Question

If $A$ and $B$ are mutually exclusive events, then

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

Answer 1

Answer: (A)

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

Answer 2

Since $A$ and $B$ are mutually exclusive events $\therefore A\cap B = \phi$ $\Rightarrow P\left(A\cap B\right) = 0$ $\Rightarrow P\left(A\cup B\right) = P\left(A\right)+P\left(B\right)$ $\Rightarrow P\left(A\right) +P\left(B\right)\le \left(\because P\left(A\cup B\right)\le1\right)$ $\Rightarrow P\left(A\right)\le 1-P\left(B\right)$ $\Rightarrow P\left(A\right) \le P\left(\bar{B}\right)$