Trying to solve this question: Probability of ball ownership I got at an expression for the solution, P:
$$\frac{P}{M} = (1 - \frac{1}{M+N})^{N(1-\frac{P}{M})+M}$$
Where M, N are parameters.
The problem is, I got P both in the left side and in the exponent in the right side, and I have no idea how to simplify it.
My goal is to have a simple approximation formula for P, as a function of M and N.
(I already solved it for several values of M, N using numeric methods, but I want a simple formula).
The Lambert W-function can be used to solve this.