Let $X$ be the random variable given by "number of chips drawn to get a white chip"
We need to compute $p_X(k)$, namely the probability for $X=k$.
Clearly $p_X(1)=\frac{1}{2}$.
Then $p_X(2)=\frac{1}{2}\cdot\frac{1}{3}$.
This is where it gets tricky
$p_X(3)=\frac{1}{3}\cdot\frac{1}{4}$
In general, $p_X(k)=\frac{1}{k(k+1)}$.
So $$E(X)=\sum_{k=1}^\infty kp_X(k)=\sum_{k=1}^\infty\frac{1}{k+1}$$
This diverges since it is a harmonic series.
So the average is infinite.
