Question

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

• A. Assertion is True, Reason is true; Statement-2 is a correct explanation for Statement-1
• B. Assertion is True, Reason is True; Statement-2 is NOT a correct explanation for Statement-1
• C. Assertion is True, Reason is False
• D. Assertion is False, Reason is True

Answer 1

The correct option is D Assertion is False, Reason is True

Case I
If $P(E\cap H_i)=0$ then $P\left(\frac{H_i}{E}\right)=P\left(\frac{E}{H_i}\right)=0$.
If $P(E\cap H_i)\ne 0$ we get
$P\left(\frac{E}{H_i}\right)$
$=\frac{P(E\cap H_i)}{P\left(H_i\right)}$

Or
$P\left(\frac{E}{H_i}\right).P\left(H_i\right)=P(E\cap H_i)$ ...(i)
And
$P\left(\frac{H_i}{E}\right)=\frac{P(E\cap H_i)}{P\left(E\right)}$
$=\frac{P\left(\frac{E}{H_i}\right).P\left(H_i\right)}{P\left(E\right)}$

Or
$P\left(\frac{H_i}{E}\right).P\left(E\right)=P\left(\frac{E}{H_i}\right).P\left(H_i\right)$ ...(ii)
Now
$P\left(\frac{H_i}{E}\right)>P\left(\frac{E}{H_i}\right).P\left(H_i\right)$
$P\left(\frac{H_i}{E}\right)>P\left(\frac{H_i}{E}\right).P\left(E\right)$
Or
$1>P\left(E\right)$ which is true.
However if we consider case I, we get that the inequality may not always be true.

The reason comes from the definition of probability that is always true.
Hence assertion is not always true.