tion if, therefore, any other person should be willing to purchase his chance, he must give for it the half of 51, or 21. 108. This is one of the most simple cases before, however, we proceed, it may be proper to give some definitions introductory to the doctrine. Def. 1. The probability of an event is the ratio of the chance for its happening to all the chances for its happening or failing: thus, if out of six chances for its happening or failing, there were only two chances for its happening, the probability in favour of such an event would be in the ratio of two to six; that is, it would be a fourth proportional to 6, 2, and 1, or 4. For the same reason, as there are four chances for its failing, the probability that the event will not happen will be in the ratio of 4 to 6, or, in other words, it will be a fourth proportional to 6, 4, and 1, or Hence, if the fractions expressing the prbabilities of an event's both happening or failing be added together, they will always be found equal to unity. For let a be the number of chances for the event's happening, and b the number of chances for its failing, the probability in a the first case being and in the sea+b cond case 1. Having therefore determined the probability of any event's either happening or failing, the probability of the contrary will always be obtained by subtracting the fraction expressing such probability from unity. Def. 2. The expectation of an event is the present value of any sum or thing, which depends either on the happening or on the failing of such an event. Thus, if the receipt of one guinea were to depend on the throwing of any particular face on a die, the expectation of the person entitled to receive it would be worth 38. 6d.; for since there are six faces on a die, and only one of them can be thrown to entitle the person to receive his mo ney, the probability that such a face will be thrown being (according to Def. 1.) it follows, that the value of his interest before the trial is made, or, which is the same thing, that his expectation is equal to one-sixth of a guinea, or 3s. 6d. Were his receiving the money to depend on his throwing either of two faces, his expectation would be equal to two-sixths of a guinea, or 7s. And, in general, supposing the present value of the money or thing or by a+b ᎪᏞ according as it depends either on a+b' the event's happening, or on its failing. Def. 3. Events are independent, when the happening of any one of them does neither increase nor lessen the probability of the rest. Thus, if a person undertook with a single die to throw an ace at two successive trials, it is obvious (however his expectation may be effected) that the probability of his throwing an ace in the one is neither increased nor lessened by the result of the other trial. Theor. The probability that two subsequent events will both happen, is equal to the product of the probabilities of the happening of those events considered separately. Suppose the chances for the happening and failing of the first event to be denoted 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 denoted by d, and those for its happening only by c; then will the probability of the happening of each of those events, separately considered, be (according to Def. 1) and respectively. Since it is necessary 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 improbability of the former. Were it certain that the first event would happen, in other a words, were a = b, or = 1, the expectation on the second event would be = But if a is less than b, and the expectation on the second event is restrained to the contingency of its having happened the first time, that expectation will be so much less than it was on the former supposition as a is less than unity. probabilities of those events separately considered," and therefore, if a always denote the probability of its happening, and the probability of its happening and failing, the fraction will express the probability of its happening n times successively, and (by Def. 1) the fraction bn n an will express the probability of its failing 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 invariable 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 counters. If the number of the white be 6, and the number of the red be four, it is evident, from what has already been shown, that the probability of taking out a red one the first time will be 4: but the probability of taking it out the 2d time will be different; for since one counter has been taken out, there are now only nine remaining; and since, in order to the 2d trial, it is necessary that the counter taken out should have been a red one, the number of those red ones must TO have been reduced to 3. Consequently, the chance of drawing out a red counter the 2d time will be 3, and the pro. bability of drawing it out the first and 4 X 3 2d time will (by this theorem) be 10 X 9 2 by the preceding corollary, be × } = {} %, On these conclusions depend all the computations, however complicated and labo rious, in the doctrine of chances. But this, perhaps, will be more clearly exemplified in the two following problems, which will serve to explain the principles on which every other investigation is founded on this subject. Prob. 1. To determine the probability that an event happens a given number of times, and no more, in a given number of trials. Sol. 1. Let the probability be required of its happening only once in two trials, and let the ratio of its happening to that of its failing be as a to b. Then, since the event can take place only by it happening the first, and failing the second time, the probability of which is = a +b b ab a+b or by its failing the 29 a+b first and happening the second time, the probability of which is ba a+b2 2ab the sum of will be 2 these two fractions, or the probability required. а -т a ab a+b) 2. Let the probability be required of its happening only twice in three trials. In this case, the event, if it happens, must take place in either of three different ways: 1st, by its happening the first two, and failing the third time, the probability of which is ; 2dly, by its a + b) failing the first, and happening the other two times, the probability of which is baa ; or, 3dly, by its happening the first and third, and failing the second time, the probability of which is a + b3. The sum of these fractions, therefore, or 3 baa will be the required probability. By the same method of reasoning, the probability of its happening only once in three trials, or, which is the same thing, of its failing twice in three 3bba. trials, may be found equal to a+b a+b 3. aba Let the probability of the event's happening only once in four trials be required. In this case it must either happen the first and fail in the three succeeding trials; or happen the second and a+b) 4. 