When only the last one will do

An unknown positive number of items arrive at independent uniformly distributed times in the interval [0,1] to a selector, whose task is to pick online the last one. We show that under the assumption of an adversary determining the number of items, there exists a game-theoretical equilibrium, in other words the selector and the adversary both possess optimal strategies. The probability of success of the selector with the optimal strategy is estimated numerically to 0.352917000207196.