A one-dimensional packing approach is used to obtain limiting results for inter-crack distances after multiple fracture of a long brittle-matrix composite with continuous aligned fibres. The results may also be appropriate for applications of the Rényi car-parking model in which there is a reduced probability of cars parking bumper to bumper.

We showed in [2] that if an object of initial size x (x large) is subjected to a succession of random partitions, then the object is decomposed into a large number of terminal cells, each of relatively small size, where if Z(x, B) denotes the number of such cells whose sizes are points in the set B, then there exists c, (0 < ≦ 1), such that Z(x, B)x−c converges in probability, as x → ∞, to a random variable W. We show here that if a parent object of size x produces k offspring of sizes y1, y2, ···, yk and if for each k x - y1 - y2 - ··· - yk (the ‘waste’ or the ‘cover’, depending on the point of view) is relatively small, then for each n the nth cumulant, Ψn (x, B), of Z(x, B) satisfies Ψn (x, B)x-c → κn (B), as x → ∞, for some κn(B). Thus, writing N = xc, Z(x, B) has approximately the same distribution as the sum of N independent and identically distributed random variables (The determination of the distribution of the individual appears to be a difficult problem.) The theory also applies when an object of moderate size is broken down into very fine particles or granules.

The Palasti conjecture on the asymptotic mean proportion of coverage is verified for the sequential random packing of rectangular cars with sides parallel to rectangular boundaries in the models of Rényi and Solomon. The extension to n dimensions is given. An extension to a random car size model is indicated.

Recent research on topics related to geometrical probability is reviewed. The survey includes articles on random points, lines, line-segments and flats in Euclidean spaces, the random division of space, coverage, packing, random sets, stereology and probabilistic aspects of integral geometry.

At each stage of the packing of a closed interval K, a random number of random open intervals (the packing objects) are placed in that part of K which is as yet unoccupied. No overlapping between the packing objects is allowed. The packing prescription is such that the packing process terminates after at most a finite number of stages. Attention is focused on the final configuration, K = K– + G, where G is a random open subset of K, and is that part of K which is eventually occupied by packing objects, while K–, a random closed subset of K, is that part of K which remains unoccupied.

Recent research on topics related to geometrical probability is reviewed. The survey includes articles on random points, lines, flats, and networks in Euclidean spaces, pattern recognition, random coverage and packing, random search, stereology, and probabilistic aspects of integral geometry.