The Johnson–Nyquist formula for spontaneous voltage fluctuations across a conductor is rederived for metals at a tutorial level within the framework of the free‐electron approximation. The spontaneous current fluctuations are linked directly to the occupancy fluctuations in the Fermi–Dirac distribution function for the conduction electrons and to the electron‐scattering processes that are responsible for the electrical resistivity. It is also shown that an externally applied Ohmic current through a metal does not affect its thermal noise. The quantum limit of thermal noise is discussed briefly.