Probability

If p represents the probability of the happening of an event in one trial and q the probability of its failing, the probability that it will happen exactly r times in n trials is n(n — - - - - ( n — r 1) r I The probability that an event will fail exactly r times in a trials is — 1) - - - (a —r +1) In the expansion of ( p n, viz.

+ the terms represent respectively the probabilities of the happening of the event exactly a times, n— 1 times, a —2 times, and so on, in a trials. Hence the most probable number of successes and failures in a trials is given by the greatest term in the corresponding series. E.g. the probability of throwing an ace in one trial with a die is and of failing to do so is Also (t + hence the bility of throwing an ace -f times in 4 throws is the probability of throwing an ace 3 times in 4 throws is the probability of throwing an ace 2 times in 4 throws is the probability of throwing an ace 1 time in 4 throws is the probability of throwing an ace no time in 4 throws is Since the last fraction is the largest, the ease of no ace in 4 throws of a die is more probable than that of 1, 2, 3, or 4 aces.

A problem in life insurance, a subject to which the theory of probability has been of indispensa ble service, will serve to show the applications of the subject. A table of mortality gives the num bers alive at each successive year of their age, out of a given number of children born. If A. and A°+, be the minthers in the table correspond ing to the nth and (n 1)th years of age. the inference from the table is, that of A. individuals now alive, and of a years of age. , will live one additional year at least. Ilenee, the chance that any one of them die during the year is An-A0+1 Calling this 1 — p, p is the chance that any one of them will survive the year. Of two individuals, one a years old, and the other a', what are the chances that (a) only one lives a year? (b) one, at least, lives a year? lc) both do not live a year? Calling the individuals A and B. the chance of A living out the year is p, and the chance of his dying within the year is I For B these are p' and 1 — p'. Hence that A lives and B dies the chance is p (1 — p',1. That B lives and A dies the chance is p'(1—p).

Hence the answer to (a) p p' —2pp'. The second case includes, in addition to the conditions of (a I. the chance that both survive, which is pp'.

]fence the answer to (b) is p p' — pp'. In the third case the chance that both live a year is pp'. hence the chance that both will die is 1 — pp".

The theory of probability also furnishes a measure of expectation. The law of expectation in its simplest form may lie stated thus: The value of a contingent gain is the product of the sum to be gained into the chance of Winning it. Suppose A, B, and C have made a pool, each sub scribing $1, and that a game of pure chance (i.e. not dependent on skill) is to be played by them for the $3. What is the value of the expectation of each? By the conditions, all are equally likely to win the pool, hence its contingent value must he the same to each : and, obviously, the sum of these values must represent the whole amount in question. The worth of the expectation of each is therefore $1. That is. if A wishes to retire from the game before it is played out, the fair price which B or C ought to pay him for his share is simply $1. But this is obviously i3 of

$3, i.e. the value of the pool multiplied by his chance of getting it.

Another very important. application of the theory of probability is to the deduction of the most probable' value from a number of observa tions, each of which is liable to certain accidental errors. In a set of such observations. the proba ble error is a quantity such that there is the same probability of the true error being greater or less than it, and this probable error has been shown to be least when the sum of the squares of the errors is a minimum. The method for ob taining this least error is called the method of least squares. See LEAST SQUARES, or.

The doctrine of probabilities dates as far hack as Fermat and Pascal (1654). Huygens (1657) gave the first scientific treatment of the subject, and Jakob Bernoulli's Ars Co/JP-chi/id( (posthu mous, 1713) and De Aloivre's Doctrine of Chances (1718) raised the subject to the plane of a branch of mathematics. The theory of errors may he traced back to Cotes's Opera .11iscellanca (posthu mous, 1722). but a memoir prepared by Simpson in 1755 (printed 1756) first applied the theory to the discussion of errors of observation. Laplace (1774) made the first attempt to deduce a rule for the combination of observations from the principles of the theory of probabilities. He rep resented the law of probability of errors by a curve y = 0 (x), x being any error and y its probability, and laid down three properties of this curve: (1) It is symmetric as to the Y-axis; (2) the X-axis is an asymptote, the probability of the error oo being 0; (3) the area inclosed is I, it being certain that an error exists. He deduced also a formula for the mean of three observations. Among the contributors to the general theory of probabilities in the nineteenth century have been besides Laplace, Lacroix (1816), Littrow (1833), Quetelet ( 1853 ) , Dedekind ( 1860) , 11 elmert (1872). and Laurent (1873). On the geometric side the influence of Miller and The Educational Times has been marked. The literature of the subject is very extensive. Of the recent works, besides those already mentioned, the following are among the most important: Czuber, Gcometrischc Wahrscheinlichkriten Mid Mittehcertc (Leipzig, 1884; French trans., Paris, 1902) ; Herz. Maw scheinlichkcits and .4usglciehanysrechnung (Leip zig, 1900) ; Kries, Die Principien der Wahrschein lielikeits-Reclinung (Freiburg, 1886) ; Poinear6, ('alcul des probabilite's (Paris, 189(1) : Whitworth, Choice and Chance (Cambridge. 3d ed., 1878). The following earlier works are also well known: Cournot, Exposition de la theoric des chances et des yrobribilitJs (Paris, 1843) ; De 31organ. Es say on Probabilities (London, 183S) ; Lacroix, eleMcntaire (lit calcul des probabilites (Paris. 1833) ; Poisson, Recherches la pi.oba des 'ailments (Paris, 1837). For the early history of the subject, consult Todhunter, His tory of the Mathematical Theory of Probability (Cambridge, 1S65). For the later history and for a brief essay on the theory, consult Merriman and Woodward, Higher Mathematics (New York, 1890).