A circle covering problem and DNA breakage (Q912462)

Points are put at random on a circle of length one and marked with 0 or 1 by independent Bernoulli trials. The random variable N is considered which is defined as the random number of points needed for getting a pair of differently marked points on the circle the distance between them being less than some critical distance d. Using a combinatorical argument, a simple proof for an explicit formula for the probabilities \(P(N>n)\) is given. In the case when the ratio of 2d to the circumference of the circle is small, an approximation of \(P(N>n)\) is given, including an upper bound of the error, where a Poisson approximation result for U- statistics is used.