Suppose a person has an expectation of receiving n, depending on an event the probability of which is N. Then(by definition 5)the value of his expectation P, and therefore if the event fail, he loses that which in value is P; and if it happens he receives N, but his expectation ceases. His gain therefore is N-P. Likewise since the probability of the event is N, that of its failure(by corollary prop. 1)is >N. But An is ton as P is to N-P, i.e. the probability of the event is to the probability of it's failure, as his loss if it fails to his gain if it happens PROP 3 The probability that two subsequent events will both happen is a ratio com- pounded of the probability of the Ist, and the probability of the 2d on suppo- sition the lst happens Suppose that, if both events happen, I am to receive n, that the probabilit both will happen is N, that the 1st will is n (and consequently that the 1st will not is -N) and that the 2d will happen upon supposition the lst does is Then(by definition 5)P will be the value of my expectation, which will become b is the lst happens. Consequently if the lst happens, my gain is b-P, and if it fails my loss is P. Wherefore, by the foregoing proposition, n is to n, i.e. a is to N-a as P is to b-P. Wherefore(componendo inverse)a is to N as P is to b. But the ratio of p to N is compounded of the ratio of p to b, and that of b to n. Wherefore the same ratio of p to N is compounded of the ratio of to N and that of b to n, i.e. the probability that the two subsequent events will both happen is compounded of the probability of the lst and the probability of N, and the probability of both together be n, then the probability of the 2d on supposition the lst happens is PROP 4 If there be two subesequent events be determined every day, and each day the probability of the 2d is s and the probability of both f, and I am to receive N if both of the events happen the lst day on which the 2d does: I say, according to these conditions, the probability of my obtaining n is f. For if not, let the probability of my obtaining n bef and let y be to r as n-b to N. The since x is the probability of my obtaining n(by definition 1)a is the value of my expectation. And again, because according to the foregoing conditions the lst day I have an expectation of obtaining n depdening on the happening of both events together, the probability of which is f, the value of this expectation is P. Likewise, if this coincident should not happen I have an expectation of being reinstated in my former circumstances, i. e. of receiving that which in value is a depending on the failure of the 2d event the probability of which(by cor prop 1)is NN or 3, because y is to r as N-b to N. Wherefore since r is the thing expected and the probability of obtaining it, the value of this expectation isSuppose a person has an expectation of receiving N, depending on an event the probability of which is P N . Then (by definition 5) the value of his expectation is P, and therefore if the event fail, he loses that which in value is P; and if it happens he receives N, but his expectation ceases. His gain therefore is N − P. Likewise since the probability of the event is P N , that of its failure (by corollary prop. 1) is N−P N . But N−P N is to P N as P is to N − P, i.e. the probability of the event is to the probability of it’s failure, as his loss if it fails to his gain if it happens. P R O P. 3. The probability that two subsequent events will both happen is a ratio com￾pounded of the probability of the 1st, and the probability of the 2d on suppo￾sition the 1st happens. Suppose that, if both events happen, I am to receive N, that the probability both will happen is P N , that the 1st will is a N (and consequently that the 1st will not is N−a N ) and that the 2d will happen upon supposition the 1st does is b N . Then (by definition 5) P will be the value of my expectation, which will become b is the 1st happens. Consequently if the 1st happens, my gain is b − P, and if it fails my loss is P. Wherefore, by the foregoing proposition, a N is to N−a N , i.e. a is to N − a as P is to b − P. Wherefore (componendo inverse) a is to N as P is to b. But the ratio of P to N is compounded of the ratio of P to b, and that of b to N. Wherefore the same ratio of P to N is compounded of the ratio of a to N and that of b to N, i.e. the probability that the two subsequent events will both happen is compounded of the probability of the 1st and the probability of the 2d on supposition the 1st happens. Corollary. Hence if of two subsequent events the probability of the 1st be a N , and the probability of both together be P N , then the probability of the 2d on supposition the 1st happens is P a . P R O P. 4. If there be two subesequent events be determined every day, and each day the probability of the 2d is b N and the probability of both P N , and I am to receive N if both of the events happen the 1st day on which the 2d does; I say, according to these conditions, the probability of my obtaining N is P b . For if not, let the probability of my obtaining N be x N and let y be to x as N − b to N. The since x N is the probability of my obtaining N (by definition 1) x is the value of my expectation. And again, because according to the foregoing conditions the 1st day I have an expectation of obtaining N depdening on the happening of both events together, the probability of which is P N , the value of this expectation is P. Likewise, if this coincident should not happen I have an expectation of being reinstated in my former circumstances, i.e. of receiving that which in value is x depending on the failure of the 2d event the probability of which (by cor. prop. 1) is N−b N or y x, because y is to x as N − b to N. Wherefore since x is the thing expected and y x the probability of obtaining it, the value of this expectation is 5