PROP. 6 The probability that several independent events shall happen is a ratio com- pounded of the probabilities of each For from the nature of independent events, the probability that any one happens is not altered by the happening or gailing of any one of the rest, and consequently the probability that the 2d event happens on supposition the lst does is the same with its original probability; but the probability that any two events happen is a ratio compounded of the lst event, and the probability of the ed on the supposition on the lst happens by prop. 3. Wherefore the probability that any two independent events both happen is a ratio compounded of the lst and the probability of the 2d. And in the like manner considering the lst and events together as one event: the probability that three independent events all happen is a ratio compounded of the probability that the two lst both happen ity of the 3d. And thus you may proceed if there be ever so many such events; from which the proposition is manifest Cor. 1. If there be several independent events, the probability that the lst happens the 2d fails, the 3d fails and the 4th happens, &c is a ratio compounded of the probability of the lst, and the probability of the failure of 2d, and the probability of the failure of the 3d, and the probability of the 4th, &c. For the failure of an event may always be considered as the happening of its contrary Cor. 2. If there be several independent events, and the probability of eac one be a, and that of its failing be b, the probability that the lst happens and the 2d fails, and the 3d fails and the 4th happens, &c. will be abba, &zc. For accor ding to the algebraic way of notation, if a denote any ratio and b another abba denotes the ratio compounded of the ratios a, b, b, a. This corollary is therefore only a particular case of the foregoing Definition. If in consequence of certain data there arises a probability that a certain event should happen, its happening or failing, in consequence of these data, I call it's happening or failing in the lst trial. And if the same data be again repeated, the happening or failing of the event in consequence of them call its happening or failing in the 2d trial; and so again as often as the same data are repeated. And hence it is manifest that the happening or failing of the same event in so many differe- trials, is in reality the happening or failing of so many distinct independent events exactly similar to each other PROP. 7 If the probability of an event be a, and that of its failure be b in each single trial, the probability of its happening p times, and failing g times in p+q trials is Eapbq ife be the coefficient of the term in which occurs aPbq when the binomial a+66+q is expanded For the happening or failing of an event if different trials ind pendent events. Wherefore(by cor. 2. prop. 6. the probability that the event happens the lst trial, fails the 2d and 3d, and happens the 4th, fails the 5th. &c. (thus happening and failing till the number of times it happens be p and theP R O P. 6. The probability that several independent events shall happen is a ratio com￾pounded of the probabilities of each. For from the nature of independent events, the probability that any one happens is not altered by the happening or gailing of any one of the rest, and consequently the probability that the 2d event happens on supposition the 1st does is the same with its original probability; but the probability that any two events happen is a ratio compounded of the 1st event, and the probability of the 2d on the supposition on the 1st happens by prop. 3. Wherefore the probability that any two independent events both happen is a ratio compounded of the 1st and the probability of the 2d. And in the like manner considering the 1st and 2d events together as one event; the probability that three independent events all happen is a ratio compounded of the probability that the two 1st both happen and the probability of the 3d. And thus you may proceed if there be ever so many such events; from which the proposition is manifest. Cor. 1. If there be several independent events, the probability that the 1st happens the 2d fails, the 3d fails and the 4th happens, &c. is a ratio compounded of the probability of the 1st, and the probability of the failure of 2d, and the probability of the failure of the 3d, and the probability of the 4th, &c. For the failure of an event may always be considered as the happening of its contrary. Cor. 2. If there be several independent events, and the probability of each one be a, and that of its failing be b, the probability that the 1st happens and the 2d fails, and the 3d fails and the 4th happens, &c. will be abba, &c. For, according to the algebraic way of notation, if a denote any ratio and b another abba denotes the ratio compounded of the ratios a, b, b, a. This corollary is therefore only a particular case of the foregoing. Definition. If in consequence of certain data there arises a probability that a certain event should happen, its happening or failing, in consequence of these data, I call it’s happening or failing in the 1st trial. And if the same data be again repeated, the happening or failing of the event in consequence of them I call its happening or failing in the 2d trial; and so again as often as the same data are repeated. And hence it is manifest that the happening or failing of the same event in so many differe- trials, is in reality the happening or failing of so many distinct independent events exactly similar to each other. P R O P. 7. If the probability of an event be a, and that of its failure be b in each single trial, the probability of its happening p times, and failing q times in p+q trials is E apbq if E be the coefficient of the term in which occurs apbq when the binomial a + b| b+q is expanded. For the happening or failing of an event if different trials are so many inde￾pendent events. Wherefore (by cor. 2. prop. 6.) the probability that the event happens the 1st trial, fails the 2d and 3d, and happens the 4th, fails the 5th. &c. (thus happening and failing till the number of times it happens be p and the 7