Top: schematic diagram of the Kepler-139 system. The circles representing the planets are spaced logarithmically according to the planets’ semimajor axes, with radii proportional to ﹩\sqrt{{R}_{p}}﹩ (with R_p estimated for planets f and e based on the mass–radius relation of J. Chen & D. Kipping 2017). Bottom left: secular (Laplace–Lagrange) evolution of the Kepler-139 planets’ orbital inclinations, assuming the outer giant is inclined by 3° with respect to the inner system. The gray time ranges are when planets d, b, and c have sufficiently low mutual inclinations to all be observable as transiting planets from a single line of sight (i.e., P(b, c, d transit)  >  0). Bottom right: the time-averaged transit probability for planet f, given that planets d, b, and c transit, as a function of the outer giant’s initial inclination. The dynamical influence of an inclined outer giant lowers the transit probability of planet f by a factor of a few. Dashed lines indicate the transit probability for planet f in the absence of an outer giant.