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ACTUARIAL SOCIETY 
 EXAMINATIONS IN 1905 
 
 QUESTIONS AND SOLUTIONS 
 REPRINTED FROM RECENT ISSUES OF 
 
 THE AIHERICAN UdDEliWHiTEll 
 
 AND 
 
 THE FUNDAMENTAL 
 PRINCIPLES OF PROBABILITY 
 
 BY 
 
 Robert Henderson, B. A., F. I. A., F. A. S. 
 
 MCM VI 
 
 Thrift Publishing Company 
 141 Bkoadwat, New Yokk City 
 
 ^ OF THE 
 
 UNIVERSITY 
 
 OF 
 

 GEHtRI^^ 
 
 Copyright, 1906 
 
 BY THE 
 
 Thrift Publishing Company 
 
PREFACE 
 
 J^R. ROBERT HENDERSON, the author of 
 the solutions of Actuarial Society examination 
 questions of 1905, presented in this book, and of the 
 accompanying essay on Probability, was born May 
 24th, 1 87 1, at Russell, near Ottawa, Canada. He 
 was graduated from Toronto University, with the de- 
 gree of B. A. in 1891, and was a University Fellow in 
 mathematics for one year. In 1892, he went into the 
 Dominion Insurance Department, Ottawa, Canada, 
 under the Actuary of the Department, Mr. A. K. 
 Blackadar, F. I. A., F. A. S. He remained until 
 July, 1897, when he became associated with the Actu- 
 arial Department of the Equitable Life Assurance So- 
 ciety of the United States, of which Society he was 
 appointed Assistant Actuary in September, 1903. 
 
 In 1896, Mr. Henderson took his final examination 
 and became a Fellow of the Institute of Actuaries. He 
 was enrolled as an Associate of the Actuarial Society 
 in 1900, and was admitted on examination as a Fellow 
 in 1902. By appointment of the President of the 
 Actuarial Society, he was a member of the Committee 
 on Examination for the years 1903, 1904 and 1905, 
 the last year as Chairman. At the annual meeting of 
 the Society,May 18, 1905, he was elected a member of 
 the Council to serve for three years. To the 'Trans- 
 actions of the Actuarial Society, his principal contribu- 
 tions have been, "Frequency Curves and Moments'*, 
 
 167288 
 
May 19, 1904, Vol. VIII, pp. 30-42; and "Note on 
 Limit of Risk", May 18, 1905, Vol. IX, pp. 40-46. 
 Mr. Henderson has given the solutions of the 
 questions in the examination of 1905 of the Actuarial 
 Society in a clear, concise and logical manner. His 
 essay on " The Fundamental Principles of Proba- 
 bility" is logical rather than mathematical in charac- 
 ter and is written along the lines of the most recent 
 treatises. In the belief that these solutions and this 
 essay will be interesting and useful to many persons 
 they are put in permanent form in this book. The 
 utility of the book is enhanced by the fact that the left 
 hand pages are blank for the convenience of those who 
 desire to make additions, criticisms or annotations as to 
 the particular subjects under consideration. 
 
 The American Underwriter 
 
CONTENTS 
 
 PAGE 
 
 Introduction 9 
 
 Syllabus of Actuarial Society Examinations 15 
 
 Questions and Solutions — 
 
 I Associate, Section A iQ 
 
 II Associate, Section B 33 
 III Fellow 45 
 
 The Fundamental Principles of Probability — 
 
 I The Measurement of Probabilities 71 
 
 II The Combination of Probabilities 75 
 
 III Expectation, or Mean Values 85 
 
 IV Repeated Trials 89 
 
UNIVERSITY 
 
 OF 
 
 INTRODUCTION 
 
 rr^HE following solutions of the questions proposed in the 
 •^ Actuarial Society's examination of 1905 were origin- 
 ally prepared for publication in The American Underwriter 
 and they are now brought together in more permanent form in 
 the hope that they be of some assistance to students preparing 
 for the examinations of the Society. In this connection the 
 author considers it proper to point out, especially in view ot 
 his connection with the examination committee of that year, 
 that the solutions given are intended only as specimens and not 
 as the only appropriate answers to the questions. 
 
 Perhaps a few words as to the principles adopted in prepar- 
 ing these solutions may not be out of place. Where the ques- 
 tions referred to subjects taken up in text-books which were 
 likely to be in the hands of students and others a mere refer- 
 ence was in most cases considered sufficient. With regard to 
 questions in algebra — in particular — references are made to the 
 text-book in which the author could most readily verify the ref- 
 erences and to which he himself most frequently refers. The 
 subjects of these questions will, however, be found taken up in 
 any standard treatise on the subject and the student will refer 
 to the particular treatise most convenient to himself. 
 
 In certain cases where an alternative treatment of a subject 
 taken up in the text-books seemed worthy of notice it has been 
 incorporated in the solution. 
 
In the discussion of questions of a general nature the en- 
 deavor has been, so far as possible, to look at the questions from 
 a fresh viewpoint. In all cases the answers have been made as 
 concise as possible and such as might be written by a candidate 
 working under a time limit. 
 
 • A copy is included of the syllabus under which the examina- 
 tion was held. It is to be noted, however, that a change has 
 been made in this syllabus which is intended to go into effect 
 at the spring examinations in 1907. This change consists in 
 adding to the final examination the subjects of Life Insurance 
 Bookkeeping, Investments and Banking and Finance, and div- 
 iding the examination into two sections which may be taken in 
 different years. 
 
 The opportunity is taken in collecting these solutions in book 
 form to append a brief exposition of the fundamental principles 
 of the theory of probability which was suggested by the two 
 questions on that subject appearing in the examination papers. 
 This essay is intended to show the connection of the mathemat- 
 ical theory of the subject with the logical theory as developed 
 by the best modern writers on that subject. 
 
 Robert Henderson 
 
 11 
 
ACTUARIAL SOCIETY 
 EXAMINATIONS IN 1905 
 
SYLLABUS OF EXAMINATIONS 
 
 Associate 
 
 Section A 
 
 1. Arithmetic, elementary Algebra and the principles of 
 double-entry Bookkeeping. 
 
 2. The following subjects in advanced Algebra: 
 
 a. Permutations and Combinations. 
 
 b. Binomial Theorem. 
 
 c. Series. 
 
 d. Theory and use of Logarithms. 
 
 e. The elements of Finite Differences, including in- 
 terpolation and Summation. 
 
 3. The elements of the Theory of Probabilities. 
 
 4. Compound Interest and Annuities Certain. 
 
 5. Elementary plane Geometry.* 
 
 6. Practical examples in the foregoing subjects. 
 Section B 
 
 1. The application of the Theory of Probabilities to Life 
 Contingencies. 
 
 2. Theory of Annuities and Assurances, including the theoiy 
 and use of Commutation Tables and the computation of prem- 
 iums for usual contract. 
 
 3. Valuation of ordinary forms of policies. 
 
 4. Practical examples on all the above, involving Joint as 
 well as Single lives. 
 
 15 
 
5. General nature of Insurance contracts. 
 
 6. The outines of the history of Life Insurance. 
 
 7. The source and characteristics of the principal Mortality^ 
 Tables. 
 
 Fellow 
 
 1. Methods of constructing and graduating Mortality Tables 
 and the use of the formulas of Gompertz and Makeham. 
 
 2. Methods of loading premiums to provide for expenses and 
 contingencies. 
 
 3. Valuation of the liabilities and assets of Life Insurance 
 companies. 
 
 4. The assesment of expenses and the distribution of surplus. 
 
 5. Practical treatment of cases of alteration or surrender 
 of Life Insurance contracts. 
 
 6. Application of the Calculus of Finite Differences and of 
 the Differential and Integral Calculus to Life Contingencies. 
 
 7. Laws of the United States and Canada relating to Life 
 Insurance. 
 
 8. Insurance of Under-Average Lives and extra premiums 
 for Special Hazards. 
 
 *Note for Canadian candidates. Euclid Books I, II and III will be consid- 
 ered sufficient for this subject. 
 
 17 
 
ERRATA. 
 
 Page 23, Question 6, line 4 — Insert 8 after word 
 
 "computing-." 
 Page 25, Question 8, line 2 — Read "annual " in place 
 of "anual." 
 Question 10, line 4 — read numerator "2401." 
 Page 29, Solution 13 (a), line 4 — Make "j,-, read as 
 index. 
 Solution 13 (a), line 5 — Make " n" curtate. 
 Page 31, Solution 14, next line to bottom — Insert 
 
 "angle" before "P AB\" 
 Page 33, Solution 1, line 7 — Read "4 + i" in place of 
 
 "/ 1." 
 Page 41, Solution 10, line 2, — Read ".02186 X '' before 
 the fraction in place of ".02186 +•" 
 Question 11, line 1 — Read A\y^. 
 Solution 12, line 4 — Read "M./' in place of 
 
 "M.r." 
 
 Page 43, Solution 14 (a), line 3 — Read "sixty" in place 
 of "twenty-three." 
 
 Page 45, Solution 1, line 11 — Insert "by " after 
 
 " existing." 
 Page 53, Solution 11, line 7 — Read "(1 +/) " in place 
 
 of "(^+y)." 
 Page 57, Solution 15, line 7 — Read "?/i" in place of 
 
 Page 61, Question 22 (b), line 2 -- Read "4" in place 
 
 of "/^." 
 Page 91, Second paragraph, line 6 — Read " ;/ -|- 1 " in 
 
 place of " n — 1." 
 
ACTUARIAL SOCIETY EXAMINATIONS 
 
 Solutions of Questions in Examination for Admission as Asso- 
 ciate OR Fellow, Held April 13 and 14, 1905 
 
 N. B. — Actuarial Society or American notation 
 
 Associate— Section A 
 
 V I. (a) Simplify ^ ~ ^ + tit 
 
 (d) Extract the square root of 1,012,766,976. 
 
 {a) I. II. 
 
 {b) 31,824. 
 
 2. (a) Prove that a quadratic equation can have only two roots, 
 
 (b) Solve the equation. 
 
