In the present exposition we are concerned only with the simple Galton-Watson process, initiated by a single ancestor (Harris (1963), Chapter I). Let denote the probability generating function of the offspring distribution of a single individual. Our fundamental assumption, which holds throughout the sequel, is that fj ≠ 1, j = 0,1,2, …; in particular circumstances it shall be necessary to strengthen this to 0 < f0 ≡ F(0) < 1, which is the relevant assumption when extinction behaviour is to be considered. (Even so, our assumptions will always differ slightly from those of Harris (1963), p. 5.)

Suppose that, e.g., in a guessing experiment, the chance of a ‘hit’ at each ‘attempt’ is the constant p, 0 < p < 1. Let q = 1 - p. In a large number n of attempts the ‘average’ number of hits is pn. Let m = (1 + k)pn.

Many problems in geometrical probability have been surveyed in the monograph by Professor M. G. Kendall and myself (Kendall and Moran (1963)). In 1966 I published a note in J. Appl. Frob. 3 describing researches which had either been carried out since that book appeared, or which had escaped our notice. The present paper attempts to describe still further investigations classified roughly as to whether they are concerned with points, lines, planes, estimation of area and length, and coverage.

The theory of storage processes, originally formulated by Moran [1] in 1954, has developed in the past fourteen years into a minor subfield of Applied Probability, closely allied to queueing theory. While dam models with discrete inputs are analogous to queueing processes, the essentially continuous nature of water inflows has distinguished generalized storage processes from queues. Indeed, some of the most complex of storage problems have arisen in the case of continuous flows.

Many problems arising in the physical and social sciences relate to processes which happen sequentially. Such processes are usually investigated by means of the theory of stationary stochastic processes, but there have been some attempts to develop techniques which are not subject to the conceptual difficulties inherent in the probabilistic approach. These difficulties stem from the fact that in practice one is often restricted to a single record which, from the probabilistic point of view, is only one sample from an ensemble of possible records. In some instances such a viewpoint seems artificial, and for some time series it is questionable whether any objective reality corresponds to the idea of an ensemble of possible time series. For example, as noted in Feller (1967), a theory of probability based on a frequency interpretation cannot meaningfully attach a probability to a statement such as “the sun will rise tomorrow”, because to do so one would have to set up a conceptual universe of possible worlds.