# Seemingly simple urn stochastic process

This seems like a simple model, but I'm getting a bit stuck on it. Suppose I have an urn with $w$ white balls and $b$ black balls. At each turn, I draw a ball, note its color, and retrieve a ball with the opposite color, paint it the drawn ball's color, and put both back in the urn. (Notice the population is constant). This continues until all balls are the same color.

Here's where I got stuck. Let $b$ denote the number of black balls, $N$ the total number of balls, and $p(b)$ the probability black "fixates" in the urn, given $b$ black balls present.

$$p(b) = \frac{b}{N}p(b+1) + \frac{N-b}{N}p(b-1)$$

such that $p(N) = 1, p(0) = 0$.

However, this doesn't seem well defined because using this expression, $p(b-1)$ is a function of $p(b)$ creating an infinite recursion.