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

.