The current document is directed to gaming dice, including physical dice and virtual dice represented on an electronic display or viewing screen. Certain of the currently disclosed gaming dice have approximately ovoid, prolate spheroid, or ellipsoid shapes that are modified to include planar surfaces. Other of the disclosed gaming dice have angular shapes, including cubes, tetrahedrons, octahedrons, and other regular shapes that are modified to provide nonuniform probabilities of position landings.