Donal Neilson and Peter Sutcliffe where apprehended as a result of chance encounters after months of investigation had made little headway. Vehicle stops, and checks made on the occupants are also a rich sources of crime detections. Stop and search of people on the street is also a very useful tool in detecting crime. All of these methods could be classed as having an element of chance about them, so luck can't be ruled out.
However, I think many more crimes are detected as a result of investigation. Usually the starting point is some form of evidence left at the scene which gives away the identity of the perpetrator. The police have a very good intelligence database, some people would argue too good, on people. Some of whom have no criminal record, so that is controversial. This mountain of information coupled with clues to the identity of the suspect leads to many more detections than chance alone.
With shrinking numbers the police need to work more efficiently. CCTV, comprehensive intelligence databases, internet monitoring, mobile telephone tracking and stop search powers all help in this regard.