Question

If A and B are mutually exclusive events then

• A. $P\left(A\right)\le P\left(\overline{B}\right)$
• B. $P\left(A\right)\ge P\left(\overline{B}\right)$
• C. $P\left(A\right)<P\left(\overline{B}\right)$
• D. None of these

Answer 1

The correct option is A

$P\left(A\right)\le P\left(\overline{B}\right)$

We know that

$P(A\cap B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

$\Rightarrow P(A\cup B)=P\left(A\right)+P\left(B\right)$ [From (1)]

$\Rightarrow P\left(A\right)+P\left(B\right)\le 1$ [P(A \cup B) \leq 1]

$\Rightarrow P\left(A\right)\le 1-P\left(B\right)=P\left(B\right)$

$\therefore P\left(A\right)\le P\left(B\right)$

Hence, the correct answer is option (a).