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 .