 ^8 U — b) jx — c) jx — c) (x — a) ^ 
 
 (a — b){a — c)'^ (b- c) {b — a) 
 
 (a) Let a and ^ be two roots of the equation, then since x — a and 
 X — /J are factors the equation must take the form a {x — a) 
 {x — ^) = o. If, then, any other value :v of x satisfies this equation 
 we have a {y — oi) {y — /?) = o which is impossible unless a = o/\n 
 which case the equation would vanish identically. Therefore, the 
 equation cannot have more than two roots. 
 
 (b) Substituting a for x in the equation it becomes 
 
 a^ \--l \--^\ ^o = a^ 
 {a — b) {a — c) 
 
 Since this is an identity a satisfies the equation, similarly b 
 satisfies it and the equation does not vanish identically since x = c 
 does not satisfy it. As the equation is of the second degree and does 
 not vanish identically it has only two roots and these we have seen 
 to be a and b. 
 
 19 
 
/ 3. (a) Given that the Binominal Theorem is true when the index 
 is a positive integer, prove that it is also true for negative or frac- 
 tional indices. 
 
 (d) Expand (i — xT"^ . 
 (a) Eor proof of first part of question refer to C. Smith's Algebra, 
 2d Edition, Article 283. 
 (d) By the Binomial Theorem. 
 
 I 1.2 1.2.3 
 
 + etc. 
 = I + 2 a; + 3 ic2 + 4 :j;3 _f. g^c. 
 
 / 4. Expand e'^ in ascending powers of x. 
 
 Hence, where Cq, Ci, etc., are the coefficients of the various powers 
 of x in (i + xf prove that Cq (a + w)*" — c\ (a + n — if^ -\-c* 
 (a + n — 2)*" — etc., vanishes if m is less than n and is equal to \n if 
 m is equal to n {m and n being both integral). 
 
 e* = \ +^+-r- + -i ^- etc. 
 
 12 |3 
 
 For proof refer to C. Smith's Algebra, 2d Edition, Article 302. 
 
 ^0 ia + n^ -c^{a + n — if + ^^ (a + « — 2)^ — etc., is \m 
 times the coefficient of x^ in the expansion of Cq e (a + «)* _ 
 ^^^(a + «-l)^ + ^2 ^(^ + "-2^^- etc. 
 
 or. of^^^(^^-i)«. 
 
 But (^* — I) = i*; + 1^ + etc. 
 
 .'. (^* _ i)« = ^« + ^ ^« + i + etc. 
 ^ 2 
 
 and e^^ = i + a x + etc. 
 
 .•.e^'={e''-ir = x-^ + {a +^)^« + i + etc. 
 
 So that the coefficient of all powers of x less than n is zero, and 
 that of x^ is unity. Hence the proposition follows. 
 
 5. How are probabilities measured? Give instances where addi- 
 tion, subtraction, multiplication and division of probabilities, re- 
 spectively, give inter pre table results, stating in each case the relation 
 of the resulting probability to those operated upon. 
 
 21 
 
The measure of the probability of an event happening under given 
 circumstances is the limit of the ratio of the number of times the 
 event happens under the given circumstances, to the number of 
 times the given circumstances occur, when the number of cases 
 is indefinitely increased. The probabilities of two mutually exclu- 
 sive events may be added together, the result giving the probability 
 that either one or the other will happen. If two events be so related 
 that the first cannot happen unless the second does, the probability 
 of the first may be subtracted from that of the second, the result 
 giving the probability that the second will happen unaccompanied by 
 the first. The probability of an event may be multiplied by the 
 chance that if it happens a second event will happen also, the result 
 being the probability that both will happen. Conversely the proba- 
 bility of a compound event may be divided by the probability of one 
 of its components, the result being the probability that if that event 
 happens the other will happen also. 
 
 6. // i is the effective annual rate of interest, j the nominal rate 
 convertible m times a year, d the rate of discount and 8 the force of 
 interest; express each in a series of ascending powers of each of the 
 others. Give an approximate method of computing d* for a given 
 rate of interest. 
 
 6. If i is the effective annual rate of interest, / the nominal rate 
 convertible m times a year, d the rate of discount and S the force of 
 interest ; express each in a series of ascending powers of each of the 
 others. Give an approximate method of computing d for a gi-en 
 rate of interest. 
 
 See Institute of Actuaries' Text Book, Part i, new edition, Chap- 
 ter I, Article 30, 
 
 From the expressions for i and d in terms of 8 we have 
 
 i + d = 26 -\ h etc. 
 
 or approximately d = — - — 
 
 also id= 6' + — 5* + etc. 
 
 so that approximately S = V i d. 
 
 7. Having given the value of an annuity certain for a term of 
 years, deduce a formula for fiiiding the approximate rate of interest. 
 
 Suppose a is the value of the annuity and « the period, and let i 
 be the rate of interest. Let also a^, near to a, be the value at some 
 
 1 1 • .u 1 « — ^1 d a^ 
 
 known rate Zi, then we have approximately 
 
 di-i 
 
 but ix «! = I — (I + ^i) 
 
 .-.ai ^i^^= «(i + z\)-" + ^= nv^^"^' 
 
 . «+ I 
 
 a a, ai — n Vi 
 
 or -r^ = — — 
 
 ^H i^ 
 
 «i — ^ . . 
 
 whence i — t^ = t ^ -~~ or i = t^ + t\~ 
 
 a, — n v^^ ~^^ a, 
 
 23 
 
 /0 
 
 ( UNI 
 
 ^ OF The 
 
 UNlVERSi 
 
 OF 
 
 y 
 
/ 
 
If desired this rate may be used as a basis for a further approxi- 
 mation. 
 
 8. Define Revenue Account, Balance Sheet and Inventory. 
 
 What items in the anual statement of a life insurance company 
 are ordinarily the result of an inventory? 
 
 The Revenue Account shows on one side the amount of the policy 
 reserve and the surplus at the beginning of the year and the income 
 earned during the year, and on the other side the expenditure in- 
 curred during the year, the balance being the amount of the policy 
 reserve and surplus at the end of the year. 
 
 The Balance Sheet is made up of the balance of the other ac- 
 counts showing, on one side, the capital stock, if any, the policy 
 reserve, the liabilities outstanding on the various accounts and the 
 surplus, and on the other side the assets on hand 
 
 An Inventory is a computation of the total amount of a given kind 
 of assets or liabilities made by valuing the individual items. Interest 
 and rents due and accrued and market value of bonds and stocks 
 over book value are ordinarily determined by inventory. The lia- 
 bility for reserve may also be considered as so determined. 
 
 *- 9. Fhid the value of ^C^, the number of combinations of n 
 different things taken r at a time. 
 
 Prove that, if x and y be any two positive integers, then will 
 x-4-yC„ = ^C„ + x^n-^^ y^ ^ + «C„_, yC^ + + yC„. 
 
 See C. Smith's Algebra, 2d Edition, Articles 244 and 248. 
 ^ 10. Given log^o2 = .30103 afid log^oS = .47712, calculate th€ 
 logarithms of all numbers from 4 to 10 inclusive, using, where 
 necessary, a finite difference formula. 
 From these values calculate log^ „ f lU • 
 We have 
 
 log 4 = 2 log 2 = .60206 
 
 Jog 5 = log 10 — log 2 = I — .30103 = .69897 
 
 log 6 = log 2 -I- log 3 = .77815 
 
 log 8=3 log 2 = .90309 
 
 jog 9 = 2 log 3 = .95424 
 
 log 10 = 1. 00000 
 
 To determine log 7 we have 
 
 log 45 = log 5 + log 9 = 1. 65321. 
 
 log 48 = log6 + log 8 = 1. 68124 
 
 log 50 = log 5 + log 10 = 1.69897 '^ 
 
 log 54 = log6 + log 9 = 1.73239 
 Whence interpolating by Lagrange's formula- 
 log 49= _ Vlog45 + flog 48 4^ ^ log 56 -sV log 54 = 1.69020 I 
 
 log 7 = i log 49 = .84510 \ 
 
 Also we have log \\^\ = 4 log 7 _ log 3 — 3 log 2 — 2 = .00019 ^ 
 
 11. (a) Define central differences. 
 
 Kb) Expand « + « x-n and x^n x — n jn terms of u 
 22 * 
 
 and its central differences . 
 
 25 
 
(a) Central differences are the differences of a function re- 
 ferred to the central value of the variable. As there is no value 
 of the odd orders of differences corresponding to the values of the 
 original function the mean of the two adjacent values is used. 
 
 (d) We have, 
 u , = u + na + n^ . , n {n* — i) , «' («* — i) 
 
 '+" ' ' K ' ~]i — "' ~\1 — 
 
 dx + etc. 
 
 x — n X *|2 |_3_ * 
 
 - «'(''' -'"> d^ + etc. 
 
 I_4_ 
 
 where a^, d^, c^, d^, etc are the successive central differences 
 corresponding to u^ 
 
 Whence "'+>• + "-" = «,+ £!«, + "' '"• - " d^ + etc. 
 
 2 * J2 "^ l_4_ 
 
 2 /J_ 
 
 12. (fl) r/i^ probability of the happening of a certain event is 
 p, and the probability of its failure is q; what is the probability of 
 its happening exactly r times in n trials? 
 
 (b) for what value of ris this probability the greatest? 
 
 (c) Does this maximum probability increase or decrease as n 
 increases ? 
 
 (a) The chance of the event happening on any particular r trials 
 and failing on the remainder n — r is ^'' q"~^- But the number of 
 
 % 
 
 different combinations of r trials is -; ; so that the total 
 
 I r \n — r 
 
 \n_ 
 
 probability is ; >, »- n^—^ 
 
 I r \n — r ^ ^ 
 
 (b) This value is the greatest for the largest value of r for which 
 
 \n \n ^ , , 
 
 I r \n — r \ r— i \n — r 4- i 
 
 {n — r + i) p> rq, or r{p ■\- q)<{n ■\- i) p 
 
 or, since p ■}- q = i, when r is the integral part of (n + i) p. 
 
