Integral equation theory was employed to study continuum percolation and clustering of adhesive hard spheres based on a "connectedness-in-probability'' criterion. This differs from earlier studies in that an "all-or-nothing'' direct connectivity criterion was used. The connectivity probability may be regarded as a "hopping probability'' that describes excitation that passes from one particle to another in complex fluids and dispersions. The connectivity Ornstein-Zernike integral equation was solved for analytically in the Percus-Yevick approximation. Percolation transitions and mean size of particle clusters were obtained as a function of connectivity probability, stickiness parameter, and particle density. It was shown that the pair-connectedness function follows a delay-differential equation which yields analytical expressions in the Percus-Yevick theory.