Top panels in each row: magnitude distribution of the real VLA-COSMOS 3 GHz counterparts (solid black line), and the simulated background (red dashed line) for the COSMOS2015 (top row), i band (middle row), and IRAC (bottom row) catalogs. The range of allowed separations between the radio sources and their counterparts is indicated above each panel (see text for details). Bottom panels in each row: ratio between the background and real magnitude distributions, defined as the false match probability (p_false), as a function of magnitude. The dotted horizontal line shows the p_false threshold of 0.2 beyond which counterparts are discarded, while the dotted vertical line shows the effective magnitude cut corresponding to p_false = 0.2. Poisson errors are also indicated.