Conductance-elongation curves of gold atomic wires are measured using a scanning tunneling microscope break junction technique at room temperature. Landauer's conductance plateaus are individually identified and statistically analyzed. Both the probabilities to observe and the lengths of the two last plateaus (at conductance values close to 2e(2)/h and 4e(2)/h) are studied. All results converge to show that the occurrences of these two conductance plateaus on a conductance-elongation curve are statistically independent events.