Water does not run out of a dropper unless its rubber bulb is pressed because of the difference in liquid pressure and atmospheric pressure. There are two forces acting on the water inside the dropper. One is the atmospheric pressure outside the dropper and the other is the liquid pressure inside the bulb. Water is held inside the dropper against atmospheric pressure. When the bulb is pressed, the liquid pressure increases. When it becomes greater than atmospheric pressure, the water will come out of the dropper.