Current-induced magnetic domain wall motion is attractive for manipulating magnetization direction in spintronics devices, which open a new era of electronics. Up to now, in spite of a crucial significance to applications, investigation on a current-induced domain wall depinning probability, especially in sub-nano to a-few-nanosecond range has been lacking. Here we report on the probability of the depinning in perpendicularly magnetized Co/Ni nanowires in this timescale. A high depinning probability was obtained even for 2-ns pulses with a current density of less than 1012 A m-2. A one-dimensional Landau-Lifshitz-Gilbert calculation taking into account thermal fluctuations reproduces well the experimental results. We also calculate the depinning probability as functions of various parameters and found that parameters other than the coercive field do not affect the transition width of the probability. These findings will allow one to design high-speed and reliable magnetic devices based on the domain wall motion.