 27 
 
(c) Putting now (« + i) for n the maximum probability is either 
 
 |« -f I . , \n + I ■ 
 
 P'' g"-^ + ^or . . _ , p^'+^g*'-' the 
 
 \r \n — r + I | r -»- i | n — r 
 
 ratio of which to the preceding is either ^ or -^ -^ 
 
 n — r + I r + 1 
 
 both of which are less than unity since (« + i) P lies between r and 
 
 r -\- I and consequently {n -\- i) q between n — r and n — r -f- i. 
 
 / 13. (o) What is the value of an annuity certain for n years, 
 
 payable p times per annum at the nominal rate j convertible m times 
 a year. 
 
 ^ (b) An annual annuity running n years is worth twenty-five 
 year's purchase; one running 2n years is worth thirty years' pur- 
 chase. What is the rate of interest f 
 
 (a) We have where i is the effective rate of interest per annum ; 
 (p) I . X i ^^ I — e/» 
 
 "' ^ /[(i + O'-iJ 
 
 but I + z = f I + -^ \ , so that 
 
 
 '[(■*i)7-J 
 
 (w) I — Z/*" 
 
 Where/ is equal to m, this becomes a — - = 
 
 n J 
 
 (b) U a — \= — : — = 25 
 
 u\ J 
 
 and 
 
 2MJ 
 
 = ^- 
 
 y 
 
 ^n 
 
 = 30, 
 
 
 
 
 
 
 a 
 
 have - 
 
 2«| 
 
 
 I — 
 
 ^^n 
 
 I 
 
 + 
 
 2/" 
 
 6 
 
 
 (//O 
 
 
 I - 
 
 - 2/" 
 
 5 
 
 n 
 
 4 . C^') 
 
 therefore i — z/« = - = 25/, since rt — = 25 
 
 therefore/ = -^, or the nominal rate of interest, convertible with the same 
 frequency as the annuity is payable, is 3^ per cent. 
 
 29 
 
y lA. On the side A B, produced if necessary^ of a triangle 
 A B C, A C is taken equal to A C] similarly <?« A C, A B is taken 
 tqual to A B, and the line B^ C is drawn to cut B C in P. Prove 
 that the line A P bisects the angle BAG. 
 
 A 
 
 Let the construction be made as described and suppose 
 C is ia A B produced and consequently B in A C. 
 
 Then in the triangles A B C and A B ^C^ since A B=A ^' 
 A C= A C^ and angle B A C= angle ^' A c/ the triangles 
 are equal in every respect, or angle A B C wangle ABC, 
 and angle A C B=angle A c/ B'' 
 
 Also since angles CT B P and C B P are supplementary 
 to the angles ABC and ABC they are equal. 
 
 Also since A B = A B and A C = A C therefore 
 B C^ = B^ C. 
 
 Then in the triangles P B C^and P B^ C, B C^ = B'^ C 
 angle P B C^= angle P B^ C and angle P d B-=angIe P C B/ 
 therefore P B = P B^ 
 
 Then in the triangles P A B and P A b{ P B -= P B^ 
 A B = A B^ *nd P A is common therefore angle P A B «. P A B^ 
 or P A bisects the angle BAG. 
 
 31 
 
Associate— Section B 
 
 y 1. What does the mortality table represent, and of what columns 
 does it usually consist? 
 Express in terms of the mortality table the probabilities: 
 
 (a) That one of the lives (x) and (y), will survive n years and 
 the other will fail within n years. 
 
 (b) That at least one of the lives will fail in n years. 
 (c)l That both will survive n years. 
 
 (d) That both will die within n years. 
 
 (e) That the first death will happen in the n^h year from the 
 present time. 
 
 It represents the history as regards mortality of a representative 
 group of people, and indicates the proportion which dies in each year 
 of age and the proportion which survives to the end of each year. 
 The mortality table proper consists of the column of /^ showing the 
 proportion of those attaining the initial age which survives to each sub- 
 sequent age, and ths column of d^ showing the proportion of the same 
 group which dies in each year, so that necessarily d^ =/^ — / ^+l 
 
 (a) ^p^(i-,Py) "r ^Py{i-^p^) 
 
 = nPx + nPy — ^nPx • nPy 
 
 ^ x-\-n ^ y+n ^x-\-n: ^y+n 
 
 ° I, ^ ly ~^ '«•', 
 
 ^ r-^n : ^y-\-n 
 
 (*> ^-nPx:nPy=^ ^- 1 } 
 
 *« • ''y 
 
 ^x+n : ^ y-^fi 
 
 <^) nPx:nPy= ^ ^ 
 
 ilx— ^x-f «^ ^^y- ^y+n^ 
 
 ''X : ''y 
 
 ^x + ti ^y-\-n ^x±n^Jy±n 
 
 = I — — + ; ; 
 
 l^ ly Ix : ^ 
 
 (^) n-lPx.n-lPy — ftPx- nPy 
 
 _ ^x + n — l-^y-\-n — l ~ ^ x+ n : ^ y -{- n 
 
 33 
 
2. Express the value of a^ m terms of the mortaltty table. 
 Prove that a^ < ar^\ . 
 
 If annuities of a unit be issued on each of l^ lives aged x the total 
 amount to be paid at the end of the first year is /<p^i, at the end the 
 second year l^j^^y and so on. So that equating present values we have 
 /;, fl^ = z/ /,^_i + z/^ /^ + 2 + etc. 
 
 t' /- . 1 + z/^ ^x + 2 + etc. 
 or a ^ = — t-L± 
 
 I, 
 
 We know that A^ > z/ ^* For the former is the arithmetic 
 
 and the latter is the geometric mean of the present values of the 
 death claims. Therefore 
 
 I — (I + /) A_ I _ z, ^* . 
 a^ = -, ? < i ^ — ; that is < a^-j. 
 
 3. On De Moivre's hypothesis find the value of a life annuity in 
 terms of the expectation of life. 
 
 Does this formula give results sufficiently close for practical useT 
 On De Moivre's hypothesis, if n is the complement of life, we 
 
 o n 
 have, since the decrements are supposed uniform, ^x~~o 
 
 Also, A, = -^ so that 
 
 I - A^ n-a~\ 2^^ — a,/^| 
 
 d net ^1 ^ 
 
 Owing to the lack of conformity of mortality tables with 
 De Moivre's hypothesis this formula does not give results suffi- 
 ciently close for practical purposes. 
 
 4. Deduce formulas for the net single and annual premiums for a 
 survivorship annuity payable to (x) after the death of (y). 
 
 The payment under the survivorship annuity will be due at the end 
 of any year if (x) be alive and (y) be dead. The net single premium 
 for the benefit is therefore, 
 
 - ^^''uPx-^^"" nP. ■ nPy 
 
 = a — a, 
 
 X X y 
 
 The annual premium is ordinarily payable during the joint lives so 
 that denoting its value by n we have 
 
 or ?r = = — I . 
 
 35 
 
5. Having given the values of D^, N^ and S^, show how the 
 values of C^, M^ and Rj. may be ascertained. 
 
 Find the annual premium for whole life insurance, all premiums 
 paid to be returned at death, assuming that the premium is loaded by 
 a percentage and a constant. 
 
 = D.-d N. 
 
 M. = ^C, = 7;:S D.-2D,^, = t,N^-N,+ 
 
 X 
 
 Let TTj = office premium required and tc = net premium where 
 
 7t^={l -\- k)7C + c 
 
 Then ;r N^ = AI^ + R^l (I + k)7C +c^ 
 
 or TT I N^ - (I + ^; R^ I =^ M^ + c R, 
 M. + c R, - 
 
 ^ .r a. fc^ ^ ^ (I + ^) M, + c N, ^' + ^^ n; + ^ 
 ''^i = (I + k) n + c = = — 
 
 N,-(i+fe)R, • f,^u,^^ 
 
 (I + ^) P^ + c 
 
 K 
 
 i-(i +^)j^ 
 
 6. Investigate the relation between the rates of mortality in two 
 tables giving the same net premium reserves on ordinary life policies. 
 What conclusion would you draw regarding the effect on such re- 
 serves of an increase in the rate of interest? 
 
 We have for all values of n 
 
 |n-iV^+P^ [(I + V) = P^^n_, „ V, +^^ + „_,- 
 
 n^ X +^;. + n-i(l-nV^). 
 Suppose any other table of mortality gives the same terminal re- 
 serve values and denote net premiums and rates of mortality according 
 to this second table by accented letters then we have 
 
 ]n-:V,+Pi [(I +^-)=„V,+9i + „_,(l-„V,). 
 
 Subtracting we get (P^- P^) (i 4- *) = (5i + „_ , - ^i^ „ _ j) 
 or Pi-PJ(i +i)(i +a^) = (5i^„_,-g, + „_,) 
 
 (I +«;c + n). 
 
 37 
 
for all values of n so that 
 
 (^i — g^) (i + ax + i) must be a constant 
 
 k 
 or q^. can be expressed in the form q^ + — T — 
 
 X -f- I 
 
 An increase in the rate of interest is equivalent so far as annuity- 
 values are concerned to a proportionate reduction of the values 
 of pj. or, for adult lives, a decreasing addition to the value of q^ 
 whereas in order to give the same reserves an increasing addition 
 to q^ is required, consequently we should expect that an increase in 
 the rate of interest would in general lower the reserves. 
 
 7. State the leading points of difference between policies now 
 issued in America and those issued fifty years ago. 
 
 Fifty years ago policies were issued on comparatively few forms, 
 principally ordinary life, the many special forms which are now such 
 a feature of the business being then unknown. The many privileges 
 and options as to surrender values, dividends, etc., now granted, 
 were then absent from the contract which on the other hand con- 
 tained many restrictions and prohibitions relating to travel, occupa- 
 tion, and residence. Deferred dividend policies were first issued in 
 1S68. 
 
 8. Fmd the values ofQly] Q J. . y^ff) and e^ . y^) . 
 
 Among those cases where (x) actually survives (y), what is the 
 average duration of the period of survival f 
 
 See Institute of Actuaries' Text Book, Part 2, Chapter 4, Articles 
 3, 13 and 16. 
 
