772 PROBABILITY the same equation of differences as (2). Its solution is equation (3), in which, if we determine the constants by the conditions ^ = 0, w., = 0, 3 =2, and divide by 2", we find for the probability of a run of three of either event during n trials K , Kl , _u n _ n + 1 (, , n -yi~ - 2 ^^TJ 1 " ~ Comparing this result with (6) we find that the chance of a run of two heads in n trials is equal to the cliance of a run of three, of eitJicr heads or tails, in n + 1 trials. 7. If an event may turn out on each trial in a + b ways, of which a are favourable and b unfavourable (thus a card may be drawn from a pack in fifty-two ways, twelve of which give court cards), and if we consider the probability that during n trials there shall occur a run of at least p favourable results, it is not difficult to see that (it r denoting the number of ways this may occur in r trials) n + P +i = (a + b) u n+p + ba f {( + &)"-}, as Un+p+j includes, besides (a + b)u n+p , those cases in which the last p trials are favourable, the one before unfavourable, and the n preceding containing no such run as stated. We will not enter on Laplace s solution of this equation, or rather of one equivalent to it, especially as the result is not a simple one (see Todhuiiter, p. 185). 8. Let the probability of an event happening in one trial be p, that of its failing q ; we have seen (art. 4) that, if a large number N of trials be made, the event is most likely to happen ;;N times and fail </X times. The chance of this occurring is, however, extremely small, though greater than that in favour of any other proportion. We propose now to examine the probability that the proportion of successes shall not deviate from its most probable value by more than a given limit that is, in fact, to find the probability that in N trials the number of times in which the event happens shall lie between the two limits pNr. Let 7rt=^N, n = qS, which are taken to be integers. The proba bility of the event happening m times is the greatest term T of the expansion (1), viz., IN The calculation of this would be impracticable when N, m, n are large numbers, but Stirling s theorem gives us 1.2. 3 . . . x = x*+le-* < /!fa , very nearly, when x is large ; and by substituting in the preceding value of T, and reducing, we easily find T= - Now the terms of the expansion (1) on either side of T are n(n-l) & T+ _n_P T , T m q m(m-V) q* . (m + lXm+2) 3 2 m+lg l+T +^+lp + (n+l)(n+-2)p^ + W- But if x is much greater than a, x- a = xe x nearly, so that n(n-l)(n-2) . . . (s terms)- e- - -^-^ } = n , e - s ^^ . also ( m + 1 )( m + 2) . . . (s terms) = m c James500 (talk) Hence the sth term before T in (9) is ^ r /> -L. < T* > OI 6 OIIIM Ov., 1 The sth term after T is Now the probability that the event shall happen a number of times comprised between m + r and m-ris the sum of the terms in (9) from the rth term before T to the rth term after T. (N. B. , though r may be large, it is supposed small as compared with N, m, or n.) Now the sth term before T+ the sth term after T = 2e~^T r i m - n . -2,wnen*--^- is small. Taking then each lie corresponding term after T, and putting since e term before T for shortne.-s 2mn (10). we have for the required probability Pr = 2(iT + Tc~ at + Te- 22 " 2 + Te- 32 " 2 + . If we now consider the curve whose equation is and take a series of its ordinates corresponding to x=0, a, 2a, 3a . . . . ra, where a is very small, and if A be its area from x = Q to 2 = ra, then A = (first + last ordinates) + sum of intermediate ordinates a 2 . . p r -A -I- last ordinate, or Pr"f-/ e * <fce+ ==e~ . . . (11). 9. We refer to the integral calculus for the methods of com puting the celebrated integral fe x *dx, and will give here a short table of its values. Table of the Values of the Integral I = - / T e~ x dx. T I T I T I T I o-oo o-ooooo 2 22270 1-3 93401 2 4 99931 01 01128 3 32863 1-4 95229 2-5 99959 02 02256 4 42839 ! 1-5 96611 2 6 99976 03 03384 5 52050
- 1-6
97635 2-7 99986 04 04511 6 60386 1 17 98379 2-8 99992 05 05637 7 67780 1-8 98909 2-9 99996 06 06762 1 -8 74210 1-9 99279 3 99998 07 07886
- -9
79691 2-0 99532 00 i-ooooo 08 09008 i-o 84270 21 99702 09 10128 1 1 88020
- 2-2
99814 1 11246 1-2 91031 i 2 3 99886 If the value of I is 5, or J, T= 4769. 10. The second term in formula (11) expresses the probability that the number of occurrences of the event shall be exactly m + r or in-r, or more correctly the mean of these two pro babilities. It may be neglected when the number of trials N is very great and the deviation r not a very small number. We see from the foregoing table that when r _ = 3 it becomes practically a certainty that the number of occurrences will fall between the limits mr. Thus, suppose a shilling is tossed 200 times in succession ; here = . If therefore r= 30, it may be called a certainty that head will turn up more than 70 and less than 130 times. In the same case suppose we wish to find the limits mr such that it is an even chance that the number of heads shall fall between them, if the second term of (11) be neglected, we see from the table that ra=r = 48, . . r=4 8 ; so that the probability that the number of heads shall fall between 95 and 105 is 52 + -- e -k = 57 nearly, rather more than an even chance. 11. Neglecting the second term of (11), we see that p r depends solely on the value of ra, or that of TV ; so that, if the number of trials N be increased, the value of r, to give the same probability, increases as the square root of N; thus, if in N trials it is practically certain (when ra=3) that the number of occurrences lies between ^Nr, then, if the number of trials be doubled, it will be certain that the occurrences will lie between 2jjNr/2. In all cases, if N be given, r can be determined, so that there is a probability amounting to certainty that the ratio of the number of occurrences to the whole number of cases shall lie between the limits Now if N be increased roc >/N ; so that these limits are G being a constant. Hence it is ahvays possible to increase the number of trials till it becomes a certainty tliat the proportion of occurrences of the event ivill differ from p, its probability on a single trial, by a quantity less than any assignable. This is the celebrated theorem given by James Bernoulli in the Ars Conjectandi. (See
Todhunter s History, p. 71.)
