  The phenomenon of molecular binding, where two molecules, referred to as a
receptor and a ligand, bind together to form a ligand-receptor complex, is
ubiquitous in biology and essential for the accurate functioning of all
life-sustaining processes. The probability of a single receptor forming a
complex with any one of L surrounding ligand molecules at thermal equilibrium
can be derived from a partition function obtained from the Gibbs-Boltzmann
distribution. We extend this approach to a system consisting of R receptors and
L ligands to derive the probability density function p(r;R,L) to find r bound
receptor-ligand complexes at thermal equilibrium. This extension allows us to
illustrate two aspects of this problem which are not apparent in the single
receptor problem, namely, a) a symmetry to be expected in the equilibrium
distribution of the number of bound complexes under exchange of R and L and b)
the number of bound complexes obtained from chemical kinetic equations has an
exact correspondence to the maximum probable value of r from the expression for
p(r;R,L). We derive the number fluctuations of r and present a practically
relevant molecular sensing application that benefits from the knowledge of
p(r;R,L).
