Question

Let $H_1,H_2,H_3,...,H_n$ be mutually exclusive and exhaustive events with $P\left(H_i\right)>0;i=1,2,3,...,n.$ Let E be any other event with $0<P\left(E\right)<1.$

STATEMENT-1: $P(H_i\mid E)>P(E\mid H_i)\cdot P\left(H_i\right)$ for $i=1,2,3,...,n.$

STATEMENT-2 $\sum _{i=1}^{n}P\left(H_i\right)=1.$

• A. Statement -1 is True, Statement-2 is True; Statement -2 is a correct explanation for Statement -1
• B. Statement -1 is True, Statement -2 is True; Statement -2 is Not a correct explanation for Statement -1
• C. Statement -1 is True, Statement -2 is False
• D. Statement -1 is False, Statement -2 is True

Answer 1

Answer: (B)

Statement -1 is True, Statement -2 is True; Statement -2 is Not a correct explanation for Statement -1

$Statement1:$ If $P(H_i\cap E)=0$ for some $i$, then
$P\left(\frac{H_i}{E}\right)=P\left(\frac{E}{H_i}\right)=0$
If $P(H_i\cap E)\ne 0$ for $\forall i=1,2,...,n$, then
$P\left(\frac{H_i}{E}\right)=\frac{P(H_i\cap E)}{P\left(H_i\right)}\times \frac{P\left(H_i\right)}{P\left(E\right)}$
$=\frac{P\left(\frac{E}{H_i}\right)\times P\left(H_i\right)}{P\left(E\right)}>P\left(\frac{E}{H_i}\right).P\left(H_i\right)$ $[\because 0<P\left(E\right)<1]$
Hence, statement $1$ may not always be true.
$Statement2:$ Clearly $H_1\cup H_2\cup ...\cup H_n=S$ (Sample Space)
$\Rightarrow P\left(H_1\right)+P\left(H_2\right)+...+P\left(H_n\right)=1$