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158    ESSAY ON PROBABILITIES.
yet introduced into the affairs of common life, though many cases occur in which it might be made useful. But many things which are only demonstrable by the higher branches of mathematics are looked upon as useless by those who do not understand them ; nor is this result of ignorance only to be looked for among the uneducated. While the Reform Bill was in its progress through the House of Commons, a method was suggested by a man of science, with whom the government advised upon the subject, for estithe relative importance of boroughs by considering their population and contributions to the revenue combinedly. This method, to the efficiency of which most of those who examined it gave strong testiwas ridiculed by some members of the house, partly because it involved decimal fractions, and partly because another and a more simple (but palpably wrong) method gave, in that particular case, nearly the same results. When legislators are neither able to see that erroneous methods may sometimes lead to truth, being not therefore one bit the less erroneous, nor that the truth of a result is the same, whether decimal or common fractions be employed, it is little to be wondered at if useful applications of abstract reasoning are looked upon with suspicion and introduced with difficulty.
CHAPTER VIII.
ON THE APPLICATION OF PROBABILITIES TO LIFE CON
TINGENCIES.
WHEN questions connected with life contingencies were first considered, it was regarded as most deliberate gambling to be in any way concerned in buying or selling such articles as annuities, or any interests depending upon them. Before we can well enter upon
ON LIFe CONTINGeNCIeS.    159
the question of the truth or falsehood of the preceding notion, it will be necessary to ask what laws the duration of human life follows, and whether it folany laws at all ? Take two separate hundreds of persons, each aged twenty, is there any reason to conclude that the united lives of all the first hundred will make an amount of years nearly equal to that of the second ?
In order to try this point, I shall take another question, yet more unfavourable to the result which I wish to establish. In 100 persons all aged twenty, we know that there is but a very slight chance that any given one of them shall reach the age of eighty; and we may consider it a certainty (or of an extremely high probability), that none of them will see the age of a hundred and twenty. We will consider it therefore as given, that no one shall live to the last-mentioned age, and we will even suppose that all ages of death between 20 and 120 are equally probable. This of course very much increases our chance of fluctuation : but even with this supposition it is not very great.
Let us suppose a lottery in which there are counters marked with every possible number or fraction inter_ mediate between 0 and E : so that the drawing may have any mark whatsoever. If then we draw out 100 counters, the least possible amount of drawings will be 0, the greatest 100 times E : and if all drawings be equally probable, we have no reason to suppose that our amount will exceed 50 times E, which does not equally apply in favour of its falling short of that quantity. That we shall have exactly 50 times E, is an event of which the chance is infinitely small : but that the amount shall lie between limits which are tolerably near 50 times E, is very probable.
PROBLEM. Let there be counters, in equal numbers, with every possible mark between 0 and E. What is the probability that the average of n drawings shall not differ from the half of E, one way or the other, by more thank.
RULE. Multiply k by the square root of six times
160    eSSAY ON PROBABILITIES.
n, and divide the product by E. Call the quotient t; then the value of H (Table I.) is the probability required.
EXAMPLe. In 600 drawings, each of which may be any thing between 0 and 100, required the probathat the average of all the drawings shall be between 50 + 5 and 50 — 5.
n = 600, E = 100, k = 5 ; the square root of 6 times 600 is 60, and 5 times 60 divided by 100 is 3. The first table does not contain values of t higher than 2 : an event being almost certain, or of a very high prowhen t is equal to 2. Table II., however, furnishes us with an extension of Table I. ; the Ii oppoto any value of t in that table being always nearly the H which belongs to half that value of t. Consethe H belonging to t = 3, is the K belonging to t = 6. But the second table only goes to t = 5 ; in which case K is •999. It is then more than 999 to 1 that the average of the 600 drawings is within the limits specified. If we take k = 1, in which case t = •6, we find it is 3 to 2 that the average is contained between 49 and 51.
If then there were 600 infants born, and if it were the law of human life that any individual is as likely to die at one age as another, for any age not exceeding 100 years, even then, and with so much more scope for fluctuation than is actually found, it would be more than 999 to 1 against the average life of the 600 infants exceeding .55, or falling short of 45 years ; and more than 3 to 2 that the same average should fall between 49 and 51 years. If such be the case, it is obvious that the chances of fluctuation are much diminished by the superior chances of death happening at some periods of life rather than at others ; as well as by the smaller limits of human life, which need not for any practical purpose be supposed to extend as far as one hundred years.
