Schemes which have been proposed for this purpose differ radically in many respects. One question is how to express and represent the force of the claim of a particular expectation to be taken seriously, what we shall call the standing of an expectation. The methods of such expression fall first into two classes. On one hand we have an analogue of statistical probability. If some operation, defined by setting bounds to the variability of certain circumstances in which it shall take place, is performed repeatedly, and if the distinct results which this operation can have are exhaustively divided under a fixed list of headings, we may be able to discover approximately in what proportion of the total number of performances the result has fallen under this or that heading. Each heading may be called a contingency,. and the ratio of its occurrences to any total of performances is its probability. If each heading can be assigned a value, this value can be multiplied by the probability and the products of these multiplications for all the different headings can be added together. The total is ·called the mathematical expectation of the value of a series of many repetitions of the operation.