ALMA images are about 2000x2000.  Its resolution is 0.01 arcsec.  It has a planar angle field of view of about 20 arcsec horizontally, and 20 arcsec vertically, for a solid angle field of view of 9.4e-9 steradian, or 7.5e-10 of the entire sky.<p>Mike Brown implies his napkin-math yields 200000 Earth-scale planets in the outer solar system.  If they have uniform density, that would be 15915.5 per steradian.  That would give a probability of 1.5e-4 for such a planet to be in the field of view of any given ALMA image.<p>So ALMA would have to take 20000 non-overlapping images to have a 95% chance of imaging one such planet.  I was unable to find a public source for the total amount of sky ALMA has surveyed over the last 2.5 years, but a mean 22 non-overlapping images per 24 hours at maximum resolution doesn&#x27;t seem impossible.<p>If we assume that ALMA had only a 1% chance of capturing an image of such an object in 20000 tries, that yields a probability of 5.0e-7 per image, 53 planets per sterad, or 672 planets total.<p>With only a 0.001% chance, it&#x27;s 5.0e-10 per image, .053 per sterad, and 0.67 in the entire sky.  That&#x27;s more reasonable, but it also means that ALMA either did not actually see an Earth-sized planet, or that it got <i>extraordinarily lucky</i> this time, as the only planet-sized object out there crossed its field of view once during its 2.5 years in operation.  It would be like a sasquatch photobombing just one of your 20000 vacation pictures.<p>In order to see distant objects better, you have to either narrow your field of view or build a bigger telescope.  As such, we really have no idea how many discrete, dark, cold objects are out there from 50-1000 AU, because no one has surveyed the entire sky with enough detail to resolve those objects in the spectrum viewed by the telescope.<p>For reference, 200000 Earth-sized planets would be approximately equal to the mass of the sun.  The entire Oort cloud is estimated at maybe 5 Earth masses total.<p>So if there are cold Earth-sized planets out there beyond the heliopause, the chances of us detecting any of them visually from Earth are vanishingly small.  They are too cold for infrared imaging, have too little mass and too much distance to affect more visible objects via gravity, and too little volume to detectably occlude a brighter object in the background.