Given a population ofNunits, it is required to draw a sample ofndistinct units in such a way that the probability πi, (i= 1, …,N) for theith unit to be in the sample is proportional to its ‘size’xi.One way to achieve this is as follows: TheNunits in the population are listed in a random order and theirxiare cumulated and a systematic selection ofnelements from a “random start” is then made on the cumulation. The mathematical theory associated with this procedure has been presented in [1], where with the help of asymptotic theory, compact expressions for the variance of the estimate of the population total are derived. These expressions contain probabilitiesPii′ (i≠i′i, i′ = 1, …,N) that unitsiandi′ both are in the sample. Forn= 2,N= 3 andn= 2,N= 4, exact formulae are derived in [1], but for othernandNonly approximate formulae are obtained. It is the purpose of this paper to derive an exact formula forPii′ for any values ofnandN.The argument makes use of the solution given in [1] for the casen= 2 andN= 4. The new formula forn= 2 is derived in Section 1 and illustrated in Section 2 by a numerical example forN= 10. An exact formula for generalnis derived in Section 3 and examplified in Section 4.