File:EB1911 Probability - line meets contour.jpg

English: If a random line meet a closed convex contour of length L, the chance of it meeting another such contour, external to the former, is p = (X − Y)/L, where X is the length of an endless band enveloping both contours, and crossing between them, and Y that of a band also enveloping both, but not crossing. This may be shown by means of Legendre's integral above; or as follows:— Call, for shortness, N(A) the number of lines meeting an area A; N(A, A′) the number which meet both A and A′; then (see figure) N(SROQPH) + N(S′Q′OR′P′H′) = N(SROQPH + S′Q′OR′P′H′) + N(SROQPH, S′Q′OR′P′H′), since in the first member each line meeting both areas is counted twice. But the number of lines meeting the non-convex figure consisting of OQPHSR and OQ′S′H′P′R′ is equal to the band Y, and the number meeting both these areas is identical with that of those meeting the given areas Ω, Ω′; hence X = Y + N(Ω, Ω′). Thus the number meeting both the given areas is measured by X − Y. Hence the theorem follows.