Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 2 / 9F / c / Solution

Urn $i$ has $2^i$ balls, so a draw from it is white with probability $2^{-i}$. Given that the first ball is white, Bayes theorem gives the posterior probabilityAfter replacement, the second draw is conditionally independent of the first given the urn. HenceConsequently $\boxed{p(n)\to 1/3}$.