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1855.] 197

On the Calculation of Annuities, and on some Questions in the
Theory of Chances. By J. W. Lubbock, Esq., B.A.*

[Extracted, by permission of the Author, from the Transactions of the
Cambridge Philosophical Society.]

1. ±HE object of the following investigation is to show how the
probabilities of an individual living any given number of years are
to be deduced from any table of mortality. All writers (with the
exception of Laplace) have considered the probability of an indi-
vidual dying at any age to be the number of deaths at that age
recorded in the table, divided by the sum of the deaths recorded at
all ages. This would be the case if the observations on which the
table is founded were infinite; but the supposition differs the more
widely from the truth the less extended are the observations, and
cannot, I think, be admitted where the recorded deaths do not
altogether exceed a few thousand, as is the case in the tables used
in England. The number of deaths on which the Northampton
Tables are founded is 4,689 (Price, vol. i. p. 357). The tables of
Halley are founded upon the dea'ths which took place at Breslau,
in Silesia, during five years, and which amounted to 5,869.

If a bag contain an infinite number of balls of different colours
in unknown proportions, a few trials or drawings will not indicate
the proportion in which they exist in the bag, or the simple pro-
bability of drawing a ball of any given colour; and not only the
probability of drawing a ball of any given colour, calculated from a
few observations, will be little to be depended on, but it will also
differ the more from the ratio of the number of times a ball of the
given colour has been drawn, divided by the number of the pre-
ceding trials, the fewer the latter have been.

Laplace (Theor. Anal, des Probabilites, p. 426) has investi-
gated the method of determining the value of annuities. He there
says — " Si l'on nomme y„ le nombre des individus de Page A dans
la table de mortalite dont on fait usage, et y x le nombre des indi-
vidus k l'age A + x, la probabilite de payer la rente a la fin de

l'annee A + x sera — ." This hypothesis coincides with that I have

y<>

before alluded to, as adopted by all other writers. Laplace, how-
ever, means this as an approximation, for he has investigated
differently the probability of an individual of the age A living to
the age A + a (p. 385 of the same work). He there considers two
cases only possible ; but as an individual may die at any instant

* Now Sir John William Lubbock, Bart., F.R.S., &c.

198 On the Calculation of Annuities, [April

during life, I think it may be doubted whether this hypothesis of

Captain John Graunt was the first, if I am not mistaken, who
directed attention to questions connected with the duration of life.
He published a book in 1661, entitled Observations on the Bills
of Mortality, which contains many interesting details, although it
is written in the quaint style which prevailed in those times. In
this book, amongst other tables there is one showing in 229,250
deaths how each arose ; and another showing of 100 births " how
many die within six years, how many the next decad, and so for
every decad till 76" — which is in fact a table of mortality, and is
probably the first ever published.

After Captain Graunt, Sir W. Petty published his Essays on
Political Arithmetick. Halley, however, was the first who calcu-
lated tables of annuities : he took the probabilities on which they
depend, from a table of mortality founded on the deaths during
five years at Breslau. Since his time a great number of writers
have treated of these subjects, of whom a notice may be seen in
the Encyclopaedia Britannica, or in the Report from the Committee
on the Laws respecting Friendly Societies, 1827 (p. 94). It is to
be regretted that those who have published tables of mortality
should generally not only have altered the radix or number of
deaths upon which the table is constructed, but also the number
of deaths recorded at different ages, in order to render the decre-
ments uniform; this is the case particularly with the Northamp-
Payments, vol. i. p. 358). For if observations were continued to a
sufficient extent, they would probably show that some ages are
more exposed to disease than others — that is, they would indicate
the existence of climacterics, of which alterations such as these
destroy all trace.

I annex four tables,* which I have calculated, with the assist-
ance of Mr. Deacon, from the Tables of Mortality for Males and
Females at Chester, given by Dr. Price (vol. ii. p. 392). The first
two tables show the probability of an individual at any age living
any given number of years, as well as the expectation of life at any
age. The last two show the value of £1 to be received by an
individual of any age after any number of years, and the value of
an annuity. The difference between these values for a male and
female, is very great, and shows that tables which would be appli-
cable for the one would not be for the other.

