In the classical theory of differential games, the duration of a game is, as a rule, specified and is a deterministic quantity. Nevertheless, mathematical models of conflict, which ends at a random time, become more relevant in terms of applications. Differential games with a random ending are an extension of the class of differential games with prescribed duration. In this article a 2-person differential game is considered, in which the distribution functions of random ending moments of the game process are different for players, and these functions are independent random variables of discrete type. In article a special case of a differential game with a random ending moment was investigated , the distribution function has discontinuities of the first kind.