Let \(H_1, H_2, \ldots, H_n\) be mutually exclusive events with \(P\left(H_i\right)>0, i=1,2, \ldots, n\). Let \(E\) be any other event with \(0 < P(E) < 1\).
Statement I \(P\left(H_i / E\right)>P\left(E / H_i\right) P\left(H_i\right)\) for \(i=1,2, \ldots, n\).
Statement II \(\sum_{i=1}^n P\left(H_i\right)=1\)

Statement : I If \(P\left(H_i \cap E\right)=0\) for some \(i\), then
\[
P\left(\frac{H_i}{E}\right)=P\left(\frac{E}{H_i}\right)=0
\]
If \(P\left(H_i \cap E\right) \neq 0\) for \(\forall i=1,2, \ldots . n\), then
\[
\begin{aligned}
P\left(\frac{H_i}{E}\right) & =\frac{P\left(H_i \cap E\right)}{P\left(H_i\right)} \times \frac{P\left(H_i\right)}{P(E)} \\
& \left.=\frac{P\left(\frac{E}{H_i}\right) \times P\left(H_i\right)}{P(E)}>P\left(\frac{E}{H_i}\right) \cdot P\left(H_i\right) \quad \text { [as } 0 < P(E) < 1\right]
\end{aligned}
\]
Hence, Statement I may not always be true.
Statement II
Clearly, \(H_1 \cup H_2 \cup \ldots . \cup H_n=S\) (sample space)
\[
\Rightarrow \quad P\left(H_1\right)+P\left(H_2\right)+\ldots+P\left(H_n\right)=1 .
\]
Hence, Statement II is true.