When to stop drawing from 2 white and 3 black

An urn holds $2$ white and $3$ black balls. Jesse draws balls one by one without replacement and may stop whenever he likes (including before the first draw). Each white ball drawn earns him $\text{\char36}1$ and each black ball drawn costs him $\text{\char36}1$. Under the best stopping rule, what is his expected net result?

A gain of $\tfrac15$ dollars. Let $V(w,b)$ be the value with $w$ white and $b$ black balls left: $V=\max\{0,\ \tfrac{w}{w+b}(1+V(w-1,b))+\tfrac{b}{w+b}(V(w,b-1)-1)\}$. Working up from small urns ($V(1,1)=\tfrac12$, $V(1,2)=0$, $V(2,2)=\tfrac23$, $V(2,1)=\tfrac43$) gives $V(2,3)=\tfrac15$, so drawing is worthwhile.