Probability 4 Question 14
14. If $P\left(u _i\right) \propto i$, where $i=1,2,3, \ldots, n$, then $\lim _{n \rightarrow \infty} P(W)$ is equal to
(a) 1
(b) $\frac{2}{3}$
(c) $\frac{1}{4}$
(d) $\frac{3}{4}$
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Answer:
Correct Answer: 14. (b)
Solution:
- Here, $P\left(u _i\right)=k i, \Sigma P\left(u _i\right)=1$
$$ \begin{aligned} \Rightarrow \quad k & =\frac{2}{n(n+1)} \\ \lim _{n \rightarrow \infty} P(W) & =\lim _{n \rightarrow \infty} \sum _{i=1}^{n} \frac{2 i^{2}}{n(n+1)^{2}} \\ & =\lim _{n \rightarrow \infty} \frac{2 n(n+1)(2 n+1)}{6 n(n+1)^{2}}=2 / 3 \end{aligned} $$
