Essentials of Mathematical Methods in Science and Engineering, 2nd Edition

CHAPTER 18PROBABILITY THEORY AND DISTRIBUTIONS

Probability theory is the science of random events. It has long been known that there are definite regularities among large numbers of random events. In ordinary scientific parlance, certain initial conditions, which can be rather complicated, lead to certain events. For example, if we know the initial position and velocity of a planet, we can be certain of its position and velocity at a later time. In fact, one of the early successes of Newton's theory was its prediction of the solar and lunar eclipses for decades and centuries ahead of time. An event that is definitely going to happen when certain conditions are met is called certain. If there is no set of conditions that could make an event happen, then that event is called impossible. If under certain conditions an event may or may not happen, we call it random. From here, it is clear that the certainty, impossibility, and randomness of an event depends on the set of existing conditions. Randomness could result from a number of reasons. Some of these are the presence of large numbers of interacting parts, insufficient knowledge about the initial conditions, properties of the system, and also the environment. Probability is also a word commonly used in everyday language. Using the available information, we often base our decisions on how probable or improbable we think certain chains of events are going to unravel. The severity of the consequences of our decisions can vary greatly. ...