To suppose that the duration of human life is regulated by no laws, would be to make an assumption of a most monstrous character, a priori, and most evidently
ON LIFE CONTINGeNCIES.    161
false. For it is a law, were it the only one, that no individual shall attain the age of 200 years. So much is known to all; but to those who consider the subject more closely, by the aid of recorded facts, it may be made as evident as the existence of a limit to human life, that the casualties of mortality are distributed among mankind in so uniform a manner, that the average existence of a thousand infants will differ very little from that of another thousand born in the same country and station of life. It is true that differences of race, climate, manner of living, &c., &c., produce marked effects upon the duration of life; which is no more than might be expected : but it is equally true that the notorious individual uncertainty of life cannot be disin the results of observations made upon masses of individuals.
There are various results of observation, which are called tables of mortality, which differ only in the methods of presenting the same sets of facts. Firstly, we have what may be called tables of the numbers living. These show, for a given number born, how many attain each year of age. Thus, in the Carlisle table, opposite to 0 and 50, we find 10,000 and 4397, indicating that, according to observations made at Carlisle, the proporof those born to those who saw their fiftieth birth-day, was that of 10,000 to 4397. Again, opposite to 60, we find 3643, meaning, that of 4397 persons aged 50, 3643 attain the age of 60. Secondly, we have tables of yearly decrements, in which the same number of per-sons are supposed to be alive at every age, and the pro-portion who die in the next year is set down in the table. Thus in the government annuity tables, opposite to 50 and 60, we find 161 and 315, meaning that, according to the observations from which these tables were constructed, of 10,000 persons aged 50, 161 died before completing the next year of life; and of 10,000 persons aged 60, 315 died before attaining the age of 61. Thirdly, we have tables of mean duration of life (comcalled expectation of life), which show the average
M
162    eSSaY ON PROBABILITIeS.
number of years enjoyed by individuals of every age. This, in another variety of the Carlisle tables, opposite to 50 and 60, we find 21.11 and 14.34 ; meaning that, according to these tables, persons aged fifty live, one with another, 21.11 years more, and persons aged 60, 14.34 years more.
Until observations of human mortality become more extensive and correct, I prefer the tables of mean duration to all others. The events of single years are subject to considerable error, and generally present such varieties of fluctuation, that it has become usual to take some arbitrary and purely hypothetical mode of introducing regularity. This practice cannot be too strongly condemned, since the tables thereby lose some of their value as representations of physical facts, without any advantage ultimately gained. For if by using the raw result of experiments, tables of annuities were rendered unequal and irregular, it would be as easy, and much more safe, to apply the arbitrary method of correction to the money results themselves, than to introduce it at a previous stage of the process. It is not, however, a matter of much consequence as to the annuities, &c., deduced from the tables : and as yet, the rudeness of the original observations renders the effect of any such alteration not so great as the probable errors of the observations themselves.
The mean duration of life is approximately calculated as follows. Suppose (taking an instance from the Car-lisle tables) that 75 persons are alive at the age of 92, of whom are left at the successive birthdays, 54, 40, 30, 23, 18, 14, 11, 9, 7, 5, 3, 1, 0. Consequently. in their 93d year, 54 persons enjoy a complete year of life, and 21 die, whom we may suppose, one with another, to live through half the year, and 54 years and 21 half years make 64 years, which is the total life of 75 per-sons for that year. Proceeding in this way, we find that there are,
in the 93d year 54 + of 21 years.
94th    40 + 3 of 14
ON LIFe CONTINGENCIES.    163
    95th    30 + z Of 10
    96th    L'3 + of 7
    97th    18 + z of 5
    98th    14 + 2 of 4
    99th    11 + of 3
    100th    9 + of 2
    101st    7 +-of 2
    102d    ,5 + of 2
    load    3+iof 2
    104th    1 + 1 of 2
    105th    0 + 1 of 1 Total, 215 + i of 75
Hence 75 individuals, aged 92, enjoy 215 + s of 75 years, and each has, one with another, the 75th part of this, or 3.37 years.