* See page 207, and the note there.

1855.] and on some Questions in the Theory of Chances. 199

I have also subjoined a table comparing the values of annuities
calculated from observations at Chester (according to the hypo-
thesis of probability I have assumed), with some which have been
calculated from observations at other places. Until lately, the
Government of this country granted annuities, the price of which
depended on the price of stock, which renders their tables com-
plicated. I have given their values of a deferred annuity for five
years, compared with those I have calculated from the observations
at Chester : it will be seen that the former are much too high.

2. Suppose a bag to contain a number of balls of p different

colours, and that, having drawn m x -\-m 2 + m 3 + m p balls, m x

have been of the first colour, m 2 of the second colour, m 3 of the
third colour, m p of the p th colour. If oc X) x 2 , x 3 . . . x p are the
simple probabilities of drawing in one trial a ball of any given
colour, the probability of the observed event is x™y x x 2 m > • • • • x

x p m p, multiplied by the coefficient of Xi m tx 2 m i x p m r in the

development of {x x + x 2 . . . + x p ) m i +m z- •■■+ m P . The event being
observed, the probability of this system of probabilities is
x^i x ar 2 m * • • • • x x t m p, divided by the sum of all possible values
of this quantity.

The probability in n x -\-n 2 . . .. + n p subsequent trials of having
w, balls of the first colour, n? of the second, n p of the p & , is a
fraction of which the numerator is the sum of all the values of
^.^,+w, x# 2 "» 4 +» s . . . .xx p m p +n p, and of which the denominator
is the sum of all the values of j," 1 ! xx 2 m * .... xx p m p, multi-
plied by the coefficient of ^"i x #2% .... xx p n p in the develop-
ment of (Xi+X 2 + X 3 ....+X p ) n l +n *~~ +n P.

Since x x + x 2 . . . .+x p =l, if x u x 2 , &c. be all supposed to
vary from to 1, and all these values to be equally possible a
priori, the numerator will be found by integrating the expression

Xl m \+"\ x a^™ 2 " 1 "" 2 - • •(! — *i — x 2— «3 — x r -\T r + " p d*\ xdx 2 ...x dx p _ x

first from x p _ 1 =0 to x p ^ 1 = l— x x — x 2 — x p _ 2 , then from

x p - 2 =0 to :rp_2=l — x y — Xp-z, and so on. The deno-
minator will be found in the same way.

If the coefficient of x x m \ x x 2 m z . . . x x p m p in the development
of (Xi + x 2 . . . . +x p ) n i +n f"+ n p be called C, these integrations
give for the probability required

(m i + l)(m 1 + 2)(m l + 3)...(m 1 +n i )(m 2 + l)(m i + 2)...(m 2 +n 2 )...

Cx

(mi + mi+nii + m p +p)(m 1 + m 2 +m 3 + m p +p+l)

(m p +l)(m p +2) (m p +n p )

(mi+m 2 +p+n l + n 2 +n 3 — 1)'

200 On the Calculation of Annuities, [April

or if the product (m p +l) (m p + 2) .... (m p + n p ) be denoted by
[m p +l]"p, which is the notation used by Lacroix (Traite du
Calcul Differentiel, vol. iii. p. 121), the probability required is

c . Qi + lM^ + l]" 2 - •••[>,+ 1]V

[j», + W 2 m p +^]"l+"2+"3 • • •+",

This probability is the same as if the simple probability of drawing
a ball of the p & colour were m p + 1, with the difference of notation.
When n 2 , n 3) « p _i, &c. = 0, and n p = 1, this expression
gives for the chance of drawing a ball of the p 01 colour

m p + l

mi + m 2 .... + m p +p'

and the probability that the index of the colour drawn is be-
tween n— 1 and n + q + 1 is

m n +m n+l .... m n+t +q
mi + mz .... +m p +p

If we suppose the law of the possibility of life to be such that p
cases or ages are possible — a priori, m lt m n , &c. will be the number
of recorded deaths in a table of mortality at those respective ages,
and the chance of an individual living beyond the » th age will be

