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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).

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    "Is there a simpler approximation to $W$?" - sure, you can take $W(x)\approx \frac{ex}{1+\left(\frac1{e-1}-\frac1{\sqrt 2}+\frac1{\sqrt{2ex+2}}\right)^{-1}}$ for instance. (The approximation is due to Serge Winitzki.)2012-04-30

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The Lambert W-function can be used to solve this.