Failure probability and average strength of disordered systems

Using a new recurrence-relation method, we have calculated the failure probability and average strength of random systems of up to linear dimension L=5000. We find a 2 deep minimum in the failure probability at an optimal sample size (${\mathit{L}}_{0)}$. As the applied stress decreases the depth of this minimum grows exponentially and ${\mathit{L}}_0$ increases algebraically. At large sample sizes the average strength exhibits a logarithmic size effect, in contrast to recent suggestions of algebraic scaling in related models.