I don't agree that a Jazz bass can sound like a Precision. You can mod or redesign a P bass as much as you like, but nothing else sounds as good as the standard P bass, which is why in reality they have ceased to evolve. They don't need to.
But like any 2 pickup bass, the wide tonal pallet of a Jazz will give you enough rope to hang yourself. The main reason for a bad sounding Jazz is the guy who twiddled the knobs.
Don't over analyse the sound of a Jazz bass. Most of the players listed in the previous posts were almost certainly playing stock basses (except Bobby Vega). They sound good because, amongst other things, their good ears can get great sounds out of their instruments. We have as much of a chance if we work harder at it.