Independent Events. In the doctrine of probability, two or more events are said to be independent, when the occurrence of one of them will not influence the happening or the failing of the other; so that if there be two bags, each containing nine white balls and one black ball, and a person be required to draw a single ball from each bag, the event of the drawing from the one will not at all influence the event of the drawing from the other; and such two events are therefore said to be independent of each other. If we inquire what is the probability (or more properly the measure of the probability) that a white ball will be drawn from each of the two bags spoken of, we must, consistently with what has been advanced, first consider how many possible cases can occur in the two drawings, and we shall perceive that in those two drawings each of the 10 balls in one bag may be combined with each of the 10 balls in the other, and, consequently, that 100 different cases are possible. As, however, there are but nine white balls in the one bag which can combine with each of the 9 white balls in the other (9x 9) so as to produce the event inquired about, there are but 81 cases 81 100 out of the 100 possible cases (or of the whole number) that favour the drawing of a white ball from each bag: and •81, will be the fraction expressing the probability of drawing a white ball from each bag. This may possibly be made more evident by showing that of the remaining 19 possible cases not one favours the production of that event of which it was required to find the probability. It is manifest that, as each white ball in each bag could be combined with each white ball in the other bag, so could the black ball in each bag be combined with each white ball in the other, that is, each black ball could be combined with either of the 9 white balls in the other bag: these cases amount to 9+9= 18. The occurrence of either of these 18 cases would cause the event to fail, of which the probability was first required; and if the black ball from each bag should be drawn, the event would also fail, which case, added to the 18 adverse cases last mentioned, would make in the whole 19 possible cases unfavourable to the production of the event in question. These 19 unfa 81 100-) tainty. But is the fraction that would be produced by mul 100 tiplying together the two separate probabilities of drawing a white ball from either of the bags without reference to the other, that is 9 9 X 81 10 × 10= 100; and if we make m to represent the number of cases favourable to the production of the event indicated, and n the number of cases unfavourable to it, we shall have this expression to represent the probability of drawing a white ball from each bag, viz., m m m+ n m + n = m2 (m+n); and this expression will be equally true, be the values of m and n what they may. The probability of the happening of any three independent events may be shown to be equal to the product of the three separate probabilities. For if there were a third bag, like the two already described, containing nine white balls and one black ball, and the probability of a white ball being drawn from each were required, we should see first that any one of the 100 possible cases which might occur in drawing a ball from each of two bags could be combined with each of the ten balls in the third bag; or that (100 × 10) 1000 possible cases could occur in drawing a ball from each of the three bags. It is also evident that the only way in which that event could occur, of which we are now inquiring the probability, must be by a combination of some one of the 81 cases favourable to drawing a white ball from each of the two bags with the drawing of a white ball from the third bag, and as those 81 cases could combine with any one of the nine white balls in the third bag, there are, of the 1000 possible cases, (81 × 9=)729 cases favourable to the drawing a white ball from each bag. The truth of this may be made clear by showing that the remaining 271 possible cases are all unfavourable to the production of the event under consideration. Thus, if any one of the 19 cases unfavourable to drawing a white ball from each of two bags be combined with any one of the ten different possible drawings from the third bag, the event under consideration would fail. This combination includes (19 x 10) 190 cases; and if any one of the 81 cases favourable to drawing a white ball from each of two bags be combined with the possible case of drawing a black ball from the third bag, the event would also fail. These combinations are (81 x 1=) 81 in number, which, added to the 190 adverse cases last mentioned, make together 271 adverse possible cases, and with the 729 favourable cases, account for all, the 1000 possible cases that can occur in one drawing from each of the three bags. The probability, therefore, of drawing a white ball from But this is the product each of the three bags is 729 1000' or ⚫729. of the three fractions, which represent the separate probabilities of drawing a white ball from any one bag, viz. If we represent these fractions by the symbols already used, we shall have this expression, As the like reasoning may be applied to any other number of independent events, we deduce therefrom this important theorem, viz., that the probability of the occurrence of any number of independent events is equal to the product of the separate probabilities of the events considered one by one. Money value of probability. If a bag contain m + n slips of paper, each having written on it the name of a different individual, and the condition exist that the person whose name may be first drawn shall receive £p, then in estimating the probability of any particular individual gaining the £p, we should see that in the drawing there would be m + n possible cases, one only of which would favour a particular individual gaining the £p; the probability of any one individual gaining would therefore be If all the m+n persons 1 were to sell their expectations to another person, the purchaser would be sure to gain the £p; and the value of all the m + n expectations would evidently be just £p ( m + n = × p); but m + n each expectation would be of the same value with all the others, and consequently would be worth Р 1 = m + n xp: that is, the value of each individual's expectation would be worth the probability of his name being the one drawn, multiplied by the contingent gain. If A should purchase the expectations from m individuals, and B should purchase the expectations from n individuals, there would be m cases favouring A's gaining the £p; and the total value of the probability of his gaining the £p would x p. The same reasoning will show that the value of m be B's expectation would be n m + n p; and as this will be true whatever may be the numerical values of m, of n, and of p, we conclude generally that the money value of the expectation of an individual, as respects the happening of a particular event, is equal to the product of the probability that such event will happen into the sum to be received, should the event occur. Human Mortality. Let it now be assumed that a sufficient numper of observations had been made to determine how many persons out of a given number born alive, say 100,000, had, on an average, been dead at the end of each year until the whole had died, then a Table, representing the numbers remaining alive at the end of each year, would be what is ordinarily termed a Table of Mortality, and such a table will furnish us with the means of measuring the probability that an individual of any given age will live to the end of a specified term. It should, however, be remarked, that any deductions to be made from such a table will be true only in proportion to the dependence we may place on the rate of mortality in future being the same as that which had been observed to have previously taken place among the number observed upon. All the computations now made to regulate the values of annuities and assurances on lives are founded on the assumption that, in like circumstances, a like rate of mortality will actually prevail. The method by which the Table of Mortality has been formed, will be hereafter explained. It may, for our present purpose, be taken for granted that it correctly indicates the rate of mortality which will hereafter prevail. On reference to the table it will be seen that of 8263 persons alive at age 20, 66 would die in the following year, and that, consequently, there would be 8197 persons living at age 21. Now, if it be required to determine the probability that a person aged 20 would live one year, or to the age 21, we should say (assuming that the facts would correspond with the mortality table) that there were 8263 possible cases by which the event might happen or fail, that is, that there were 8197 cases which favoured the person being alive, or the happening of the event, and 66 cases which favoured his being dead, or the failing of the event. It has been shown, that the probability of the happening of an event, is in the ratio of the number of cases by which it may happen, to the number of cases by which it may either happen or fail. The probability in question will therefore be expressed by 8197 8197 the fraction 8197+66 8263' = the numerator of which, it will be observed, is the number living at age 21, and the denominator the number living at age 20. Again, if we desire to find the probability, that a person aged 20 will live 20 years, or be alive at age 40, we shall, on reference to the table, see that of the 8263 persons alive at 20 years of age 6575 persons attained the age 40, the difference (or 1688) having died in the interim; we, therefore, say, that there are 6575 cases which favour the person aged 20 living 20 years longer, and 1688 cases which favour his failing to do so; the probability of his living 20 years will therefore be expressed by the fraction 6575 6575 6575+1688 8263' = which, like the former, has for the numerator the number living at the higher age, and for the denominator the number living at the younger age: and as the like reasoning may be applied to every case embraced in the mortality table, we may conclude generally, that the probability of a life of a given age, living a specified term, will be properly represented by a fraction, having for its denominator the number living at the younger age, and for its numerator the number living at an age older than the given age by the specified term. |