 In a large number N of cases the number where (a:) survives {y) 
 
 is N QJ. and the total number of years included in all the periods of 
 such survival is N e^ , , = N {e^ — e^^. Therefore, the average 
 
 penod IS — p— . 
 
 9. Frotn general reasoning find the value of A^ i?i terms of a^. 
 
 Express also its value in terms of the mortality table and in com- 
 mutation symbols. 
 
 See Institute of Actuaries' Text Book, Part II, Chapter VII, Arti- 
 cles 32, 33. 41, 42, 43 and 44. 
 
 10. Give formulas for the ordinary life, limited payment life and 
 endoii'ment assurance annual premium rates in terms of annuity 
 values and also in terms 'jf commutation columns. 
 
 Given P^. = .02186, P^-| = .0^20 and i = .03; find ^ ^. 
 
 See Institute of Actuaries' Text Book, Part II, Chapter VII, Arti- 
 cles 61, 69, 73, 79 and 80. 
 
 30 
 
 OF THE 
 
 UNIVERSITY 
 
We have „P, = ^::=^ = A, (P "j -^ d) = (P^j +rf) 
 
 I + ax:n—l\ ^x '^ "■ 
 
 (I + i) P-1 + « 1-03 X .04220 + .03 
 = P = .02186 % TT- 
 
 ^ (I + /) P^ + « '^ 1.03 X .02186 + .03 
 
 .0734660 
 
 = .02186 X 1 = .03058 
 
 .0525158 
 
 11. Find an expression for A ' and show how to express in 
 
 terms of simpler functions the values of A ^ A ^^^ and AJ — 
 
 See Institute of Actuaries Text Book, Part II, Chapter XIII, Ar- 
 ticles 24, 26 and 27. 
 
 12. Write down formulae for the value of a limited payment life 
 policy by the prospective method and by the retrospective method, 
 explaining each. State under what conditions they give identical 
 results. 
 
 Prospectively „V = =; where m is. 
 
 the premium period. 
 
 • , w '^.(Nx-N:, + „)-(M,-M,+,) 
 Retrospectively „V = — 
 
 The difference between these two values is ■ 
 
 Dx+« 
 
 which vanishes if 7t^ = -tt — >^ , that is, if the premium 
 
 X X -f m 
 
 valued is the net premium for the policy on the basis of valuation. 
 Otherwise, the two will give different results, unless the retrospec- 
 tive formula be properly modified. 
 
 13. State as briefly as possible the main points to be covered in 
 drawing up a policy of insurance on the limitedpayment life plan. 
 
 The special points to be covered in drafting a limited-payment life 
 polic3% as distinguished from other forms, are that the face of the 
 policy is payable on receipt of satisfactory proofs of death of the 
 assured and that the assurance is conditional on the payment of the 
 specified premium annually in advance during a specified period or 
 until the prior death of the assured. These are the distinguishing 
 features of a limited payment life policy, the other privileges and 
 conditions being in general common to this and other forms. The 
 conditions embody such restrictions and limitations regarding oc- 
 
 41 
 
cupation, travel, residence, suicide, and proof of age, as may be 
 considered advisable, and usually incorporate by reference the ap- 
 plication as part of the contract. The privileges refer to such 
 features as incontestability, grace in payment of premiums, rein- 
 statement, change of beneficiary, loans and surrender values. 
 
 14. (a) What are the leading features which distinguish the O^ 
 table from the Actuaries' table? 
 
 (b) What was found to be the experience of policies with and 
 without profits respectively and of endowments as compared with 
 whole-life policies f 
 
 (o) As regards source the O^ table is distinguished by being 
 the result of the recent experience of. a large and homogeneous class 
 of risks, being the experience of twonfirfthrcc British offices during 
 the years 1863 to 1893 on male lives assured under ordinary life 
 policies, while the Actuaries' table is based on the whole experience 
 of seventeen offices up to 1839. As regards methods, the O** table 
 was made up on lives, while the Actuaries' table was made up on 
 policies, except that in some companies duplicates in the same 
 company were eliminated. As regards results, the O^ table shows a 
 relatively very low rate of mortality at the younger ages as com- 
 pared with the Actuaries' table. The difference in rates of mortality 
 becomes relatively very small as the age advances. The expectation 
 by the O^ table is about five per cent, higher at age 20 than by the 
 Actuaries table. At age 35 this difference is reduced to about two 
 and one half per cent, and remains at that figure beyond age 70. 
 
 (b) In the British offices experience, the mortality on lives as- 
 sured with profits was much lighter than on those assured without 
 profits, and the experience on endowments was much more favor- 
 able than on whole life policies. 
 
 43 
 
Fellow- 
 
 I. Draft a memorandum of instructions to be followed by 
 clerks in taking out the mortality experience of a company, the result 
 to be exhibited in the form of select tables. 
 
 From proper record enter on cards the age at entry, date of issue 
 and date and mode of termination of risk (if terminated). Include 
 only terminations occurring before the anniversary in the year 1904. 
 In the case of terminations otherwise than by death, record, as the 
 duration, the difference between the calendar years of entry and 
 exit. In the case of deaths add one year to the number of com- 
 plete policy years which have elapsed between date of issue and 
 date of death. Take the age at entry at nearest birthday. Sort cards 
 into deaths, other terminations (withdrawals) and existing, then 
 sort each lot by age at entry. Sort the deaths and withdrawals at 
 each age according to duration, and the existin^calendar year of 
 issue (the duration being the difference between that year and the 
 year of termination of observations). Record on a schedule for each 
 age at entry the total number for each duration, separately for the 
 deaths, withdrawals and existing. (The cards should contain further 
 information such as kind of policy, residence, etc., if required for the 
 purposes of subdivision). 
 
 2. On the assumption of Makeham's law, find the value of A^^ in 
 terms of the constants and A^y Hence give approximately the 
 value of A J.y in terms of A ^^,. 
 
 We have by Makeham's law, 
 
 /' 
 
 + t 
 
 A + Bc* + * 
 and>Uy^^= A + B(7^ + ^ 
 
 where A = — log^ s. 
 .-. c^i^^j^t + log^s) = c'i^y^t + log^s) 
 multiplying then through by 
 v^ f p^y and integrating with respect to i, we get 
 
 ovcyAiy = c'A^; + {c^-cy)iog^s'^^y 
 
 or ic' + c:>')Ai,, = c^' A^^ + (c' - cy) log^ s'a ^ ^ 
 = c-A^y +{c^-cy)^fll^ii-A^y) 
 
 45 
 
But a;^ = yAi^and A3.3, = -^ A ^^ approximately. 
 Hence substituting and reducing we have 
 
 3. ff'i^a/ various forms of investment are now open to life insur- 
 ance companies ? Discuss their relative desirability . 
 
 Railroad Bonds : They yield a fair return, are available in con- 
 siderable quantity, when well selected furnish ample security, and 
 are readily convertible. 
 
 Government and Mufiicipal Bonds : The interest return, except 
 for small issues, is comparatively low, but they are sometimes useful 
 for deposit purposes. 
 
 Mortgage Loans on improved property, with a good margin, are a 
 safe investment and yield a return somewhat higher than do first- 
 class bonds. 
 
 Stocks : Great care in selection is necessary. Some first-class 
 railroad, bank and trust companies stocks however, furnish a safe 
 and remunerative investment. 
 
 Loans on Policies : This is one of the most desirable forms of 
 investment, but only a limited proportion of the funds can be invested 
 in this way. 
 
 4. Show how the reserve on any policy at the end of n years can 
 be expressed in terms of the net premium, and the commutation 
 symbols for age at entry a7id age attained. Hence show how policies 
 of all kifids a7id durations may be grouped together by attained age 
 for valuation purposes. 
 
 Let „V be the reserve at the end of n years. S the sum assured 
 and re the net premium, then we have 
 
 N^-N^ + n ^ M^-M^+n 
 
 or„V=7r-— S. 
 
 ^x^n ^x-\-n 
 
 This expresses the reserve in terms of the net premium and com- 
 mutation symbols for age at entry and age attained. This equation 
 can also be expressed in the form 
 
 nV= S.A^_^„- ;r (I 4- a^^J + 
 
 ^x + n 
 
 47 
 
D, 
 
 = S. A,^^- «-(! + a,^J +6.- 
 
 TT N, — S. M, 
 where fl = — and z is some convenient advanced age. 
 
 z 
 
 If then the quantities S. ic and 9 be recorded once for all on each 
 policy a classification may be made by age attained and an account 
 kept of the totals of the three functions for each group. These can 
 be multiplied by the respective valuation functions A^^^^, i + ^x-\-n 
 
 and — and a summation will give the valuation required. If the 
 
 ^x+n 
 
 conditions of the policy are such that the value of S or of tt changes 
 after the policy is in force, a corresponding change in B will have to 
 be made; the new value of 6 being such that at the date when the 
 change takes effect the value on the new set of constants will be the 
 same as on the old. 
 
 5, Under what conditions is the declaration of a simple rever- 
 sionary bonus of a uniform percentage of the face of the policy for all 
 ages, durations, and kinds of policy , equitable f 
 
 Profits may for practical purposes be divided into two parts, one to 
 be apportioned in proportion to the reserve on the policy and the 
 other in proportion to the effective loading, so that the equitable 
 share of a policy may be expressed in the form 
 
 al-\-b.y=^al-\-b 
 
 I -A, 
 
 for ordinary life policies ; for endowments the corresponding 
 changes in the formula should be made. If the proposed method 
 is to be fair, this should be capable of expression in the form 
 ^ A^_l_„ where c is constant for all values of x and n that is 
 
 al-b ^ _ ^ + ^ __ ^ . A,^„ = c A, + „ for all values of x 
 
 ^x b ^x 
 
 and n. Hence a I — b = o ox I = — . —r or the effective 
 
 I — A^ ad 
 
 b 
 loading is a constant percentage of the net premium, also * = c. 
 
