I'm trying to approximate the number of lottery cards where the probability of the jackpot according to the birthday paradox is not lower than the number of cards divided by the number of all probable lottery variations:
$1 - e^\frac{-n^2}{2M} \ge \frac{n}{M}$,
where $n$ is the number of lottery cards and $M$ is the number of all probable variations. Now reordering this inequality gives:
$\frac{n^2}{2M} - \frac{n^2}{2M}e^\frac{-n^2}{2M} \le \frac{n^3}{2M}$
$\frac{-n^2}{2M}e^\frac{-n^2}{2M} \le \frac{n^2(1-n)}{2M}$
$\frac{-n^2}{2M} \ge W(\frac{n^2(1-n)}{2M})$
Besides not being sure about the direction of the inequality in the last line, I have no clue how to go on with the solution. The number of $M$ is known, $n$ needs to be expressed somehow. I only need hints and keywords on how to move on.