RULE. To find the mean duration of life from a table of the living at every age out of a given number born, add together th u numbers in the table for all the ages above the given age, divide by the number at the given age, and add half a year to the result.
The preceding rule is mathematically incorrect, being only an approximation to the truth, even supposing the tables perfectly correct. The error of computation may be found, nearly, as follows. Divide the number who die in the year next following the given age by twelve times the number in the table at that age, and diminish the result of the preceding rule by the quotient. Thus, in the instance before us, 21 divided by 12 times 75 is •02, so that 3.35 is nearer the truth. This error, however, is immaterial for practical purposes.
A more important question is that of the degree of confidence which may be placed in tables of mean duration, the errors of observation being supposed to be as likely to be positive as negative. In order to estimate this, we must compute the mean square of the duration of life; that is, multiplying the time which each individual lives by itself, we must add the results together and divide by the whole number of individuals. To make a rough approximation to this in the case before us, remember that
m
164    ESSAY ON PROBABILITIeS.
21 individuals live n a year    x = a giving 41
14    3 —    x 3 = .3 — 1 za
10    2 s —    2 sx R s - -
2so4
    7    3 7 —    3x 3=V    949
            4
    5    8 —    7 x 3= 61    40s
    4    11    Ilx11=1Yl    44
    3    1 9 —    , (1 1 = 1160    
    2    2    2    4    :7
    2    IS —    IS    ~5    450
        x    _25    
        24    
    O    2    2    2    4
    t7 —    17x11 _Rag    578
    Z    2    2    4    4
    2    1Q —    19    19_ -- 361    77
        x    
    2    2    2    2 4    4
    21 —    2l x 21 = 441    882
    2    2    2    4    4
    2    2a —    29 x 2J = 52S    1058
    s    2    2    4    —a
    1    25 —    25 x 25=62s    625
    2    2    2    4    4
75    ( Average square 21.5)    a4s1
RULE. From the mean square of the duration of life at any age, subtract the square of the mean duration at that age : divide the difference by the number of lives of the given age from which the table was made, and extract the square root of the quotient. Take four tenths (more correctly •39894, of this square root, which gives the mean risk of error, and •67 of the square root gives the probable error.
Suppose that in the case before us, the number of lives aged 92 was 40 *, from which the preceding table was made. We have then,


Mean square of durations    21.5
Square of 3.37, the mean duration 11.36
40)10.14
.254  •254 = •504 •504 x •67 = •33 of a year, the probable error.
The same process may be applied to any other case, and the result of the whole is, that observation of a number of lives which is not very great, will be suffi


* This is nearly the number of lives at that age among those from which the Carlisle table was formed, but the arbitrary help introduced from other tables at the older ages, on account of presumed insufficiency of data, makes the result of this example of no greater value than a numerical instance arbitrarily chosen.
ON LIFE CONTINGENCIES.    165
cient to give the mean duration of life with considerable approach to exactness. This is confirmed by the results of various tables, from which it appears that when the individuals composing an observation are of the same country, and under the same general circumthe results of such tables come very near to each other.
The reader who desires to know the history of tables of mortality should consult the articles MORTALITY and ANNUITIES in the new edition of the Encyclopaedia Britannica, both from the accurate pen of Mr. Milne, the author of the Carlisle tables. I cannot, in this work, pretend to give more than a slight summary of results connected with life contingencies, such as may guide the reader who understands the main points of the theory of probabilities to safe conclusions.
From some tables made from observations at Breslau, De Moivre concluded that the following hypothesis, namely, that of 86 persons born one dies every year till all are extinct, would very nearly represent the mortality of the greater part of life, and that its errors would nearly compensate one another in the calculation of annuities. The Northampton tables of Dr. Price, which have been used by most of the insurance offices, very nearly represent this hypothesis at all the middle ages. But both give much too large a mortality for the circumstances of the last half century, as is proved by all the tables which have been lately constructed. The greater part of the difference, I have no doubt, is due to the real improvement of life which has taken place, from the introduction of vaccination, more temhabits of life *, better medical assistance, and greater cleanliness in towns. We may now state, as a much nearer approximation to the mortality of the
* I must be understood, here, as speaking particularly of the middle classes, in English towns and cities. Most of the tables have a majority of this class, and there is not any very precise information on the mortality of the labouring classes, or in the inhabitants of the country as distinfrom those in towns. With regard to the point on which this note is written, all old persons remember the time when what we should now call hard drinking was almost universal.
