55. Suppose a prime to b, and let the quantities be divided by b; prove that the sum of the quotients arising from any two terms equidistant from the beginning and end will be a-1, and that the sum of the corresponding remainders will be b. 56. If any number of square numbers be divided by a given number n there cannot be more than different remainders. n 2 57. Express generally the rational values of x and y which satisfy 140x y3. = 58. If r the radix of a scale of notation be a prime number r+1 greater than 2, there are different digits in which square 2 numbers terminate in that scale. 59. If any number n can be resolved into the sum of p squares, 2 (p − 1)n can be resolved into the sum of p (p-1) squares. 60. If n be any positive integer 22"+15n-1 is divisible by 9. 61. If P, denote the sum of the products of the first ʼn numbers taken r together, 1 + P, + P2+ ... + P1-, is a multiple of [n. 2 62. Shew that the 100th power of any number is of the form 125n or 125n + 1. LIII. PROBABILITY. 714. If an event may happen in a ways and fail in 6 ways, and all these ways are equally likely to occur, the probability of its happening is a a + b' and the probability of its failing is This may be regarded as a definition of the meaning of the word probability in mathematical works. The following explanation is sometimes added for the sake of shewing the consistency of the definition with ordinary language. The probability of the happening of the event must, from the nature of the case, be to the probability of its failing as a to b; therefore the probability of its happening is to the sum of the probabilities of its happening and failing as a to a+b. But the event must either happen or fail, hence the sum of the probabilities of its happening and failing is certainty. Therefore the probability of its happening is to certainty as a to a+b. So if we represent certainty by unity, the probability of the happening of the event is repre 715. Hence if p be the probability of the happening of an event, 1-p is the probability of its failing. 716. The word chance is often used in mathematical works as synonymous with probability. 717. When the probability of the happening of an event is to the probability of its failing as a to b, the fact is expressed in popular language thus; the odds are a to b for the event, or b to a against the event. 718. Suppose there to be any number of events A, B, C, &c., such that one must happen and only one can happen; and suppose a, b, c, &c., to be the numbers of ways in which these events can respectively happen, and that all these ways are equally likely to occur, then the probabilities of the events are proportional to a, b, c, &c. respectively. For simplicity let us consider three events, then A can happen in a ways out of a+b+c ways and fail in b + c ways; therefore, by Art. 714, the probability of A's happening is and the probability of A's failing is b+c a+b+c' a a+b+c' Similarly the probability of B's happening is b a+b+c' and the probability of C's happening is C a+b+c 719. We will now exemplify the mathematical meaning of the word probability. If n balls A, B, C,..., be thrown promiscuously into a bag and a person draw out one of them, the probability that it will 1 be A is ; the probability that it will be either A or B is n The same supposition being made, if two balls be drawn out the probability that these will be A and B is number of pairs of balls is the same as the number of combinations 1 of n things taken two at a time, that is, n(n-1); and one pair 2 is as likely to be drawn out as another; therefore the probability 1 2 of drawing out an assigned pair is 1÷n(n − 1), that is, 2 n(n - 1) Again, suppose that 3 white balls, 4 black balls, and 5 red balls are thrown promiscuously into a bag, and a person draws out one of them; the probability that this will be a white ball is the probability that it will be a black ball is probability that it will be a red ball is But suppose 5 12' two balls to be drawn out, and estimate the probabilities of the different The number of pairs that can be formed out of 12 things cases. is 1/2 × 12 × 11, that is, 66. The number of pairs that can be formed out of the 3 white balls is 3; hence the probability of drawing two white balls is 3 Similarly the probability of draw 66 6 ing two black balls is ; and the probability of drawing two red 66 Also since each white ball might be associated with each black ball, the number of pairs consisting of one white ball and one black ball is 3 x 4, that is, 12; hence the probability of 12 drawing a white ball and a black ball is Similarly the proba 66 15 The sum 66 bility of drawing a black ball and a red ball is bability of drawing a red ball and a white ball is of the six probabilities which we have just found is unity, as, of course, it should be. We will give one example from a subject which constitutes an important application of the theory of probability. According to the Carlisle Table of Mortality, it appears that out of 6335 persons living at the age of 14 years, only 6047 reach the age of 21 years. As we may suppose that each individual has the same chance of being one of these survivors, we may say that 6047 is the proba bility that an individual aged 14 years will reach the age of 21 years: and of 21 years. 720. 288 6335 is the probability that he will not reach the age Suppose that there are two independent events of which the respective probabilities are known; we shall proceed to estimate the probability that both will happen. Let a be the number of ways in which the first event may happen, and 6 the number of ways in which it may fail, all these ways being equally likely to occur; and let a' be the number of ways in which the second event may happen, and b' the number of ways in which it may fail, all these ways being equally likely to occur. Each case out of the a+b cases may be associated with each case out of the a+b′ cases; thus there are (a+b)(a′ +b′) compound cases which are equally likely to occur. In aa' of these compound cases both events happen, in bb' of them both events fail, in ab' of them the first event happens and the second fails, and in a'b of them the first event fails and the second happens. Thus, T. A. 27 απ' (a + b)(a + b) bb' (a + b)(a' + b') ab' is the probability that both events happen, is the probability that both events fail, (is the probability that the first happens and the (a + b)(a+b)\ second fails, a'b (is the probability that the first fails and the (a+b)(a+b) second happens. { Thus if p and p' be the respective probabilities of two independent events, pp' is the probability of the happening of both events. 721. The probability of the concurrence of two dependent events is the product of the probability of the first into the probability that when that has happened the second will follow. This is only a slight modification of the principle established in the preceding article, and is proved in the same manner; we have only to suppose that a' is the number of ways in which after the first event has happened the second will follow, and b the number of ways in which after the first event has happened the second will not follow, all these ways being supposed equally likely to occur. 722. In like manner, if there be any number of independent events, the probability that they will all happen is the product of their respective probabilities of happening. Suppose, for example, that there are three independent events, and that p, p', p" are their respective probabilities. By Art. 720, the probability of the concurrence of the first and second events is pp'; then in the same way the probability of the concurrence of the first two events and the third is pp' xp", that is, pp'p". Similarly the probability that all the events fail is (1 − p)(1 − p)(1-p'). The probability that the first happens and that the other two fail is p(1 − p)(1 − p′) ; · and so on. |