On a hot day, droplets form on the outside of a cold bottle taken from the fridge

Why

Maanasi's guess makes sense at first glance — there IS water inside the bottle, and now there's water outside it, so 'it must have come from in there' looks like the simplest explanation. But look at where the droplets actually appear: spread all over the outside surface, even if the bottle is sealed shut with a cap that never opened. The real source is the air itself — ordinary air always carries some invisible water vapour in it, and on a hot day the air right around a cold bottle gets chilled by the cold glass. Cold air cannot hold as much vapour as warm air, so the extra vapour condenses back into tiny liquid droplets right on the cold surface, the same way your breath fogs a cold window in winter. Test: dry the outside of a cold bottle completely with a cloth and set it down — new droplets reappear within minutes, with no leak anywhere. Her brother is right: those droplets are water vapour from the air condensing on the cold surface, not water escaping from inside.