Expected no of Moves to get a Subregion Ai , from k subregions of a land with respective areas.

Question

Given $K$ regions on a land with subregions of areas A1,A2,A3....Ak , a random sub region is given to a farmer one at a time , what is the expected no of moves that any farmer gets the land Ai or say A1.

Answer 1

Let $E_{i}$ be the probability that land $A_{1}$ is allotted in $i^{th}$ move. Observe that $E_{i}$ = $1 \over k$.

This is because we have to allocate $A_{1}$ in $i^{th}$ move, so the first $i-1$ moves should allocate any land other than $A_{1}$.

$$E_{i} = {k-1 \over k} {k-2 \over k-1}..{1 \over k-i+1} = {1 \over k}$$

$$E(E_{i}) = { 1.E_{1} + 2.E_{2}+ .. + k.E_{k}} = {k+1 \over 2}$$