Ah! It's not just a potential probability, but how do we know? The double split experiment! In this experiment the particle's wave splits into two waves as it goes through the two slits and interacts with itself to form an interference pattern. The particle exists physically as a wave spread out along the length of the detector (the universe?) and then "picks" a certain single point to be when asked to be a solid particle, according to the wave's peaks - in a way that is probabilistic. Could it have an underlying mechanism? Yes, however we have yet to discover it - and so it appears that all particles really actually do spend a lot of their time existing as unintuitive waves.
So since the double split experiment exists: The particles must really actually be travelling as real particle waves, exactly as light can travel as a wave (after all, photons are particles too).
Kinda mind blowing that the universe works this way.