u 3
166    eSSAY ON PROBABILITIES.
middle classes, that from the age of 15 to that of 65, the average may be represented as follows : - of 100 persons aged 15, one dies every year till the age of 65. But the mean duration of life will serve to give a more precise idea, and a simple rule may be given, which will, for rough purposes, represent the Carlisle table between the ages of 10 and 60. Of persons aged 10 years, the average remaining life is 49 years, with a diminution of 7 years for every 10 years elapsed ; thus of persons aged 20 years, the average remaining life is 49-7 or 42 years ; at 30 years of age, 35 years. The following list of tables will be followed by some notice of each.
    
Z.'.    G    c    U    -    U    O    s    m
    O    m    2        ,~    W    G    U    O
    ~'n    a            81        G    
    K    z                        
    0    43    25.2    -    -    38'7    -    -    -    -    -    -    0
    5    40.5    40.8    -    -    51.3    -    -    48.9    54.2    5
    10    38    39.8    -    -    48.8    48.3    45.6    51.1    10
    15    35.5    86.5    -    -    45.0    45.0    41.8    47.2    15
I    20    33    33.4    36.6    41.5    41'7    38.4    44.0    20
    25    30.5    30.9    34.1    37.9    38.1    35.9    40'8    25
    30    28    28.3    31.1    34.3    34.5    33.2    37.6    30
    35    25.5    25.7    27.7    31.0    30.9    30.2    34.3    35
    40    23    23.1    24.4    27.6    27.4    27.0    31.1    40
    45    20.5    20.5    21.1    24.5    23'9    23.8    27.8    45
    50    18    18.0    17.9    21.1    20.4    20.3    24.4    50
    55    15.5    15.6    15.1    17.6    17.0    17.2    20'8    55
    60    13    13.2    12.5    14.3    13.9    14.4    17.3    60
    65    10'5    10.9    9.9    1P8    11.1    11.6    14.0    65
    70    8    8.6    7.8    .    9.2    8.7    9.2    11.0    '70
    75    5.5    6.5    6.2    7.0    6.6    7.1    8.5    75
    80    3    4.8    5.0    5.5    4.8    4.9    6.5    80
    85    0.5    3.4    4.0    4.1    3.4    3.1    4.8    85
    90    -    -    2.4    2.9    3.3    2.6    2.0    2'8    90
    95    -    -    0.8    1.4    35    1.1    P2    P6    95
    100    -    cu    -    .    -    2'3    -    -    -    cu    -    -    100
ON LIFE CONTINGENCIES.    167
            C    . s    .—.    _    ro    L y    c    `a            
            ro    `oy    U    U            
            _    _                
            5    5    •447    '400    80    50    •136    '093                
                10    '415    383    85    55    •113    094            
            5    15    •420    '331    90    60    '097    •113                
            SO    20    423    •375    95    65    '037    •147                
            S5    25    417    '372    45    5    •252    •177                
            40    30    409    '366    50    10    •206    •146                
            L5    35    '402    '360    55    15    '201    •135                
            0    40    '394    '350    60    20    '193    •119                
            5    45    '385    •329    65    25    '172    '110                
            60    50    •376    •315    70    30    '148    '097                
            65    55    •360    '323    45    35    •124    •081                
            0    60    '339    322    80    40    '102    •075                
            5    65    '317    •303    85    45    •082    •059                
            O    70    300    '320    90    50    •069    •052                
            05    75    •292    •332    95    55    •025    •078                
            10    80    '302    '329    55    5    •195    •125                
            15    85    161    '462    60    10    •144    •091                
            ?5    5    •377    .307    65    15    •136    '085                
            t0    10    •344    '283    70    20    •125    .071                
            15    15    '345    •279    75    25    •103    '061                
            f0    20    •343    •270    80    30    •082    '056                
            15    25    •331    '263    85    35    •064    •046                
            10    30    •317    '251    90    40    •051    '044                
            i5    35    '303    '231    95    45    '018    •049                
            60    40    •288    '212    -                
            65    45    •269    •194    6e    5    143    086                
            0    50    246    '177    70    10    '087    '054                
            5    55    '218    •177    75    15    '078    048                
            SO    60    159    1 90    80    20    •069    '040                
            5    65    '166    174    85    25    '053    •0434                
            0    70    157    191    90    30    •041    '033                
            15    75    •072    •300    95    35 _014    039                
                            
            S5    5    •314    •235    75    5    098    057                
            O    10    273    '207    80    10    •0-14    •028                
            45    15    271    20`3    85    15    '037    '024                
