How can you calculate where an earring or a similar small object, like a safety pin, lands after it falls out of your hand?

Due to your lack of earing, I’m assuming you dropped it on carpet, a better item-concealing agent than tile or hardwood or alike. therefore, some of these tips are geared more toward carpet-floored rooms.
On finding small shiny objects: a flashlight might help because you might catch a lucky glint of light, but good luck.
On finding small metal objects: strong magnet or metal detector, though expect false reading on metal detectors, as there are nails, screws, etc.. everywhere under the floor.
On finding anything else that’s small and dropped: search with your hands; a small golden object, say an earing, may blend in with the carpet, but won’t fool your hands. También, in a few weeks, check the edges of the room and around heavy furniture, as stuff that’s able to roll and/or be accidently kicked around will most likely end up in the “valleys” of the carpet. If you have hot-water heating, the object may have bounced into the little slit on those, which has happened to me before – don’t count on it though.
On finding small objects, you-have-a-thermal-imaging-camera-version – you can use a thermal camera to help locate the object, but obviously thermal cameras aren’t cheap. If you have one than you can find the object by seeing temperature difference.
On finding a cheap or durable small object – vacuum with a cloth on the front, this might be your best bet.
On the unpredictable vanishing nature of small objects: they bounce, and the angle at which they hit the floor, its center of mass, the breeze (which may slightly though notably change the angle of collision), ………….. are variables in its bouncing pattern. Por lo tanto, it’s nearly impossible to predict the item’s location without sound.