Birds on a Wire

This problem was inspired by Birds on a Wire.

If the mean density of birds is , then the probability distribution of interval lengths is . Integrating this, the probability that an interval is less than is . Squaring that gives the probability that the two intervals on either side of a chosen interval are both smaller than the chosen interval, in which case it would not be painted. Thus, the probability that an interval of length will be painted is . Thus, the probability that a link will be painted is . Likewise, the average length of an interval weighted by the probability that it is painted is . Dividing by the average length of an interval λ gives the fraction of painted wire as .