            0    20    •265    •189    90    20    '035    '012                
            5    25    249    174    95    25    •012    •029                
            10    30    '230    •158    85    5    '063    •040                
            15    35    •209    143    90    10    '022    •017                
            70    40    •186    '126    95    15    '007    '023                
            P5    45    •160    '104    95    5    •021    •036                
ON LIFE CONTINGENCIES.    1 77
The quantity found in this table is the probability of an elder life surviving the younger. The difference of ages differs in the various compartments of the table ; in the first it is ten years, in the second twenty years, and so on. The two results accompanying each pair of ages are those of the Northampton and Carlisle tables. Thus. according to the Northampton table, the chance of a life of sixty surviving one of thirty is •23 ; that of the younger surviving the elder is therefore 1 — •23 or •77. According to the Carlisle table, the same chances are •158 and •842.
Almost universally, the Northampton table gives a greater chance of the elder life beating the younger than the Carlisle. This is a consequence of that undue degree of comparative goodness which the former table gives to older lives, and to which I have already adverted.
If De Moivre's hypothesis were correct, it would be sufficient to divide the mean duration of B's life by twice that of A, and the result would be the chance which B has of surviving A, B being the elder of the two lives. This process, applied to the Northampton table, will give results very near the truth, when neither of the lives is very young. The same rule would give comparatively but a very rough guess at the result of the Carlisle table. If, however, the chance be calculated which the younger life possesses of dying in the average term of the elder, the result will be an approximation to the probability of the elder surviving the younger, when neither of' the lives is very young, and when their ages are not nearly equal. Thus, the mean duration of a life of 50 being 21 years, and the chance of a life of 30 years surviving 21 years being '769, the chance of the same life not surviving 21 years is •23] ; while in the table, the chance of a life of 50 Surviving one of 30 is •251.
I shall, in the next chapter, consider the application of the tables to pecuniary questions, and shall now
N
178    ESSAY ON PROBABILITIES.
proceed to point out the connexion between a table of mortality and one of population.
The whole number of persons inhabiting any country is in continual state of increase from births and immiand of decrease from deaths and emigration. There are few countries in which immigration and emigration produce any serious effect upon the population, and, in times of very moderate quiet and prosthe births always exceed the deaths : so that, generally speaking, the number of people alive in a given country is yearly augmented by the excess of the births over the deaths. If accurate registers of births and deaths (with the ages at death) were kept for a a century and a half, accompanied, if need were, by a register of incomers and outgoers, with their ages, the community would be in possession of a complete history of its statistical changes, from which the law of mormight be deduced, and its fluctuations noted, if any.
Again, if in any one year a complete census were made, registering the age of every individual, and if the deaths which took place in the 365 days next following the day of the census were noted, the law of mortality could be deduced. In such a case, the numbers of the living at every age would be so large that the proporof deaths among them in a single year could b safely depended on for pointing out, with great nearness, the law which regulates the mortality of large masses of people.
No such statistical means exist in this country, partly from the defective manner in which the censuses of population are made, partly from the circumstance of the registries of births and deaths having been, almost up to the time of writing this work, connected with the religious ceremonies of the established church, which has had the effect of excluding many dissenters from registration. In the absence of all specific information, recourse was had to the registers of burials, which are usually accompanied by a statement of the ago of the
ON LIFE CONTINGENCIES.    179
parties, though without any sufficient guarantee for the accuracy of the information. The hypothesis upon which alone registers of burials will give a correct law of mortality, requires that one of two alternatives should exist : either a permanent law of mortality, with a knowledge of the population in every year, and of the number of emigrants and immigrants, with their ages ; or a stationary population, with the same number of births and of deaths in each year, and a permanent law of mortality. This latter supposition is never exactly true ; but, as many societies have made a near approach to it, and as many tables have been constructed by its means, it will be worth while to explain the consequences of the supposition.
