In this study, we present a probabilistic approach for analysing runnability of a moving paper web with random defects. The paper web is modelled in the case of an open draw as an axially moving elastic plate that has an initial crack of random length. We derive a formula for the optimal velocity, at which the probability of fracture is limited. We study edge and central cracks perpendicular to machine direction (mode I cracks) and oblique central cracks (mixed mode). The crack length is modelled with the Weibull distribution. The effect of changing the value of distribution parameters and the probability of fracture on the optimal velocity is illustrated. It is found that the optimal velocity decreases when the expected value and variance of the crack length increase. The results also show the relation between the effect of edge and central cracks on the optimal velocity. The study is fundamental for rigorous analysis of the paper making process.