Del 3. Events are independent, when the happening of any one of them does neither increase nor lessen the probabi lity of the rest. Thus, if a person un dertook with a single die to throw an ace at two successive trials, it is obvious (however his expectation may be effect ed) that the probability of his' throwing an ace in the one is neither increased nor lessened by the result of the other trial.
These. The probability that two subse quent events will both happen, is equal to the product of the probabilities of the happening of those events considered se parately.
Suppose the chances for the happening and failing of the first event to be denot ed by b, and those for its happening only to be denoted by a. Suppose, in like manner, the chances for the second event's happening and failing to be de noted by d, and those for its happening only by c ; then will the probability of the happening of each of those events, sepa rately considered, be (according to Def. 1) and respectively. Since it is ne cessary that the first event should happen before any thing can be determined in regard to the second, it is evident that the expectation on the latter must be lessened in proportion to the improbabi lity of the former. Were it certain that the first event would happen, in other words, were a = b, or = 1, the expec tation on the second event would be = E. But if a is less than b, and the ex pectation on the second eventis restrain ed to the contingency of its having hap pened the first time, that expectation will be so much less than it was on the former supposition as is less than unity.
Hence we have 1 : c ac : for the true expectation in this case.
Cor. By the same method of reasoning it will appear, that the probability of Use happening of any number of subsequent events is equal to the " product of the probabilities of those events separately considered," and therefore, if u always denote the probability of its happening, and b the probability of its happening and failing, the fraction will express the probability of its happening n times suc cessively, and (by Def. 1) the fraction bn will express the probability of its fhiling n times successively.
Rem. It should be observed, that in some instances the probability of each subsequent event necessarily differs from that which preceded it, while in others it continues invariably the same through any number of trials. in the one case
the probabilities are expressed, as in the theorem, by fractions, whose numerators and denominators continually vary ; in the other they are expressed, as in the corollary, by one and the same inva riable fraction. But this perhaps will be better understood by the following examples.
1. Suppose that out of a heap of coun ters, of which one part of them are white and the other red, a person were twice successively to take out one of them, and that it were required to determine the probability that these should be red coun ters. if the number of the white be 6, and the number of the red be four, it is evi dent, from what has already been shown, that the probability of taking out a red one the first time will be 4 • but the TO" probability of taking it out the 2d time will be different ; for since one counter has been taken out, thee are now only nine remaining ; and since,. in order to the 2d trial, it is necessary that the coun ter taken out should have been a red one, the number of those red ones must have been reduced to 3. Consequently, the chance of drawing out a red coon. ter the 2d time will be and the pro. bability of drawing it out the first and 2d time will (by this theorem) be X 3lU 2 2. Suppose next, that with a single die a person undertook to throw an ace twice successively : in this case the pro bability of throwing it the first does not in the least alter his chance of throwing it the second time, as the number of faces on the die is the same at both trials. The probability, therefore, in each will be ex pressed by the same fraction, so that the probability, before any trial is made, will, by the preceding corollary, be IX A. On these conclusions depend all the com putations, however complicated and labo rious, in the doctrine of chances. But this, perhaps, will be more clearly exemplifi ed in the two following problems, which will serve to explain the principles on which every other investigation is found ed on this subject.