Buffon's Needle and the Probability of Intercepting Short‐Distance Trips by Multiple Screen‐Line Surveys

Abstract

For transport planning purposes, information on origin‐destination movements may be obtained by stopping traffic at the roadside and interviewing drivers. The roadside interview stations are usually located so that they lie on one of a number of lines (called screen lines) that cross from one side of the survey area to another. In this way, all movements with the origin on one side of a screen line and the destination on the other are intercepted. For regional surveys, a grid of screen‐lines may be used. The movements intercepted are not, however, representative of all those in the region. That is because shorter‐distance movements are underrepresented; the coarser the grid, the worse the effect. The extent of this underrepresentation is estimated in this paper by calculating the probability of intercepting a trip of given direct length, under the assumption that the screen lines constitute a uniform rectangular grid, and that trips of a given length are distributed over the region at random. The result is an extension to the Buffon needle problem. Ways in which such a result, obtained for an idealized situation, may be extended to apply to more realistic situations, are discussed: in particular it is shown how the trip‐length frequency distribution of all trips may be estimated from that of intercepted trips, using a simple formula that is applicable to a much more general pattern of screen lines.