In this paper a new concept of robustness is introduced and the corresponding optimization problem is stated. This new concept is applied to transportation network designs in which the set of scenarios arising from the uncertainty of the parameters follows a probability distribution. The p-robustness concept is aimed to problems where the feasibility of the solutions is not affected by the uncertainty of the parameters. In order to compare the solution with those of other concepts of robustness already known, some computational experiments with real data are included.