Let $$\theta \in\left(0, \frac{\pi}{4}\right)$$ and $$t_{1}=(\tan \theta)^{\tan \theta}, t_{2}=(\tan \theta)^{\cot \theta}, t_{3}=(\cot \theta)^{\tan \theta}$$ and $$t_{4}=(\cot \theta)^{\cot \theta}$$, then

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:

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 $$\mathrm{P}\left(u_{i}\right)=c$$, where $$c$$ is a constant then $$\mathrm{P}\left(u_{n} / w\right)$$ is equal to:

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 $$n$$ is even and E denotes the event of choosing even numbered urn $$\left(\mathrm{P}\left(u_{i}\right)=\frac{1}{n}\right)$$, then the value of $$\mathrm{P}(w / \mathrm{E})$$ is :