So the problem is in Tanis & Hogg, Probability and Statistical inference section 1.5 Independence of Events.
An Urn contains five balls, one marked WIN and four marked LOSE. You and another player take turns selecting a ball at random from the urn, one at a time. The first person to select the WIN ball is the winner. If you draw first, find the probability that you will win if the sampling is done with replacement.
I can't see why the probability isn't $\displaystyle\frac{1}{5}$. The book states it is $\displaystyle\sum_{k=0}^\infty(\displaystyle\frac{1}{5}\times\displaystyle\frac{4}{5}2k)$
Thanks!