When probed with current pulses, Josephson junctions and superconducting wires exhibit stochastic switching from a superconducting to a stable nonzero-voltage state. The electrical current dependence of the switching probability (the so-called S curve) or the switching-current distribution is a fingerprint of the physics governing the escape process. This work addresses the criterion of the independent switching event, which is important for the credibility of the switching measurements of superconducting wires and various Josephson junctions involving superconductor-insulator-superconductor tunnel junctions, proximity junctions, and Dayem nanobridges. Treating the Josephson junction as an electrical coin with a current-tuned switching probability, we investigate the effect of correlation between switching events on the switching statistics. We show that such a correlation originates from the thermal dynamics of the superconducting wire.