T he mechanics of tensile failure of brittle matrix composites, uniaxially reinforced with continuous strong fibers, are investigated. A representative volume element description is used to establish a statistical theory for the strength of such composites. The load carrying capacity of such composite components is limited by a phenomenon called “bundle pullout”; a concentration of fiber failures in a small volume element enables whole clusters of fibers and matrix to be pulled out of the surrounding material due to its relatively low shear strength. The probability of initiating failure by this mechanism is governed by the stochastic distribution of flaws and fiber breaks. A certain number of failures have to occur in one (any) small volume element, the size of which is dependent on the global strain applied. This mechanism explains the dependence of the strength of components on both size and shape as well as on the type of loading. The theory presented is used to calculate the strength of a certain material in tension and in both 3-point and 4-point bending of two different cross-sections. The results agree very well in both level and range with reported experiments.