 I Ay 
 
 This last condition cannot be fulfilled as A^ is a variable and b and c 
 
 are constants unless the valuation basis is varied according to age 
 at issue and kind of policy so that b would vary also, being made 
 proportional to the difference between the rate of interest assumed 
 and that actually earned. 
 
 49 
 
6. Expand -7 in ascending powers of x. 
 
 (I \ xY — \ 
 
 Hence derive an approximate formula for the sunt of a series when 
 
 only every «<^ value is k7iown . 
 
 See Institute of Actuaries' Text Book, Part 2, Chapter 22, Article 
 30, and Chapter 24, Article 20, and Transactions of Actuarial Society 
 of America, Volume 8, page 58. 
 
 7. A policy is drawn payable as follows : " unto Jane Doe, bene- 
 ficiary, wife of the insured,'' without any further provision. The 
 beneficiary dies and the insured wishes to surrender the policy. 
 Discuss the rights of the insured in such a case. 
 
 A policy so worded would at the death of a wife inure to the benefit 
 of the children, if any, and according to the law of some states could 
 not be assigned. Where an assignment is possible the children 
 should join in the release, either individually, if adult, or if minors, 
 through their general or special guardian. Where no children sur- 
 vive, her interest will pass to her heirs. 
 
 8. A set of select tables are to be graduated in which the mor- 
 tality in the early policy years at ages belozv twenty-five at entry is 
 higher in the ungraduated data than at several higher ages. What 
 method would be most satisfactory? Give details of the work of 
 applying the method you would adopt. 
 
 As it is impossible to represent, by a table graduated by Make- 
 ham's law unmodified the special feature referred to, that law should 
 not be used unless an extended use is expected to be made of the 
 resulting table in joint life calculations. No perfectly satisfactory 
 method has, however, yet been devised of applying summation grad- 
 uations, such as those of Woolhouse and Higham, to select tables. 
 The graphic method of graduation would, therefore, be the most 
 suitable. The expectations of life for the separate ages at entry 
 should be made from the ungraduated data. These will indicate 
 what groupings should be made to obtain a series as regular as pos- 
 sible. The data for each group should then be combined and -' 
 select table graduated bv the graphic method. These tables wi'' 
 indicate where it is possible to merge the select tables into the ulti 
 mate, and will furnish a guide for drawing the curves of mortality 
 for each of the early insurance years. A mathematical formula ap- 
 proximating to the facts can generally be used to advantage as a 
 basis, as then only the departure of the facts from the formula re- 
 mains to be graduated and a larger scale can be adopted. 
 
 9. Discuss the propriety of issuing single premium policies on the 
 participating plan. How would you compute rates for such policies? 
 
 There does not appear to be any conclusive objection to doing so. 
 It is true that one considerable item can be determined more ac- 
 curately in advance in the case of single premium than in the case of 
 annual premium policies, namely, expense of collecting premiums, 
 also that the lapse feature is practically eliminated. There are 
 other elements, however, which are subject to variation, and these 
 are sufficient to justify issuing the policies on the participating plan. 
 
 51 
 
As any lack of exact justice in the premium rate can be adjusted in 
 the dividends, the loading can be made to conform to the general 
 rule of the company for participating business. 
 
 10. Show what relationship must exist hetzveen the values of p^ 
 by two different tables of mortality in order that the policy values by 
 the two tables for the same rate of interest may be the same. Discuss 
 other forms of policy as well as ordinary life. 
 
 We have for any form of policy where it is the net premium and i 
 the rate of interest (^jV + «) (i + = „V + g^^^ + « _ i (Sn — «V) 
 where „ _iV and „V are the terminal reserves for the {n — i)** and 
 n*^ years respectively, and S„ the sum assured in the w year. Let 
 ^[,1 _!_„ _i be the rate of mortality in the «'* year according to some 
 other mortality table, giving the same terminal reserves but a dif- 
 ferent net premium it . Then 
 
 (n_lV + it^) (I + = „V + ^U+n-li^n-n^) 
 or, subtracting, 
 (TT^ -7t){i+t)= (/[^i + „_i - ^[,]4.„_i) (S„-„V). 
 
 The left side of the equation is constant so long as the premiums 
 do not change so that the difference in the rates of mortality must 
 be inversely proportional to the net amount at risk. In the case of 
 ordinary life policies two aggregate tables may give the same ter- 
 minal reserves (see Associate B, question 6), but for other forms 
 one at least of the tables must be in the analyzed or select form. 
 For policies which have become paid-up the left side vanishes and 
 consequently no variation in the rate of mortality is permissible. For 
 endowments in the final year, the formula calls for an infinite differ- 
 ence in the rates of mortality, showing an impossible condition. 
 
 11. Deduce a formula for the distribution of profits on the con- 
 tribution plan, and state what facts would govern you in determining 
 the constants. 
 
 Under what theory of expense assessment is a dividend earned 
 during the first policy year? 
 
 In the case of any policy we have („_iV -f it){i + — ^-p_j_„_i 
 = (i — ^^_}_„_i) „V where it is the net premium ; i the assumed 
 rate of interest and ^^ i „_ithe tabular rate of mortality. Suppose 
 now the loading is / and the expenses are e, that the actual rate of in- 
 terest earned is/ and the actual rate of mortality is ^J.^„_i then the 
 profits earned during the year are 
 
 („_iV + ;r + /-^) (f 4-7-)-^i+n_i-(i-iri+„-l)nV 
 or substituting from the equation above 
 a-e) (I + ^O + (/- i) («_iV 4- ir) + (^,+n_i-^+n-l) 
 
 53 
 
The rate of interest earned is determined by dividing the interest 
 earnings, less investment expenses, by the mean funds (reserve and 
 surplus) of the year less one-half the interest earnings. An average 
 rate should be used based on the company's recent experience. The 
 rate of mortality should be based on the company's experience over 
 a sufficient period to give stability. The expenses after deducting 
 off-setting items, such as gain from lapses, give when distributed the 
 .expense factor. 
 
 . If it is assumed that any excess of the cost of new business over 
 the profit from lapses and gain from mortality of recently selected 
 risks is properly assessable on the business at large as necessary for 
 the maintenance of the company, a dividend will be considered as 
 earned the first policy year. 
 
 12. A woman in receipt of an income under an annuity on her 
 own life desires to have the annuity changed so as to be payable so 
 long as either she or her husband survives. What action would you 
 take on the request and what circumstances would you take into con- 
 sideration f 
 
 In the case of such a request, the company should decline to make 
 the proposed change and suggest that application be made for a 
 survivorship annuity, in favor of the husband. In the consideration 
 of this application the various questions affecting the desirability of 
 the risk could be taken up in the regular way. If it is desired that 
 the annuity be continued for the full amount, such additional pre- 
 mium as is necessary being paid now to the company, the case would 
 be more favorable for action. 
 
 13. Express, in the form of an integral, the value of axyz\w 
 and lay out a schedule shozving how you would apply a formula of 
 approximate integration to its evaluation. 
 
 See Institute of Actuaries' Text Book, Part II, Chapter XV, Art. 
 29, The arrangement of work given in Article 24, with the neces- 
 sary additional column for the additional life, would be convenient. 
 
 14. Under-average lives have been assured with extra premiums 
 payable during the continuation of the contract. Under what condi- 
 tions, if at all, can the extra premium be remitted^ 
 
 Where extra premiums have been charged on account of an im- 
 pairment they should not in general be remitted. The fact of the 
 assured subsequently passing a good medical examination is not 
 sufficient ground for such action. If, however, other circumstances 
 are such as to indicate clearly that an error was made by the medical 
 examiner originally, the extra may be remitted. Also if the busi- 
 ness has been transacted under such conditions that, at the time the 
 application is made, it would be more profitable to the company to 
 remit the extra and retain the healthy life on those terms than to 
 refuse and cause a lapse, prudence would dictate its remission. This 
 could in general only occur during the first policy year. 
 
 15. Show how Woolhouse's formula for graduation was obtained 
 and draw up a schedule of, operations by which it can be mo^t con- 
 vert en tly applied. 
 
 55 
 
Woolhouse's formula for graduation was derived by considering 
 the five different series, which can be formed by taking values from 
 the original series separated by quinquennial intervals. Each series 
 is then supposed to be filled out by central difference interpolation 
 to the second order, and a final series formed by taking the aver- 
 ages of the corresponding values in these five series. We then see 
 that expressing by u i the graduated value we have. 
 
 A«x + 6) + (»V«ar-3+ M »x + 2 " l'? «x + 7 ) [ 
 
 or I2S ul= — 3 «,_7 - 2 u^_f^ +3 «^_4 + 7 u^_^ + 21 
 
 «^_2 + 24 »^_i + 25 «^ + 24 U^^i -h 21 «,^.2 -f 7 
 
 "xr3 +3«:, + 4-2«, + «-3«x + 7 
 
 Various methods have beeen suggested to simplify the calcula- 
 tions involved, but probably the simplest is that of J. A. Higham, 
 which consists in arranging the values in columnar form, summing 
 in threes and subtracting three times the result from ten times the 
 central original value, then summing the result in fives three times 
 successively and dividing by 125. See J. I. A. Vol. 31, p. 323. 
 
 16. Gross premiums are to be computed, which, after covering 
 certain initial and renewal expenses, which are partly proportional 
 to the premium and partly constant, will provide, on the assumption 
 of four per cent interest, a simple reversionary bonus of one per 
 cent per annum. Give a formula for the premium and state what 
 basis of z'aluation should be adopted. 
 
 Let ?f i be the required office premium, and suppose the initial 
 
 expenses are k 7t^ -\- a per unit assured and the renewal expense 
 
 / ^3.+^. Also suppose, to be perfectly general, the policy in question 
 
 s an endowment, with a period of n years and with premiums for m 
 years. We have then where all monetary values are on a four per 
 cent, basis 
 
 1. , „ . _, Rx^l — Ra + nH-l + n(T>x + n — M^^w ) 
 
 ItxKl + axm-l\) = Axnl + .01 g 
 
 + ^TTx + n + iiix ax iir^Tii + bax ;rrii 
 
 Rx + 1— Rx + n 4- 1 + n{\)x^n — Ma;-4-n) 
 
 Ax„l + tf + bnxn-\\ 
 
 1 r>- 
 
 A net valuation on a three and one half per cent, basis, with a 
 special reservation for expenses on limited payment policies after 
 they have become paid-up, would probably on an average distribu- 
 tion of the business be sufficient to maintain the dividends. 
 