4. Let the probability be required of its happening twice and failing twice in four trials: here the event may be determined in either of six different ways: 1st, by its happening the first and second, and failing in the third and fourth trials; 2dly, by its happening the first and third, and failing the second and fourth trials; 3dly, by its happening the first and fourth, and failing the second and third trials; 4thly, by its happening the second and third, and failing the first and fourth trials; 5thly, by its happening the second and fourth, and failing the first and third trials; or, 6thly, by its happening the third and fourth, and failing the first and second trials. Each of these probabilities being expressed by 4, it follows a2 b3 a+b 6a2b2 that the sum of them, or = a+b) press the probability required. By proceeding in the same manner, the probability in any other case may be determined. But if the number of trials be very great, these operations will become exceedingly complicated, and therefore recourse must be had to a more general method of solution. Supposing n to be the whole number of trials, and d the number of times in which the event is to take place, the probability of the event's happening d times successively, and failing the remaining nd times, ad ad. bn--d n-d will be: dx n-d a + b a+b a + b) n. But as there is the same probability of its happening any other d assigned trials and failing in the rest, it is evident that this probability ought to be repeated as often as d things can be combined in n things, which, by the known rules of combina Sol. It will be observed that this problem materially differs from the preceding, in as much as the event in that problem was restrained, so that it should happen neither more or less often than a given number of times, while in this problem the event is determined equally favourable by its happening either as often or oftener than a given number of times, so that in the present case there is no further restriction than that it should not fall short of that number. 1. Let the probability be required of an If it happens the first and fails the second event happening once at least in two trials. time, or fails the first and happens the second time, or happens both times, the event will have equally succeeded. The probability in the first case is probability in the second is probability in the third is аа ab a+b ba a+b a+b)2 ,and the probability required will be = 2ab+aa a+b) 2. Let the probability be required of its happening once in three times. Provided it has happened once at least in the first two trials, the event will have equally succeeded, whether it happens or fails in a2+2ab the third trial, and therefore will a+b2 represent the probability in this case. But it may have failed in the first two and happened in the third trial, the probability of bba which is adding this to the preceda+03; ing fraction we have2+2ab× a+b+b2 a a+b)3 a3+3a2b+3ab for the probability a+b required. In like manner the probability of its happening once at least in a3+3 ab+3abb four trials will be + a+b3 a++6 a3 b+6a2 b2+4 a b3 a+b+ least in n times will be = and In the probability of its happening once at a+bn-bn a+bn other words, since the event must happen once at least, unless it fails every time, the probability required (by Def. 1)will always be expressed by the difference between bn unity and a+b" 3. Let the probability be required of an event's happening twice at least in three trials. In this case it will succeed, if it happens the first and second, and fails the third time, if it happens the first and third, and fails the second time, if it happens the second and third, and fails the first time, or if it happens each time successively. 3a2b The first three probabilities are. and a+b13 event is to happen twice at least in four bn+nbn-la+n.” bn-2a2 to d terms, a+b1n will express the probability of the event's not happening so often as d times in n trials. Ex. Supposing a person with six dice undertakes to throw two aces or more in the first trial, what is the probability of his succeeding? In this case a, b, n, and d, being respectively equal to 1, 5, 6, and 2, the above expression will be come = times, the probability of its happening dur- 1+30+15 × 25 +20 × 125 +15 × 625 ing the first three times has been already 66 12281 46656 succeeding will be as 34375 to 12281, or nearly as three to one. Hence the odds against his We have already observed, that the doctrine of chances is particularly applicable to the business of life annuities and assurance. This depends on the chance of life in all its stages, which is found by the bills of mortality in different places. These bills exhibit how many persons upon an average out of a certain number born are left at the end of each year, to the extremity of life. From such tables the probability of the continuance of a life of any proposed age is known. 586 84 bability that he will die in that time. See MORTALITY, bills of, &c. Those who would enter more at large into this subject may be referred to the works already mentioned, or to the article CHANCES in the new Cyclopædia of Dr. Rees, a work that will be found in every library of general literature, and in which this subject is treated with great ability. Though we shall under the article GAMING refer again to the doctrine of chances, it may not be amiss to mention a deduction or two, drawn by the writer of the article just referred to, as the necessary consequences of mathematical reasoning. The first is suppose a lottery consisting of 25,000 tickets, of which 20 are to be prizes of 1000/. and upwards; a person, to have an equal chance of one of those prizes, must purchase about 870 tickets, which at 201. each is equal to 17,400Z. Again: suppose there are three prizes of 20,000l. and three of 10,000l. and a person out of 25,000 tickets has purchased 3000 of them to his own share, in hopes of gaining one of each of these capital prizes; still the chances against such an expectation will be nearly twelve to one. See GAMING. CHANCE medley, in law, is the accidental killing of a man not altogether without the killer's fault, though without any evil intention; and is where one is doing a lawful act, and a person is killed thereby, for, if the act be unlawful, it is felony. The difference betwixt chancemedley and manslaughter is this: if a person cast a stone, which happens to hit one, and he dies; or if a workman, in throwing down rubbish from a house, after warning to take care, kill a person, it is chance-medley, and misadventure: but if a person throws stones on the highway, where people usually pass: or a workman throws down rubbish from a house, in cities and towns where people are continually passing; or if a man whips his horse in the street, to make him gallop, and the horse runs over a child and kills it, it is manslaughter; but if another whips the horse, it is manslaughter in him, and chance-medley in the rider. In chance-medley the of fender forfeits his goods, but has a pardon of course. CHANCELLOR, an officer supposed originally to have been a notary or scribe under the emperors, and named cancellarius, because he sat behind a lattice, called in Latin cancellus, to avoid being crowded by the people. CHANCELLOR, Lord High, of Great Britain, or Lord Keeper of the Great Seal, is the highest honour of the long robe, being made so per truditionem magni sigilli. per dominum regem, and by taking the oaths: he is the first person of the realm next after the king and princes of the blood in all civil affairs; and is the chief administrator of justice next the sovereign, being the judge of the court of chancery. All other justices are tied to the strict rules of law in their judgment; but the chancellor is invested with the king's absolute power to moderate the written law, governing his judgment purely by the law of nature and conscience, and ordering all things according to equity and justice. The Lord Chancellor not only keeps the King's great seal; but also all patents, commissions, warrants, &c. from the King, are, before they are signed, perused by him; he has the disposition of all ecclesiastical benefices in the gift of the crown under 207. a year in the king's books; and he is speaker of the House of Lords. To him belongs the appointment of all justices of the peace throughout the kingdom. |