There are n urns each containing (n+1) balls such that the ith urn contains ‘i’ white balls and (n+1– i) red balls.

Question

Read the following passage and answer the question.

There are n urns each containing (n+1) balls such that the i^th urn contains ‘i’ white balls and (n+1– i) red balls. Let u_i be the event of selecting i^th urn, i = 1, 2, 3, ........n and W denotes the event of getting a white ball.

(i) If P(ui) \(\propto\) i, where i = 1, 2,........n, then \(\lim\limits_{n \to \infty}\) P(W) is 

(a) 1

(b) \(\frac 23\)

(c) \(\frac 14\)

(d) \(\frac 34\)

(ii) If P(u_i) = c, where c is a constant, then P(u_n|W) is

(a) \(\frac 2{n +1}\)

(b) \(\frac 1{n +1}\)

(c) \(\frac n{n +1}\)

(d) \(\frac 12\)

(iii) If n is even and E denotes the event of choosing even numbered urn (p(u_i) = \(\frac 1n\)), then the value of P(W|E) is

(a) \(\frac{n + 2}{2n+1}\)

(b) \(\frac{n + 2}{2(n+1)}\)

(c) \(\frac{n }{n+1}\)

(d) \(\frac{1 }{n+1}\)

Answer 1

(i) (b) \(\frac 23\)

(ii) (a) \(\frac 2{n +1}\) 

(iii) (b) \(\frac{n + 2}{2(n+1)}\)