Unknown Composition Bag

From a bag containing $n$ balls, all either white or black, all numbers of each being equally likely, a ball is drawn which turns out to be white; this is replaced, and another ball is drawn, which also turns out to be white. If this ball is replaced, prove that the chance of the next draw giving a black ball is $\frac{1}{2}(n-1)(2n+1{)}^{-1}$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XXXII. Probability, Examples XXXII.d, No. 16