CHANCES. 



. anonytnmu tract d-i the Laws of Chance, published 



' at London in l< 



'.'iff celebrated James B.-rnoulli is the next wri- 

 ter whew labours require \n In- particularly noticed. 

 H* began by proposing the following problem in tin- 

 r 1<>90 : " Two persons, A 



and IS, pi.iy with a die ; the condition is, that who- 

 ever gets a cert. iin imuihrr of points first wins the. 

 game. A begins by throwing the die oiwre ; then B 

 throws it once. A next throws the die twice ; af- 

 terwards B throws it twice ; then A three times, and 

 B three times ; and so on. What is the ratio of their 

 respective chances of winning " The problem re- 

 mained without an answer, until its ingenious author 

 gave one in the Lcipsic Acts for 1690. This called 

 the attention of Leibnitz to, the same subject, who 

 gave also a solution in the same Acts. Bernoulli was 

 now preparing his Ars Conjectandi, a work which, 

 besides such queiti.ms a-; were agitated by Pascal and 

 Fermat, contains a multitude of others, increasing in 

 difficulty. He has also attempted to apply his theory 

 to moral and political events. He died, however, be- 

 fore he could give his labours the degree of perfec- 

 tion he wished for ; and they did not appear until 

 the year 1715, when they were published by his ne- 

 phew Nicolas Bernoulli, who had himself treated of 

 the same theory in the Leipsic Acts for 1711, in a 

 memoir called Specimina artis conjectandi ad ques- 

 tioncs juris applicatas. 



5. In the interval between the completion of James 

 Bernoulli's discoveries and their publication, the 

 theory of chances was handled by two excellent ma- 

 thematicians, Mot'tmort and DC Moivre. The first 

 of these turned his attention to the subject, with a 

 view to compensate for the loss of Bernoulli's labours, 

 in the event of their never being published. De 

 Moivre began by communicating to the Royal So- 

 ciety, in 171 1, a memoir entitled, De Men^ura sorlis. 

 He- afterwards published, in 1716, his Doctrine of 

 Chances^ a work justly regarded as one of the moat 

 valuable tliat has ever appeared on the subject. The 

 best edition is the third, printed in 17.56, together 

 with his Treatise of Annuities on Lives, His Mfa- 

 celUmea Analytica. also contains some disquisitions 

 on ihe same subject. 



6. Mr Thomas Simpson has likewise treated of 

 this subject, in a work called the \aiure and Laws 

 of Chance, first published in 174-0. This treatise is 

 concise, and at the same time perspicuous, and, like 

 the ingenious author's other writings, is remarkable 

 for its originality. 



This cunou- branch of science has also been more 

 or less cultivated by most of the eminent mathemati- 

 cians of the last century; as by John Bernoulli,. Eu- 

 ler, Cramer, D'Alembert, Beguelin, &c. ; and at a 

 latter period by Condoreet, in his Essai stir I' appli- 

 cation de I' analyse aux decisions qui se donnent a la 

 pluralite ties roue. 



7. The application of the doctrine of chances to 

 questions connected with political economy, is by far 

 the most interesting branch of this theory. The sub- 

 ject of Life Annuities, in particular, is highly im- 

 portant. Van Hudden, and the celebrated pensiona- 

 ry of Holland, DC Witt, appear to have been the 

 earlkst who considered it ; and Sir William Petty al- 

 so turned htb att.ntion to it, but without any degree 

 of success pr b;jbly on account of his want of ma- 



TOL. V. PART II. 



thematical knowledge. Dr H alley wa< the firrt who 

 made auy considerable progress in itt improvement, 

 by constructing tablet of the probability of human 

 life, from a comparison of the bills of mortality of 

 Breslaw in Silesia. Dr Moivre carried on what Dr 

 I Iilley had begun ; and Simpson greatly contributed 

 t . the perfection of the theory. The labour* of 

 lion, the friend of !) Moivre, are also juttly en- 

 titled to notice ; in hn> Mni'n-tmtlical Repotitory, the 

 subject of annuities, aa well as the doctrine ol chan- 

 ces in general, are treated with great clearness, and 

 in a manner well auited to persons having but an or- 

 dinary share of mathematical knowledge. Indeed, 

 we have freely availed ourselves of his labours in the 

 compilation of the brief view we here give of the sub- 

 ject. 



We shall now explain some of the more useful 

 parts of this theory, and exemplify the mode of rea- 

 soning it requires by a series of problems. 



PROBLEM I. 



8. Suppose a circular piece of metal having two 

 opposite faces, the one white and the other black, is 

 thrown up, in order to see which of its faces will be 

 uppenrost after it has fallen to the ground. When, if 

 the white face be uppermost, a pereon is to be entit- 

 led to L.5,. or any other sum of money; it is requi- 

 red to determine, before the event, what chance or 

 probability that person has of receiving the L.5 ; and 

 what sum he may reasonably expect should be paid 

 to him, in consideration of resigning his chance to 

 another person ? 



SOLUTION. Since by supposition there is nothing 

 in the shape of the metal to determine one face to 

 come up rather than the other, there is an equal 

 chance for the appearance of either face ; or in other 

 words, there is one chance out of two for the' appear- 

 ance of the white face. Therefore, the probability 

 that it is uppermost may be expressed by the fraction 

 aj. And if any other person should be willing to 

 purchase this chance, the proprietor may reasonably 

 expect i of L.5 in consideration of his resigning 

 chance thereof. 



PROBLEM II. 



9. Suppose there are three cards, each of different 

 suits, viz. one heart, one diamond, and one club, laid 

 on a table with their faces downward, out of which, 

 if a person at one trial takes the heart, he is to be en- 

 titled to L.5, or any other sum of money. It is re- 

 quired to determine, before the event, what chance 

 or probability he has of winning and missing the said 

 L.5 ; and what sum he may reasonably expect to be 

 paid to him, in consideration of his resigning bis 

 chance to another. 



SOLUTION. Since it is supposed that there is no- 

 thing in the external appearance of the cards' to in- 

 duce the person to choose one rather than another ; 

 and since he is to have but one choice, it follows that 

 he has but one chance in three for obtaining the mo*- 

 ney : therefore the probability of his getting it may 

 be expressed by the fraction $-. Again, since two 

 cards remain after he has had his choice, either of 

 which may be the heart, there are two chances out 

 of three that he will miss it j and the probability 

 SA 



