Let \(P(r)\) be the cumulative probability that no particles are closer than \(r\) to a given particle. Breaking up the interval between zero and \(r\) into small intervals starting ar \(r_{k}=k \Delta r\) with width \(\Delta r\) gives

\[
P(r)=\prod_{k=0}^{r / \Delta r}\left(1-4 \pi n g\left(r_{k} \right) r_{k}^{2} \Delta r \right)
\]

since each factor represents the probability there are no neighbors in interval \(k\). This gives

\[
\ln (P(r)) \approx \sum_{k=0}^{r / \Delta r} \ln \left(1-4 \pi n g\left(r_{k} \right) r_{k}^{2} \Delta r \right) \approx-\sum_{k=0}^{r / \Delta r} 4 \pi n g\left(r_{k} \right) r_{k}^{2} \Delta r
\]

Therefore

\[
P(r)=\exp \left(-4 \pi n \int_{0}^{r} r^{2} g(r) d r \right)
\]

Finally

\[
w(r)=-\frac{d P}{d r}
\]

For an ideal gas \(g(r)=1\), so the integrals are easily evaluated.