LEMMA 1. To find the value of a sum to be received at a future time in the event of the happening of a given contingency. - Suppose that the sum of 1 is to be received in n years' time, provided that a certain event shall then happen (or shall have then happened), the probability of which is p. We have seen that the value of 1 to be certainly received in n years' time is v". In order to introduce the idea of probability into the problem, suppose that p = a+ b' so that there are a cases favourable to the happening of the assumed event, and b unfavourable, the total number of possible cases, all of which are equally probable, being (a + b). We may suppose, for instance, that there are (a + b) balls in a bag, of which a are white and b black ; and that 1 is to be received if a white ball is drawn. In order to determine the value of the chance of receiving 1 in consequence of a white ball being drawn, suppose that (a+ b) persons draw each one ball, and that every one who draws a white ball receives 1 ; then the total sum to be received is a, and the value of the expectation of all the (a + b) persons who draw is also a. But it is clear that each of the persons has the same chance of drawing a white ball, therefore the value of the expectation of each a of them is a+ -=p. This is the value of the chance of receiving 1 immediately before the drawing is made in n years' time ; the value at the present time will therefore be v"p. We may also arrive at this result as follows : The same suppositions being still adhered to, the present value of the sum a to be distributed at the end of 71 years is av"; and each of the (a + b) persons having the same chance of receiving 1, the value of the expectation of each a is - a +bv"-pv".