ∩ B) $ is not less than $P(A) + P(B) -1.$