  A particle entering a scattering and absorbing medium executes a random walk
through a sequence of scattering events. The particle ultimately achieves
first-passage, leaving the medium or it is absorbed. The Kubelka-Munk model
describes a flux of particles moving perpendicular to the surface of a
plane-parallel medium. The particle path alternates between the positive
direction into the medium and the negative direction back towards the surface.
Backscattering events from the positive to the negative direction occur at
local maxima or peaks, while backscattering from the negative to the positive
direction occur at local minima or valleys. The probability of a particle
avoiding absorption as it follows its path decreases exponentially with the
path-length \(\lambda\). The reflectance of a semi-infinite slab is therefore
the Laplace transform of the distribution of path-length that ends with a
first-passage out of the medium. In the case of a constant scattering rate the
random walk is a Poisson process. We verify our results with two iterative
calculations, one using the properties of iterated convolution with a symmetric
kernel and the other via direct calculation with an exponential step-length
distribution.
  We present a novel demonstration, based on fluctuation theory of sums of
random variables, that the first-passage probability as a function of the
number of peaks in the alternating path is a step-length distribution-free
combinatoric expression. Counting paths with backscattering on the real
half-line results in the same Catalan number coefficients as Dyck paths on the
whole numbers. Including a separate forward-scattering Poisson process results
in an expression related to counting Motzkin paths. We therefore connect walks
on the real line to discrete path combinatorics.
