Weird universe

Let's separate a whole universe in a thin shell of thickness $dr$ and radius $r$ . Probability of finding star in one of shell is

$dP=S_{effective}\dot{}n\dot{} 4\pi r^2 dr$

where $S_{effective}=\frac{\pi R^2}{4\pi r^2}$

Now the probability of finding star in first or second,.. or n-th shell is equal to sums of all probabilities in a single shells. So to be sure you have found a star your probability must be equal to $1$ so now suppose that it is going to happen in a shell of radius $r_k$ we have equation:

$\int_{0}^{1} dP=\pi R^2\dot{} n\dot{} \int_{0}^{r_k} dr$

What leads to:

$r_k=\frac{1}{\pi R^2\dot{} n}$