»>»+>»„+i .... m p +p— n

mi + nti .... +m p +p

m n + m n _i + &c. + m p is the number given by the table as living at
the n th year ; therefore, on the hypothesis of this law of possibility,
the chance of an individual living beyond the w th year is a fraction
of which the numerator is the number living at that age, +p—n,
and the denominator is the whole population on which the table
is founded, or the radix + p. The Tables I. and II. have been
calculated from this formula, from observations at Chester given
by Dr. Price (vol. ii. p. 107) : p was taken equal to 101 for a child
at birth — that is, the chances of a child living beyond a hundred
years, and of its dying in each intermediate year, were supposed
to vary from to 1, all these values being equally probable,
a priori. The value of any sum to be received after any number
of years is equal to the sum itself, multiplied by the chance of
the individual being alive to receive it : therefore these tables
give the value of unity to be received after any number of years.
Considering duration of life to be valuable in proportion to its

Cx

1855.] and on some Questions in the Theory of Chances. 201

length, the value of the expectation of life to any individual is the
sum of the chances of his living any number of years multiplied
by the intervening time ; so that if P„. be the chance of an indivi-
dual living exactly n years, the value of his expectation of life is
2nP„, which is evidently equal to SP'„, if P'„ be the chance of an
individual surviving n years : therefore the value of the expecta-
tion of life of any individual is the sum of the numbers on the
same line in Tables I. and II. The unity of expectation is here
the expectation of an individual who is certain to live exactly one
year. The Tables I. and II. give the values of contingencies
depending on a single life, without discount ; the Tables III. and
IV. are the same values, discounted at the rate of three per cent.
compound interest. These tables give the values of annuities
about six per cent, higher than those calculated from the North-
ampton, and given by Dr. Price, vol. ii. p. 54. The only tables
of annuities on female lives that I have met with are calculated
from observations in Sweden, and are given by Dr. Price, vol. ii.
p. 422 ; but they are calculated at four and five per cent, interest.
It is not to be expected, however, that tables calculated from
observations made in one country will serve in another, or even in
different parts of the same country.*

The probability of having ra, balls of the first colour in % + N
trials, the colours of the other N balls being any whatever, is

fxf^jl-x^fx^xfs (l— Xl — X2 x^-^pdx^x^ dx p _ x

fxi'ix^" 2 . . . .(1 — Xy — x 2 . . . .x p _ l )'"pdx l dx 2 . . . .dx p _i '

multiplied by the coefficient of x n \ in the development of
{ x i+y) n ^> the integrals being taken between the same limits
as before.

These integrations give for the probability required

(»»i + l)(>»i + 2). ..(wi+7tiX»»2+'»3+'»4. ..+p— l)(m 2 +m 3 +m 4 . ••+/>).
{mi+m 2 + m p +p)(m l + m 2 +m p +p+l)

(m 2 +m 3 + m 4 +jp+N— 2)

(j»,+m 2 + m 3 +p + m + N— 1)'

C being equal to k+l> ^V///.^ ' Ado P tin S the
same notation as before, this probability is equal to

* Since writing the above, I find that Mr. Finlaison has given the values of
annuities, distinguishing the 6exes, in the Report of the Committee on Friendly Societies,
1825, p. 140.

VOL. V. P

202 On the Calculation of Annuities, [April

[m l +iyi[m i +m 3 ..+m p +p— l] N

[»ii + »n 2 + TOa «i p +jh]"i +n

CQi + lJ"^-^ +m p +p— l] CT i +1

~~ [m 2 +OT 3 ....+N+i»-l]"i + '" 1+1

which probability, as before, is the same as if the simple probability
of drawing a ball of the />*** colour were »ip + l.