 S7 
 
Fellow. 
 
 17. Policies with semi-annual or quarterly premiums are usually 
 valued on the assumption that the annual premium has been paid 
 and credit is allowed in the assets for the gross deferred premium, 
 less loading. How does the reserve thus obtained compare with 
 that obtained by valuing the policies as strictly half-yearly or quar- 
 terly contracts? 
 
 See Institute of Actuaries Text Book, Part II, Chapter XVIII, 
 Articles 84 to 88. 
 
 It is to be noted in this connection that, in this country where 
 premiums are payable semi-annually or quarterly they are merely 
 instalments of the annual premium and not true semi-annual or 
 quarterly premiums ; as any unpaid balance is deducted from the 
 death claim. 
 
 18. (a) What are the three principal sources of surplus and in 
 what proportions do they generally contribute? 
 
 (b) How would you treat investment expenses in determining 
 the share of profits from each source? 
 
 (a) The question of the proportions in which the various sources 
 contribute to profits is one to which various answers may be given, 
 according to the point of view. When we remember, however, 
 that what appears as gain from surrenders and lapses, is in large 
 part merely a refund of initial expenses not yet reimbursed out of 
 loadings and that the bulk of what appears as gain from interest is 
 really interest earned on surplus, the distribution of which is defer- 
 red, it would appear that the main sources of profit, at the present 
 time, in the order of their importance, are gain from excess of 
 loading over net expenses, mortality savings and interest earned in 
 excess of the rate necessary to maintain the reserve. The two 
 former probably contribute about equally, and the last considerably 
 less than either of them. 
 
 (b) Investment expenses which are necessary for the care of 
 old investments, including changes of securities and reinvestments, 
 should be considered as a charge against interest income. 
 
 19. (a) A company grants, as surrender values under its poli- 
 cies, cash, paid-up policies or extended insurance. How would you 
 fix the relative amounts of these surrender values? 
 
 (b) What theoretical objections are there to extended insurance 
 in case of endowments? 
 
 (a) The amount of the paid-up policy to be given, should be 
 determined by deducting from the reserve on the policy a charge 
 for initial expenses not yet reimbursed. The manner of determin- 
 ing the amount of this charge should vary with the practice of the 
 company regarding dividends and other features. The balance after 
 deducting this charge should then be converted at net rates into 
 paid-up assurance. Theoretically, a further charge should be 
 made on account of the justifiable presumption that a person sur- 
 rendering his policy is a select life, but, so far as paid-up policies 
 are concerned, this is already sufficiently provided for, if the re- 
 serves used as a basis are based, for premiums as well as for other 
 factors, on an ultimate table. In converting into cash, however, ac- 
 count should ordinarily be taken of this possible selection. And in 
 
 59 
 
determining the period of extended insurance, a sufficient loading 
 should be used to compensate for the selection adverse to the com- 
 pany, which may be exercised by the assured. 
 
 (b) Applications are frequently accepted on the endowment 
 form which would not be accepted for term assurance, the expecta- 
 tion being that near the end of the period the amount at risk 
 would be small and that the effect of extra mortality occurring at 
 that time might be neglected. If however extended assurance is 
 given after a few years' premiums have been paid the company 
 would be on the risk for the full amount as the reserve would be 
 small. 
 
 20. Find an approximate expression for '»^. and show how it 
 must be modified to give the value of a^**^^- 
 
 See Institute of Actuaries Text Book, Part II, Chapter XI, Ar- 
 ticles 5 and 8. 
 
 21. What are the legal requirements as to basis of valuation in 
 order that nezu business may be transacted? 
 
 To what point must the reserve be reduced before a receivership 
 can be insisted on? 
 
 In the State of New York, a company is not permitted to transact 
 new business, if a statement of assets and liabilities, including the 
 value of its policies on the basis of the Actuaries' or Combined Ex- 
 perience Table with four per cent, interest, for policies issued prior 
 to January 1st, 1901 and on the basis of the American Experience 
 Table, with three and a half per cent, interest, for policies issued 
 since that date or on such higher basis as the company may have 
 adopted, shows an impairment of more than fifty per cent, of the 
 capital. But a receiver cannot be appointed unless the funds in- 
 vested according to law, after deducting the outstanding liabilities 
 and capital stock, fall below the reserve on the basis of the Ameri- 
 can Experience Table with four and one half per cent, interest. 
 
 See section 82 of the general Insurance Law. 
 
 22. (a) State Gompertz's hypothesis as to the law of human 
 mortality and deduce the value of l^ in terms of the constants. 
 
 (b) What modification was introduced by Makeham, and how 
 did it affect the value of ixf 
 
 See Institute of Actuaries' Text Book, Part II, Chapter VI, Ar- 
 ticles 9, 10, 14 and 15. 
 
 23. (a) Give a brief outline of the usual form of statement re- 
 quired of a life company, enumerating the various schedules. 
 
 (b) In preparing the annual statement a company wishes to 
 write off two per cent of the value of its real estate. Show two 
 zvays in which this may be done. 
 
 (a) See standard blank for returns of life companies. 
 
 (b) Any desired amount may be written off the value of real 
 estate, either by passing the amount through disbursements under a 
 special profit and loss item and correspondingly reducing the item of 
 
 61 
 
book value of real estate under assets, or by setting up an item 
 in the liabilities of a real estate depreciation fund. 
 
 24. Give the formula for the reserve on a survivorship annuity 
 with return of premiums in case of prior death of the nominee. 
 
 What basis would you adopt for valuation? 
 
 Let (x) be the assured and (y) the nominee, and let 7t be the net 
 premium and tt* the gross premiun per unit; then 
 
 
 (I + a^) = ay- a^y + 7t\lA)ai or l + 7t = ^^f:^ 
 
 Then the reserve at the end of « years will be 
 
 — 7t (l + ax+n:v + n) = (l + ^y + n) — (l + TT) (i + ax+n-.y + n) 
 I 
 
 M : + R 
 
 X + n y + n x + n y + n 
 
 i'x -r n -.y ^ n 
 
 Theoretically, an assurance table should be used for the as- 
 sured and an annuity table for the nominee, but in practice the 
 labor would be prohibitive and the Carlisle table, besides being con- 
 venient, provides for a sufficient reserve. The same rate of interest 
 should be used as for other annuities. Three and one half per 
 cent, would be sufficiently conservative. 
 
 25. What will be the probable effect upon the dividends to the 
 survivors of an extra mortality in the eighteenth to the twentieth 
 years under tzventy-year endowment policies issued on the twenty- 
 year dividend planf State the reasons for your answer. 
 
 Each extra death claim reduces the total profits to be divided, by 
 the amount of premiums unpaid less expenses and by the interest on 
 those premiums and on the amount of the claim. If this amount is 
 less than the share of profits which would have been allotted to the 
 policy at maturity, had the policyholder survived to that time, the 
 reduction in the number sharing in the profits will more than 
 counterbalance the reduction in the total profits and the share of 
 each survivor will consequently be increased. In the case of twenty 
 year endowment policies on the deferred dividend plan this would 
 generally be the case after the eighteenth j^ear. , 
 
 26. (a) A twenty-year endowment has been in force five years 
 and it is desired to change it to an ordinary life policy. What 
 allozvance should be made for the higher premium paid in the pastf 
 
 (b) What difference zuould you make in such cases as between 
 annual dividend and deferred dividend policies? 
 
 (a) In making a change such as this the maximum allowance 
 which can be made is the difference in reserves on the two policies 
 with such deduction as may be necessary on account of the unre- 
 couped initial expenses. The amount allowed should, of course, 
 
 63 
 
 
not exceed the difference between the surrender values of the two 
 policies, and a rule to allow that difference would in practice pro- 
 duce fair results. 
 
 (b) The rule in this form would apply to either annual or de- 
 ferred dividend policies the appropriate difference in treatment be- 
 ing provided for by the difference, if any, in values allowed on the 
 two forms of contract. 
 
 27. Evaluate the following integrals : 
 
 (a) /* Ji' e-^ dx. 
 
 {h) f {x - af {b - xf dx. 
 
 J a 
 
 ^ ' J O X + 1 
 
 (a) f x^ e-"" dx= 3 f x^ e-^' dx 
 Jo Jo 
 
 ' = 6 / X <?-* dx 
 
 J o 
 
 = d f e-=^ dx 
 = 6. 
 
 (b) / (x - ay {b - xf dx = 5 /'%- - a)* (b - x) dx 
 
 J a •/ a 
 
 = V,. / (.r - nf dx 
 J a 
 
 - g\, (^' - ^rf. 
 
 (c) / ^ dx = {x + 2) dx - / dx 
 
 ^ ' J o .r+l Jo J o X + 1 
 
 = ~ + 2a — loge (i + a). 
 
 28. It is found from a large number of observations that the 
 annual death rate from accident among railroad engineers averages 
 about five per thousand. State under what terms you would grant 
 an engineer a twenty-pa^jment life policy. 
 
 The addition of a little under .005 to the force of mortality at 
 each age is equivalent, so far as annuity values are concerned, to 
 adding one half per cent, to the rate of interest. Expressing, then, 
 net premiums in terms of annuity values, we see that the extra 
 mortality is provided for on ordinary life policies by taking the net 
 premium at a rate of interest one half per cent, higher as a basis 
 
 65 
 
and adding the difference in the rates of discount. And that the 
 net premium for a twenty payment life policy bears the same ratio 
 to that for an ordinary life policy as holds for the corresponding 
 premiums for normal mortality at the higher rate of interest. In 
 other words, the rule for a twenty payment life policy would be to 
 take the net premium at a rate one half of one per cent, higher and 
 increase it in the proportion which the difference in the rates of 
 discount bears to the net ordinary life premium at the higher rate. 
 This provides for an addition to the force of mortality equal to 
 the difference in the force of discount. Now the normal rate of 
 death from accident is about one per thousand so that the extra 
 force of mortality for engineers, according to the data, would be 
 about .004 or eighty per cent, of that which the above rule provides 
 for. This could be adjusted by reducing the extra in that pro- 
 portion. 
 
