What is the small hole on the casing of a bic biro for?

Air pressure does not cause the ink to flow. Gravity does that. That's why you can't write very long with the pen upside down. The weight of the column of ink pushes down and the pen's roller ball transfers the ink to the paper. However, air pressure can PREVENT ink from flowing. So, as the ink inside the pen is emptied out through the tip, the pressure would drop inside the pen, creating a relative vacuum. When the pressure difference is strong enough, the gravity is not strong enough to pull the ink out against the vaccum inside the pen. The hole allows air pressure inside the pen and outside to equalize.