There are $$n$$ urns, each of these contain $$n+1$$ balls. The ith urn contains $$i$$ white balls and $$(n+1-i)$$ red balls. Let $${u_i}$$ be the event of selecting ith urn, $$i=1,2,3........,n$$ and $$w$$ the event of getting a white ball.

If $$P\left( {{u_i}} \right) \propto i,\,$$ where $$i=1,2,3,.......,n,$$ then $$\mathop {\lim }\limits_{n \to \infty } P\left( w \right) = $$

There are $$n$$ urns, each of these contain $$n+1$$ balls. The ith urn contains $$i$$ white balls and $$(n+1-i)$$ red balls. Let $${u_i}$$ be the event of selecting ith urn, $$i=1,2,3........,n$$ and $$w$$ the event of getting a white ball.

Let $$P\left( {{u_i}} \right) = {1 \over n},$$ if $$n$$ is even and $$E$$ denotes the event of choosing even numbered urn, then the value of $$P\left( {w/E} \right)$$ is

There are $$n$$ urns, each of these contain $$n+1$$ balls. The ith urn contains $$i$$ white balls and $$(n+1-i)$$ red balls. Let $${u_i}$$ be the event of selecting ith urn, $$i=1,2,3........,n$$ and $$w$$ the event of getting a white ball.

If $$P\left( {{u_i}} \right) = c,$$ (a constant) then $$P\left( {{u_n}/w} \right) = $$

Let $$\overrightarrow a = \widehat i + 2\widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k$$ and $$\overrightarrow c = \widehat i + \widehat j - \widehat k.$$ A vector in the plane of $$\overrightarrow a $$ and $$\overrightarrow b $$ whose projection on $$\overrightarrow c $$ is $${1 \over {\sqrt 3 }},$$ is