Question

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

• 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. equal to $P(A) + P(B) + P(A \cup B )$

Answer 1

Answer: (A), (B), (C)

We know that,
$P (A \cap B) = P(A) + P(B) - P(A \cup B)$
Also, $P (A \cup B) \le 1 $
$P ( A \cap B)_{\min}$ , when $P ( A \cup B)_{\max} = 1$
$\Rightarrow P ( A \cap B ) \ge P ( A ) + P ( B ) - 1 $
$\therefore \, $ Option (a) is true.
Again $ P(A \cup B) \ge 0$
$\therefore P(A \cap B)_{\max} ,$ when $ P(A \cup B)_{\min}=0$
$\Rightarrow P(A \cap B) \le \, P(A)+P(B)$
$\therefore $ Option (b) is true
Also, $P(A \cap B)=P(A)+P(B)-P(A \cup B.)$ Thus, (c) is
also correct.
Hence, (a), (b), (c) are correct options .