Question

Assertion :$P\left(H/E\right)>P\left(E/H_i\right)P\left(H_i\right),i=1,2,3,...,n$.Let $H_1,H_2,H_3,.....H_n$ be n mutually exclusive & exhaustive events with probability $P\left(H_i\right)>0,i=1,2,3,...n$. Let E be any other event with $0<P\left(E\right)<1$ Reason: ${\sum }_{i=1}^{n}P\left(H_i\right)=1$

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
• B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
• C. Assertion is true but Reason is false
• D. Assertion is false but Reason is true

Answer 1

The correct option is B Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion

$P\left(H_i/E\right)=\frac{P(H_i\cap E)}{P\left(E\right)}$ ...............( i )
$P\left(E/H_i\right)=\frac{P(E\cap H_i)}{P\left(H_i\right)}$ ...............( i i )
From (i) & (ii), we have
$\frac{P\left(H_i/E\right)}{P\left(E/H_i\right)}=\frac{P\left(H_i\right)}{P\left(E\right)}$
$\Rightarrow P\left(H_i/E\right)>P\left(E/H_i\right)\cdot P\left(H_i\right)$
Hence Assertion (A) & Reason(R) both are individually correct but Reason (R) is not correct explanation of Assertion (A).