THE FUNDAMENTAL 
 PRINCIPLES OF PROBABILITY 
 
I. The Measurement of Probabilities 
 
 As mathematical science has to do with quantities and 
 their nlations, it is evident that before anything can be 
 done in the way of developing a mathematical theory of 
 probability, we must establish some method of determin- 
 ing the numerical measure of a given probability. It is 
 true that without doing so it is possible to lay down 
 rules of operation and deduce their necessary conse- 
 quences just as we may say that the area of a rectangle 
 is equal to the product of its length by its breadth even 
 though we do not know and have no means of ascertain- 
 ing either of those factors. So also, in probability, we 
 are able to deduce certain relations among connected 
 probabilities which hold, even though the exact values 
 of these probabilities are unknown. But in order to de- 
 termine these relations it is necessary to know the nature 
 of the quantities with which w^e are dealing and their re- 
 lation to the entities of which they are the measure. 
 
 Probability then has to do w^ith events or things which, 
 in detail and as to the individual items, do not, at some 
 specified time, fall within our knowledge. In so far as 
 anything is a matter pf knowledge it is necessarily taken 
 out of the region where probability holds sway. 
 
 Again probability has to do with the frequency with 
 which an event happens or a specified result is obtained. 
 
 71 
 
Other things being equal that event which happe n s the_ 
 more frequentlxJs_tjie rnore probable. In fact the nu- 
 ■nTericaT measure which has been universally adopted for 
 ^he probability of an event under given circumstances is 
 the ultimate value, as the number of cases is indefinitely" 
 increased, of the ratio of the number of times the event 
 happens under those circumstances to the total possible 
 number of times. From this statement it follows that the 
 greatest possible value of a probability is unity and that 
 this expresses the probability of an event which happens 
 on every possible occasion, or in other words, which is cer- 
 tain to occur. The lowest value is zero which expresses 
 the probability of an event which never occurs. We 
 also see that those events, or ways of happening of an 
 event are equally likely or have equal probabilities which 
 happen in the long run in the same percentage of the pos- 
 sible cases. An individual is said to be selected at ran- 
 dom from a group when all the individuals composing 
 the group are equally likely to be selected. 
 
 It is stated above that the measure there given has been 
 universally adopted and this holds true in spite of the fact 
 that the rule has been stated in ways which on their face 
 differ widely from that above given. The one most com- 
 monly given is that if an event can happ£rLirua_ways_and 
 fail in b ways all of which are equally likely, the probability 
 of the event is the ratio of o^to the sum of a and b. It is 
 readily seen that if we read mto this statement the meaning 
 of the words "equally likely," this measure, so far as it goes, 
 reduces to a particular case of that given above. This form 
 of stating the rule is useful in those cases where the happen- 
 ing or failure of the event can be analyzed into a definite 
 number of elementary ways of happening, all of which we 
 may, either on a priori grounds or by the conditions of the 
 problem, assume to be equally likely. For example, where 
 three balls are drawn at random from an urn containing 
 
 73 
 
seven white and three black balls, all the different combina- 
 tions three at a time which are possible, are equally likely 
 as we can go from any one combination to any other by a 
 series of interchanges or substitutions of one ball for an- 
 other each of which by the condition of the problem that the 
 drawing is made at random leaves the probability un- 
 changed. In order then to determine the probability that 
 the three balls drawn should fulfill any given condition, for 
 instance should be all white, we count the number of com- 
 binations which do fulfill it, in this case 35, and the total 
 number 120, and the ratio of these two numbers gives the 
 required probability. It is for this reason that so many 
 problems in the theory of probability reduce to an enumera- 
 tion of combinations, permutations or arrangements. But 
 the problem in every case under all its various shapes and 
 disguises is the same, namely, to determine in what propor- 
 tion of the possible cases the event, in the long run, tends to 
 happen. The nature of this tendency will be discussed in 
 the note on repeated trials. 
 
 II. The Combination of Probabilities 
 
 The next step in the development of the mathemat- 
 ical theory of probability is to determine under what 
 conditions the fundamental operations of addition and 
 multiplication, with their inverses, subtraction and divis- 
 ion, give, when performed between probabilities, inter- 
 pretable results and to ascertain the interpretation of 
 those results. 
 
 In order that two probabilities may be added together, 
 it is necessary that they should relate to the same gen- 
 eral subject. For example, the result of adding together 
 the chance of throwing six with a die and that of a man 
 aged 35 dying within a year would not be capable 
 of interpretation as a probability. Again the two proba- 
 
bilities must not have any part common to both, as other- 
 wise that part would be included twice in the sum and 
 the result again could not be interpreted. 
 
 _We thus see that, in order that their probabilities 
 may be added together, two events must be mutually ex- 
 clusive, that is, it must be impossible for the two events 
 lo happen together. When this is the case the sum of 
 the probabilities gives the chance that one or other of 
 the events will happen. This may be seen from the fact 
 that in a large number of trials, since the two events 
 never happen together, the number of times when either 
 one or the other happens is found by adding together the 
 number of times when they respectively happen. For 
 example, consider the chance of throwing more than 
 four with one throw of an ordinary six-faced die. This 
 can be done either by throwing five or by throwing six,, 
 the probability of each being one-sixth. Then in the 
 long run for each six trials five will be thrown once and 
 six once or more than four will be thrown twice, and its 
 probability is therefore one-third. 
 
 Again, if an urn contains five white, seven red and 
 eight black balls and one be drawn at random the chance 
 of draAving a white one is five-twentieths, that of draw- 
 ing a red one is seven-twentieths and the chance of draw- 
 ing either a white or a red ball is the sum of these two,, 
 or twelve-twentieths, as may be readily seen from the 
 fact that twelve of the twenty balls are either white or 
 red. 
 
 From the above reasoning it follows since subtraction 
 is the inverse of addition that the probability to be sub- 
 tracted must be included in and form a part of the prob- 
 ability from which it is to be subtracted. For example, 
 in the case above cited, if we know that the probability 
 of either a white or red ball being drawn is three-fifths 
 and that the probability of a white ball being drawn is 
 
 77 
 
one-fourth, then, by subtraction, the probability of a red 
 ball is the difference or seven-twentieths. 
 
 A prominent example of the subtraction of probabilities 
 arises in connection with the fact that since an event must 
 either happen or fail the chance of one or the other result 
 is unity. So that by subtracting from unity the probability 
 of an event happening we get the probability of its failing. 
 These two probabilities are said to be complementary to one 
 
 another. 
 
 Since a probability is a fraction of which the denom- 
 inator is the number of times an event can possibly hap- 
 pen, and the numerator is the number of times it does 
 happen, it follows that if the product of two probabilities 
 is to be itself a probability they must be so connected 
 that the denominator of one represents the same class 
 of events as the numerator of the other. In other words, 
 if one factor is the probability of an event happening, 
 the second factor must be the probability of some other 
 event also happening if the first does, and in that case 
 the product represents the probability of both events hap- 
 pening together. For example, suppose an urn contains 
 one hundred balls of which twenty are ivory and that, of 
 these twenty, five are white. Then, if one ball be drawn 
 at random the chance that it is ivory is one-fifth, and the 
 chance that if it is ivor\^ it will be white is one-fourth, and 
 we see, from the fact that of the total of one hundred 
 balls five are white ivory ones, that the probability of a 
 white ivory ball being drawn is one-twentieth which is the 
 product of one-fourth and one-fifth. Similarly, if at a 
 given age one person, on an average, in each 125 dies within 
 one year and of those who die at that age one in eight on 
 an average die from accident, then in the long run out of 
 each thousand persons eight will die within the year and 
 of these eight one will die from accident. Or the proba- 
 bility of a person dying from accident within one year is 
 
 79 
 
one one-thousandth or the product of one hundred and 
 twenty-fifth and one-eighth. 
 
 We have said that one of the factors is the probability 
 that if one event happens another event will also happen. 
 This probability may be different from the probability 
 that if the first event fails the second will happen or it 
 may be the same. In the former case, the events are 
 said to be dependent and it is necessary to attend to the 
 distinction above noted. In the latter case, the events 
 are said to be independent and, as the probability to be 
 used in the multiplication is, in this case, equal to the 
 general probability that the second event shall happen, it 
 is unnecessary to attend to the distinction. This gives the 
 rule that to find the chance that both of two independent 
 events shall happen, we multiply together their respective 
 ■ probabilities. 
 
 The rule for independent events is the one which is most 
 frequently used in practice, because in a great many cases, 
 we may, in the absence of any reason to doubt it, assume 
 that the two events are independent. This, however, does 
 not alter the fact that where two probabilities are multi- 
 plied together to give a resultant probability one of the 
 factors is always, either expressly or by implication, the 
 probability that if one event happens another will happen 
 also. 
 
 Conversely the probability of a compound event may be 
 divided by the probabilitv- of one of its component simple 
 events, the happening of which is necessary to that of the 
 compound event, and the quotient gives the probability that 
 if the simple event happens the remainder of the com- 
 pound event will also happen. Or this latter probability 
 may be known and be used as the divisor and the quotient 
 would give the probability of the simple event. For ex- 
 ample, suppose we know that at a given age the prob- 
 ability of death from accident within one year is one per 
 
 81 
 
thousand and that of the total deaths at that age one in 
 ten is by accident, then by division we find the total chance 
 of death within one year to be ten per thousand. 
 
 All the computations of probabilities are made by the 
 application direct or indirect of these four rules, and most 
 of the errors that arise in such computations come from 
 neglect to observe the restrictions and conditions under 
 which these various operations are permissible. The fol- 
 lowing is a good example of the combination in one com- 
 putation of the various elementarj^ rules. 
 
