Probability 4 Question 6
6. 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 $P\left(H _i / E\right)>P\left(E / H _i\right) \cdot P\left(H _i\right)$ for $\quad i=1,2, \ldots, n$ Statement II $\sum _{i=1}^{n} P\left(H _i\right)=1$
(2007, 3M)
Passage Based Problems
Passage I
Let $n _1$ and $n _2$ be the number of red and black balls, respectively in box I. Let $n _3$ and $n _4$ be the number of red and black balls, respectively in box II.
(2015 Adv.)
Show Answer
Answer:
Correct Answer: 6. (d)
Solution:
- Statement I If $P\left(H _i \cap E\right)=0$ for some $i$, then
$$ P \frac{H _i}{E}=P \frac{E}{H _i}=0 $$
If $P\left(H _i \cap E\right) \neq 0, \forall \quad i=1,2, \ldots, n$, then
$$ \begin{aligned} & P \frac{H _i}{E}=\frac{P\left(H _i \cap E\right)}{P\left(H _i\right)} \times \frac{P\left(H _i\right)}{P(E)} \\ & =\frac{P \frac{E}{H _i} \times P\left(H _i\right)}{P(E)}>P \frac{E}{H _i} \cdot P\left(H _i\right) \quad[\because 0<P(E)<1] \end{aligned} $$
Hence, Statement I may not always be true.
Statement II Clearly, $H _1 \cup H _2 \cup \ldots \cup H _n=S$
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$$ \Rightarrow \quad P\left(H _1\right)+P\left(H _2\right)+\ldots+P\left(H _n\right)=1 $$
Hence, Statement II is ture.
