There is a famous experiment illustrating the consequences of
the wave-description for the probability to find an electron at
a given point in space. If an electron-source is placed in front
of a metal-sheet with two narrow slits, a recording-device
behind this screen would detect a distribution of incoming
electrons that is identical to an interference-pattern of waves
passing through the double slit (as would be seen when shining
coherent, monochromatic light (a laser produces this kind of
light) on the two slits). This interference-pattern shows up no
matter how low the rate of electrons coming through. Even if
only one single electron per hour would pass through the slits,
there would, finally, be a distribution of detected electrons
given by the well-known interference-pattern. This experiment
clearly demonstrates the wave-characteristics of the electron -
or more precisely, it demonstrates that the probability to find
an electron at a given point in space and time is given by a
wave-function. It is this