Probability

There are \(n\) urns each containing \(n+1\) balls such that the \(i^{\text {th }}\) urn contains \(i\) white balls and \((n+1-i)\) red balls. Let \(u_{i}\) be the event of selecting \(i^{\text {th }}\) urn, \(i =1,2,3 \ldots, n\) and \(w\) denotes the event of getting a white ball.

If \(\mathrm{P}\left(u_{i}\right) \propto i\), where \(i=1,2,3, \ldots n\), then \(\lim_\limits{n \rightarrow \infty} \mathrm{P}(w)\) is equal to: