On the benefits of institutions that depend on the probability of events

Abstract

Let us recall here what has already been said about expectation [1]. We have seen that in order to find the advantage that results from several simple events, some of which produce a gain and others a loss, it is necessary to add the products of the probability of each favourable event and the gain that it procures, and to subtract from ^atheir^a sum that of the products of the probability of each unfavourable event and the attendant loss [2]. But whatever the advantage expressed by the difference between these sums may be, a single event composed of these simple events in no wise guarantees against the fear of experiencing a ^α loss. One might imagine that this fear ought to decrease as the compound event is repeated more and more often. Probabilistic analysis leads to the following general theorem:

By the repetition of an advantageous event, be it simple or compound, the real benefit becomes more and more probable, and it increases continually. It becomes certain under the hypothesis of an infinite number of repetitions; and on dividing it by ^bthis^b number, the quotient, or the mean benefit of each event, is the mathematical expectation itself — or the relative advantage of the event. The same thing holds for a loss that becomes certain in the long run, if the event is in the least disadvantageous.