A parabola is symmetrical with regard to symmetry axis that goes through its vertex; a branch of a hyperbola is not necessarily symmetrical.
One area where probabilities are widely used nowadays is earthquake predictions (American Geophysical Union 1995 n/p). The probabilities of certain geological events that can lead to an earthquake are estimated for a given area and used to formulate important predictions.
Another field where applications of probabilities are extremely popular is business. Probabilities are used in financial models (Myerson, R 2001 p.1) as well as in supply chain, logistics, marketing forecasts etc.