A priori probability

A priori probability is a probability that is derived purely by deductive reasoning. One way of deriving a priori probabilities is the principle of indifference, which has the character of saying that, if there are N mutually exclusive and collectively exhaustive events and if they are equally likely, then the probability of a given event occurring is 1/N. Similarly the probability of one of a given collection of K events is K / N.The a priori probability has an important application in statistical mechanics. The classical version is defined as the ratio of the number of elementary events (e.g. the number of times a die is thrown) to the total number of events - and these considered purely deductively, i.e. without any experimenting. In the case of the die if we look at it on the table without throwing it, each elementary event is reasoned deductively to have the same probability — thus the probability of each outcome of an imaginary throwing of the (perfect) die or simply by counting the number of faces is 1/6. Each face of the die appears with equal probability — probability being a measure defined for each elementary event. The result is different if we throw the die twenty times and ask how many times (out of 20) the number 6 appears on the upper face. In this case time comes into play and we have a different type of probability depending on time or the number of times the die is thrown. On the other hand, the a priori probability is independent of time - you can look at the die on the table as long as you like without touching it and you deduce the probability for the number 6 to appear on the upper face is 1/6. The following example illustrates the a priori probability (or a priori weighting) in (a) classical and (b) quantal contexts.In statistical mechanics (see any book) one derives the so-called distribution functions f {displaystyle f}   for various statistics. In the case of Fermi–Dirac statistics and Bose–Einstein statistics these functions are respectively