 A number of urns are filled with balls. In each of one- / 
 tenth of the total number of urns nine-tenths of the balls 
 are white and in each of the remainder four-tenths are 
 white. From an urn taken at random a ball is drawn at 
 random, found to be white and returned. What is the 
 chance that a second raadom drawing from the same urn 
 would give a white ball ? 
 
 Here the a priori probability of selecting an urn in which 
 nine-tenths of the balls are white is one-tenth and, conse- 
 quently the chance that such an um will be selected and 
 a white ball drawn is by multiplication nine one-hundredths. 
 Also the chance of selecting one of the others is the com- 
 plement of one-tenth or nine-tenths, and consequently the 
 chance that such an um will be selected and a white ball 
 drawn is, by multiplication, thirty-six one hundredths. 
 Hence, by addition the total chance of a white ball beings 
 drawn is forty-five one hundredths. From this we see by 
 division that if a white ball is drawn the chance that it is 
 drawn from one of the first set of urns is nine forty-fifths 
 or one-fifth and the chance that it is drawn from one of the 
 second set is four-fifths. If then a second drawing is made 
 from the same urn the chance that the urn will belong to 
 the first set and that a white ball will be drawn is nine- 
 tenths of one-fifth or nine-fiftieths, and the chance that the 
 urn will belong to the second set and that a white ball will 
 
 83 
 
be drawn is four-tenths of four-fifths or sixteen-fiftieths. 
 Hence the total probability of a white ball being drawn 
 is twenty-five fiftieths or one-half. 
 
 III. Expectation or Mean Values 
 
 Frequently in applications of the theory of probability 
 the important element in the problem is the value assumed, 
 in the various contingencies, by some variable quantity. 
 This quantity may be the number of successes in a given 
 number of trials, the distance of a point from another 
 point, a line or a plane, the duration of life, the amount of 
 money to be paid or received or the present value of such 
 amount or any other quantity connected with the result 
 of the trial, including any function explicit or implicit of 
 any of these quantities. In these cases the special point to 
 which attention is generally directed is the mean value of 
 the variable quantity, or the ultimate value, as the number 
 of trials is indefinitely increased, of the average of the 
 values of the variable resulting from the various trials. 
 In certain cases this function is also called the expectation 
 from each trial, the idea being that a certain aggregate 
 result is expected from a large ntimber of trials, and that, 
 by dividing this aggregate result by the number of trials, 
 we get the expectation from each. It is evident that this 
 average will be determined by multiplying each value of 
 the variable by the number of times it occurs and divid- 
 ing the sum of the products by the total number of trials. 
 In the ultimate limit this is equivalent, from the definition 
 of probabilities, to multiplying each value of the variable by 
 the probability of its occurrence and adding all the pro-i 
 ducts together. For example, suppose one dollar is to be 
 received if a coin, about to be tossed, should turn up head. 
 In a large number of such trials the net results would be 
 that one dollar would be received for every two trials or 
 
 85 
 
the expectation from one trial is one-half of one dollar, or 
 one dollar multiplied by the chance of head being thrown. 
 
 It follows from the way in which the mean value is ar- 
 rived at, that the mean value of the sum of the values of 
 the variable resulting from a specified number of trials or 
 the expectation from such trials may be obtained by mul- 
 tiplying the expectation from each single trial by the num- 
 ber of trials. Thus the expectation of success in one trial 
 is equal to the probability and the expected number of suc- 
 cesses in a given number of trials is the product of that 
 number into the probability. 
 
 An important consideration arises in connection with the 
 mean value of the product of two independent variables, 
 that is, of the product of two quantities whose variations 
 depend on contingencies which are independent of each 
 other. In the computation of this mean value all the pos- 
 sible values of the product and their respective probab- 
 ilities will be represented by the various terms in the ex- 
 pression for the product of the mean values of the indi- 
 vidual variables, so that the mean value of the product is 
 equal to the product of the mean values. As an example, 
 suppose a sum of money is to be received which is to be 
 determined, by tossing a penny to decide whether payment 
 will be in dimes or in nickels, and by throwing a die to 
 determine the number of coins, the number to be equal 
 to the number thrown. Then the mean value of each coin 
 is seven and one-half cents, and the mean number of coins 
 is three and one-half and the expectation therefore twenty- 
 six and a quarter cents, as may also be seen by directly 
 taking the mean of the amounts payable in the twelve 
 equally likely possible cases. 
 
 It is to be carefully noted that this rule, that the mean 
 value of the product is equal to the product of the mean 
 values, applies only when the two quantities involved are 
 entirely independent Where the variation of one quantity 
 
 87 
 
is in any way dependant on that of the other, this relation 
 cannot be depended upon. As a case in point, the mean 
 value of the square of a variable quantity is always greater 
 than the square of the mean value. This appears from 
 general considerations, but it may be also demonstrated 
 algebraically and the difference may be shown to be equal 
 to the mean value of the square of the difference between 
 the actual value of the variable and its mean value. For 
 example, in one trial the mean value of the square of the 
 number of successes is equal to the probability of success 
 since it is equal to unity multiplied by that probability 
 added to zero, multiplied by the complementary probability. 
 We have already seen that the mean value of the number of 
 successes is also equal to the same probability. The mean 
 value of the square of the departure of the actual from the 
 expected is seen by a direct computation to be equal to the 
 product of the probability into its complement, or to the 
 difference between the probability and its square, which is 
 in agreement with the statement made above. 
 
 IV. Repeated Trials 
 
 Where a number of trials are made the chance that the 
 event should happen every time is, by the principle of the 
 multiplication of probabilities, equal to the probability of 
 success at a single trial raised to a power equal to the num- 
 ber of trials. Similarly where p is the probability of suc- 
 cess at a single trial and q the complementary probability 
 of failure, the chance that of n trials, r particular ones 
 should be successful and the remainder unsuccessful, is de- 
 termined by multiplying p raised to the rth power, by q 
 raised to a power equal to the difference between n and r. 
 We thus see that the probability of the success at any par- 
 ticular r trials only, is equal to the probability for any other 
 particular r trials only, and that consequently the total prob- 
 
 89 
 
ability of exactly r successes is found by multiplying this 
 probability by the number of combinations of n things r at 
 a time. Accordingly the ratio of the chance of exactly r 
 successes to that of exactly r— 1 is compounded of the ratio 
 of the chance of success on a particular r occasions only ta 
 that of success on a particular r - 1 occasions only, which 
 is equal to the ratio of p to q, and the ratio of the number 
 of combinations of n things r at a time to the number r - 1 
 at a time, which is equal to the ratio of « -|- 1 - r to r. This 
 ratio will be greater than unity or the chance of exactly r 
 successes will be greater than that of exactly r - 1, if p 
 (n+1 -^) is greater than ^r or if r is less than p (n-f-1). 
 We thus see that the most probable number of successes is 
 defermtned hy taking the integral part of the expected num- 
 ber of successes in n -f- 1 trials and that the probability in- 
 creases regularly, without any retrograde movement up to 
 the maximum and then decreases in the same way. 
 
 We may have r successes out of n -|- 1 trials in two ways 
 only, by r - 1 successes in n trials followed by success in the 
 other or by r successes followed by failure, its probability 
 therefore is intermediate in value between those of r - 1 and 
 r successes in n trials. We thus see that the probability of 
 the most probable number of successes in w - 1 trials is in 
 general less than the corresponding probability for n trials 
 as it must be less than the greater of the probabilities of 
 some two consecutive numbers of successes unless their two 
 probabilities are equal, in which case it will be equal to 
 either. 
 
 We thus see that as the number of trials increases the 
 maximum value of the probability of an exact number of 
 successes tends to diminish sometimes halting but never 
 retrograding. It follows that the chance, that the actual 
 number of successes will be within any definite number of 
 the expected, decreases as the number of trials is increased. 
 This result appears at first sight inconsistent with the funda- 
 
 91 
 
mental definition of the measure of probability, but it is not 
 in fact so inconsistent as the definition refers to the ratio of 
 the number of successes to the number of trials and this re- 
 lates only to the number of successes. We shall proceed to 
 investigate the variations of the ratio of successes. 
 
 We have seen in the note on mean values that the mean 
 value of the square of the departure of the actual from the 
 expected number of successes in one trial is equal to p q. 
 Also the departure of the actual from the expected in n 
 trials is equal to the algebraic sum of the departure in the 
 individual trials, so that the square of the departure in n 
 trials is equal to the sum of the squares of the departures 
 in the individual trials together with twice the sum of their 
 products two and two. But these departures for the in- 
 dividual trials are independent of one another and their 
 mean values are each zero, consequently by the principle 
 established in the note on Mean Values, so also is the mean 
 value if the product of any pair of them. We thus see that 
 the mean value of the square of the departure of the actual 
 number of successes from the expected number in n trials 
 is n times the value for one trial or is equal to « p q. But 
 the departure of the actual proportion of successes from 
 the expected is found by dividing the departure of the num- 
 ber by n the number of trials, so that the mean value of its 
 square is determined by dividing n p q by the square of n 
 or by dividing p q by n. This quotient is seen to diminish 
 as n increases. 
 
 Let now h be an assigned departure and let P be the 
 probability that the actual proportion of successes differs 
 from the expected by more than h. Then it is evident that 
 the mean value of the square of the departure is not less 
 than P times the square of h. Or in other words n P is 
 not greater than the quotient of p q by the square of h. 
 It follows then that no matter how small h is, provided it 
 does not vanish, we can by taking n large enough make P 
 
 93 
 
less than any assigned quantity. We thus see that as n in- 
 creases the actual proportion becomes more and more nearly 
 certain to agree within narrower and still narrower limits 
 with the expected proportion. This is the nature of the 
 tendency to the ultimate value as the number of trials is in- 
 definitely increased. 
 
 OF THE 
 
 UNIVERSITY 
 
 OF 
 
 9» 
 
OS T«5 I^^"^ 
 
 DATE 
 
 OVERDUE. 
 
YC 23597 
 
 
 THE UNIVERSITY OF CALIFORNIA LIBRARY