The probability of events may be useful in determining the hope  or fear of people affected by their occurrence. The word expectation has various interpretations: in general it expresses the advantage to anyone who expects any benefit whatsoever, under assumptions that are only probable. This advantage, in the theory of chances, is the product of the sum expected and the probability of getting it; it is the partial sum that ought to be paid out when one does not want to rim the risks of the event, supposing that the total sum is to be shared out in proportion to the probabilities. This allocation is the only fair one when all irrelevant circumstances are set aside, because an equal degree of probability gives an equal claim to the sum expected. We shall call this advantage mathematical expectation .
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