If m 2 + m 3 . . . . + m p +p—2=M., and if n, and N are in the
same ratio as »i, and M, the chance that the number of balls of
the first colour in «i + N trials is between the limits n x and n x ±z,
by the reductions given in the Thiorie Anal, des Probabilites,
p. 386, is

1—21/ — — i—- +f — - / dze 2m 1 M(N+» 1 )(M + N+m 1 +n 1 ),

e being the number of which the hyperbolic logarithm is unity,
and the integral being taken from z=z, to ^=infinity.

The question of determining the probability that the losses
and gains of an Insurance Company on any class of life are
contained within certain limits, is precisely similar to this.

It will be seen from the formula "»*+ "Wi- •••"»»+? + g (p . 2 00,

mt + nit + m p +p xr

line 12), that if life were divided into an infinite number of ages or
intervals (in which case p is infinite), the hypothesis of possibility
remaining the same, the probability of an individual dying in any
given interval would be the given interval divided by the whole
duration of life, which coincides with that which is given by De
Moivre's hypothesis. Thus if life were supposed to extend to a
hundred years, the probability of an individual dying in any
given year would be -j-^-g, and any finite number of observa-
tions or recorded deaths would not influence the value of this
probability. As diseases and other causes producing death are
not equally distributed throughout life, the last hypothesis cannot

In order to investigate accurately the probability of death at
any age, it would be necessary to know the law of possibility.
Let <j>pX p be the probability of the possibility of x v : then the pro-
bability in the former question of having n x balls of the first colour,
w 2 of the second, &c, in n 1 + w 2 • • • • + w j> trials, is

yy l+ "'(frgi>2 » + "Kf i ig!i)...(l— x l -x i ...x„_i)'"'+ n pdx i dx 2 — dx p _ x
/*i"i(0i*i), «™s(^ 2 iF 2 ) (1 —xi —x 2 . . .x p ^) m p dxidxi —dx p ^ '

1855.] and on some Questions in the Theory of Chances. 203

is a sign of function, and this function may be either continuous
or discontinuous.

This expression must be integrated between the same limits as
before.

The coefficients of the different powers of x p in <j>pX p , or the
constants in <j>pX p , will generally be functions of the index p. If
the probability of life were known at a great many places, and if
x Pl were the value of x p at q v places, x P9 at q t places, &c, the law
of possibility might be determined approximately by considering
QpXp as a parabolic curve, of which x p is the abscissa, passing
through the points, of which the ordinates are

?1 ?2

?1 + ?2 +> &c- ' 7i + ft +> &c. '

3. In the preceding investigations, the results of the preceding
trials are supposed to be known ; it may be worth while to examine
what the probability of any future event is when the results of the
preceding trials are uncertain.

Let a bag contain any number of balls of two colours, white
and black: suppose m trials have taken place; and let e„ be the pro-
bability that a white ball was drawn the /1 th trial, /„ the probability
that a black ball was drawn.

e„+f n =l.

First let e lt e 2 . . . . e„ be all equal, and let x be the probability
of drawing a white ball. If a white ball was drawn every time
in the m trials which have taken place, the probability in «i + n 2
future trials of having n x white balls and » 2 black balls is

(«i+«2)(wi+n 2 -l)---("i + l)/^- | "''(l-^" t ^
1.2 n n faf'dx

But the probability that a white ball was drawn every time is e m ;
therefore the probability of drawing a white ball n t times and
a black ball n 2 times, on this hypothesis, multiplied by the pro-
bability of the hypothesis, is

(nt+nzXnx + nz— l)...(ni + l) „ /^ m+ °» (1— x) n 2dx
1.2 n 2 e ~JaFTx ;

and the probability of drawing n x white balls and w 2 black balls
will be the sum of the probabilities on every hypothesis, multi-
plied respectively by the probability of the hypothesis, which is

p2

204 On the Calculation of Annuities, [April

(wi + H^fa + n,-!).. .(«i + l) ( m f<r**jl-xpd*
1.2 « 2 \ e /V<fo

«/ /ar-\l—x)tb "*" 1.2 ^ faf-Xl-&yki '

This integral being taken from a;=0 to 03=1, is

(» 1 +«g)(wi+n a — l)..(wi + l) r w 2 ,n 2 — 1,^— 2 l.m+1

1.2 n 2 \m+n 1 + l.m+n 1 + 2...m+n 1 + ii2+l

ni+\.iu.iu — 1....2
+me m -]f. — 2.J! m+l.m+, &c.

