4.2.8 problem 33

Suppose $w=r=1$. The white ball is equally likely to be any of the $w+b$ balls. Also, note that the event $k$-th drawn ball is the white ball is equivalent to the event $k-1$ black balls are drawn until the white ball is drawn. Thus, for $X\sim \text{NHGeom}(1,n-1,1)$, $P(X=k)=P(k+1\text{-th drawn ball is white})=\frac{1}{n}$ for $0\le k\le n-1$.

$P(X=k)=\frac{\left(\frac{1+k-1}{1-1}\right)\left(\frac{1+n-1-1-k}{1-1}\right)}{\left(\frac{1+n-1}{1}\right)}=\frac{1}{n}$