A total of \(N\) points are dropped at random and independently of one another into a sphere of radius \(R\).

(a) What is the probability that the distance from the centre to the nearest point will be at least \(r\) ?

(b) What does the probability found in

(a) approach if \(R \rightarrow \infty\) and \(\frac{N}{R^{3}} \rightarrow \frac{4}{3} \pi \lambda\) ?

The problem is taken from stellar astronomy: in the vicinity of the sun, \(\lambda \approx 0.0063\) if \(R\) is measured in parsecs.