If the Carlisle law of mortality remained in uninoperation for a century, and if 10,000 infants were born alive in every year, the time would come when the number of the living at any age in that table would express the number alive at that age in the society in question. Thus the number of persons aged 25 would be the 5879 survivors of those who were born 25 years ago ; and the number of the living at every age and upwards would be found by multi-plying the number alive at that age by the mean duration of life in the table in p. 166. If, then, the law of mortality of such a society were required, it would be found written in the burial registers of any one year. For the numbers of births and deaths being equal, there would be found for each year 10,000 burials ; which, if the law of mortality were permanent, would be found distributed among the different ages according to the table. Hence the number, out of 10,000 born, who attain a given age, would be found by adding the number buried after that age.
But let us now suppose a population uniformly increasing from year to year, say at the rate of 2 per cent. per annum ; such a population would double itself in 35 years ; and the younger lives would always exist in a greater proportion to the older ones than
N 2
180    ESSAY ON PROBABILITIES.
would be indicated by a correct table of mortality. The burials at the younger of two ages would therefore occur in too large a proportion to those at the older. Suppose, for instance, that 350 deaths take place at the age of 40-41, and 1200 at the age of 5—6; we are not therefore to conclude, that out of 10,000 individuals born, the deaths at 40 and 5 would be as 350 to 1200: for since the population doubles itself in 35 years, those who now die aged 5, are part of twice as great a number of such lives as were of the same age 35 years ago : consequently, of the set from whom 350 died at the age of 40, 600 died at the age of 5. If, then, a table were constructed from burials alone, without paying any attention to the rate of increase of the population, the older lives would appear too good of their kind ; that is, relatively to the younger ones of the same society. This, as already observed, is the case in the Northampton table ; whereas, in the formation of the Carlisle table, proper attention was paid to the variation in question. The difference is very perceptible in comparing each of these tables with that of the insurance office which it most resembles. At 25 years of age, the mean duration of the Northtable is 30.9, and that of the Amicable 34.1. If the proportions of the mean durations remained nearly the same, (as generally happens,) then the Amicable table at 60 giving 12.5, the Northampton table should give 11.4 ; instead of which it gives 13.2.
The preceding supposes, that while the population changes, the law of mortality remains stationary. It is very unlikely that such should be the case; and observso far as it goes, tends to confirm the a priori suspicion. When provisions are cheap, or wages high,—when, in fact, it is easy to maintain a family,—marriages are more frequent, and are contracted at earlier ages. The same abundance of nourishment which tends to production, also tends to preservation, both of parents and children ; the consequence of which is, that a rapid increase of population is often accompanied by a diminution of the
ON ANNUITIES.    181
proportionate mortality. On the other hand, and from contrary causes, a diminution of the rate of population may be attended by an increase of the mortality.
As this work does not profess to enter further into statistics than is necessary to exemplify the principles of the theory of probabilities, I shall here close what I have to say on the rate of mortality, considered independently of the most important pecuniary applications. The next chapter will point out in what way money calculations are made.
CHAPTER IX.
ON ANNUITIeS AND OTHER MONEY CONTINGENCIES.

IF money could make no interest, the principles of this chapter would be simplified, and the details of calculation connected with it would be somewhat reduced in amount. It will first be requisite to point out the effect of compound interest, and to show how to make computations connected with it. The fundacalculation may be saved, for all such purposes as this work is intended to answer, by the following table, which may be described as follows. Opposite to any year in the column headed Y, and under the rate of interest in question (which is in numerals at the head), will be found, within ten shillings, the number of pounds which will, in such number of years, at such rate of interest, produce a thousand pounds. Thus, opposite to 23 years, in the column headed 3, we see 507 ; that is, 5071. (or, more strictly, something between 5161. 10s. Od. and 5171. lOs. Od.,) will, when improved at 3 per cent. for 23 years, produce 10001.
N 3