ductive of the appearance of two aces, or any other two faces. So that the chance of throwing two aces either together on two dice, or successively on one die, is only . 136. Hence the probability of two or more independent but joint events is equal to the product of the chances of each. Thus, the probability of throwing three aces successively on one die is × ×=zte. So if the proba¦ 1 216. bility, that one man, A, will live a year, be f‰, and the probability of the life of another man, B, for one year, bef, the probability, that both will live another year, is but fo fo = . Hence the concurrence of two events is less probable than the occurrence of either; and is even improbable, though each is probable and completely independent of the other. 137. From the foregoing rule it is manifest, that the joint occurrence of two or more equicasual, independent events is improbable; and the more so, the more numerous they are. For the probability of each is ; therefore the joint chance of two such events is × =; and of three such events is XX. So the concurrence of two independent, improbable facts is still more improbable. For, supposing the X = improbability of one of them to be, and that of the other, their joint improbability would bez. By the same rule, the improbability of the death of A within a year being, and that of the death of B within a year, the improbability, that both will die within a year, is of 18o. And the probability that one of the events will happen and the other fail is, as the probability of the happening of the one, multiplied by the probability of the failure of the other. So, in the above case, the probability, that A will live and that B will die, is fofo = 18%. And the probability, that B will live and that A will die, is X= 12. 138. A dependent event is one, whose existence is rendered more or less probable by the chances attending the existence of another event. When several events are connected in such a manner, that the second depends on the first, the third on the second, and so on, the probability of the first or independent event must be first ascertained; that of the second, which depends on the first, is then found, by multiplying its separate probability into that of the first; and the product will give the real probability of the second event. In the same manner we proceed to find the probability of a third or fourth dependent event. 139. Thus, suppose six white and six black balls to be placed in a box, and through a hole in the box, two balls to be successively drawn out; and let it be required to determine the probability, that both these will be white. As there are twelve balls in the box, and six of them are white, it is evident, that the probability of drawing a white ball at the first trial will be f. But the chance of doing this on the second trial will be different; for, as one of the balls has been taken out, there are but eleven remaining; and since, in order to the second trial, it is necessary to suppose, that the ball removed was a white one, the remaining number of these is reduced to five. The separate probability, therefore, of drawing a white ball at the second trial will be only ; and the chance of drawing it the first and second time will be = 1 × = 1⁄2. The separate probability of drawing out a white ball at a third trial, since two white balls have been removed, will be ; and the chance of drawing three white ones at three successive trials will be × × == . f 140. Again, W sailed for Africa in a fleet of twelve ships, three of which were lost in a storm, on the first part of the voyage. Of the crews of the nine ships, that escaped the storm, one third part perished from the hardships, they met on the voyage. We wish to ascertain the probability, that W has escaped both calamities. Now, as the chance of his having survived the hardships of the voyage depends on the event of his having escaped the storm, the probability of the last named event must be first ascertained. If this be found improbable, the second event must fail; but if it be found probable, the second event may exist, and the probability of its existence may be found by the rule already given. [No. 138.] 2 = 141. As nine ships out of the twelve survived the storm, the probability that W escaped in one of them is This being supposed, the probability of his having escaped the second danger, since only one third of those, who survived the storm, perished, is. Hence the probability of his having lived through both dangers is × == Therefore it is merely doubtful whether he survived both calamities. If only of the crew survived the second danger, then his escape would be improbable; for x = If only two out of the twelve ships were lost, and consequently ten had escaped the first danger, and of the crew had escaped the second danger, as above, then the probability of his entire survival would be 19 × 3 = 38 = ; a slight probability.* CHAPTER SEVENTH. GENERAL DESCRIPTION OF DEMONSTRATIVE REASONING. 142. The general nature of demonstrative reasoning has already been explained, in pointing out the circumstances, which distinguish it from moral, or probable reasoning. [See No. 87 to 93.] It has generally been admitted, that demonstration can be employed only about such truths as have been termed necessary, the subjects of which are not supposed to have any real existence, but to be abstractly conceived by the mind. All created beings depend on the will of their Creator. Their existence, their properties, and of course the relations, subsisting among those properties, are contin * Demoivre, Doctrine of Chances, Introduction. Kirwan, Logick. part iii. ch. 7. |