J» + »!.»» + Wi + l . . . .Wi+»]+W 2

_ (wi + w 2 )(wi+w 2 — 1) (wi + 1) 1

1.2. .. . n 2 m+2.w+3. . .w + » 1 + «2+i

|n 2 .»! 2 — 1 .n 2 — 2. ..m+ni.m+ni — 1. ..m+1 .e m +n 2 -i-l ■ ••
2.m+n x — 1. . .m+1 .m .me m ~]f+ , &c.

_,,. . . lx rf n . + " 2 .« B 3 a^i(ex+/w)"' , ,

This series is equal to , - , . ±3d— when x and

^ dx n \dy n i

y are made equal to 1, and this is equal to 1 . 2 . 3 . . . . n x . 1 . 2 . 3

. . . . ra 2 x coefficient of A"i k n *, in the development of

(i+ hp(i + A)xi + eh +/ty

(l +e h+/ky=l+m(eh+fk)+^^-(eh+/ky

m.m — l.m — 2, , „,,„
+ j-^3 («*+/*)»+ , &c

(1 + hfi(l+kp=h"iJ^+nih' , i- l k"2+ Wl '" 1 ~ 1 A"i-**-»
+ n 2 &"i/E"2- ■ + » . w^".-'^- 1 + - '"' ' "' ~ * A-i- 2 ^-' + , &c.

Coefficient of h" i A"* = 1 + Jn(n,e + n 2 /)

m.m—l( w,.^— 1 n 2 .n 2 —l\

1855.] and on some Questions in the Theory of Chances. 205

The probability required is

1.2.3 «i+«2

m+2.m+3. . . .m+ni + t^+l

l+ m (n l e+T h f)+ — | - l - e i +2n l n 2 e/+ 2 - » g f A+, &c.

If there are p different colour's, and if m trials have taken place,
and e g , p is the chance that a ball of the p th colour was drawn
the q th trial, the probability of drawing n x balls of the first colour,
n 2 of the second, n P of the p^, in n t + n 2 . . . . +n p future trials,
may be found in the same way. Let

«1,1 +«1,* + «1,I+ &c «l,n = S 1 , «l,

e l, 1 J *1, 2 "I" *1, 3 » e l, 4 < ^ C> = "2> *1>

(the sum of the products of e 1 two and two together,)

«1,1> «2,S + *1,8> «2,3+»&C. =S 1 « 1 , SAj*

and so onj then it may be shown that this probability is equal to

1.2.3.n 1+ n 2+ n 3 . ■ + n, (1+Sei) „ l(1 + S ^.. .. (1 + S ^,
>n+^ w+w 1 +n 2 .. ..+n p +p— 1

1 + (Sej), 1 + (Se 2 ), &c. being expanded by the binomial theorem,
and the indices of S written at the foot.

The method which was used for summing the series in the last
page is of very general application, and depends, in fact, on this
principle, that the generating function of the sum of any series
is» the sum of the generating functions of each of the terms of the
series.

If in the last formula « 2 , n 3) &c.=0, and if there be only two

events possible, and n x = 1, the probability required is — — ■— . In

order to apply this, suppose an individual to have asserted m
events to have taken place, of which the simple probabilities are
equal, and equal to p ; and suppose it required to find the proba-
bility of his telling the truth in another case, where the simple
probability of the event he asserts to have taken place is not
known. Let x be the veracity of the individual, the probability of

his telling the truth on this hypothesis is — ,., — . ,., r : and

° Jr px+(l—x)(l—p)

the probability of his telling the truth is the sum of the pro-
babilities of his telling the truth on each hypothesis, divided by
the number of the hypotheses.

* This is a method of notation which ohtains, but it is not meant to imply that
S 1 e 1 S 1 e 2 =S,,e 1 x S„ e 2 .

206

On the Calculation of Annuities,

[April

Suppose x to vary from to 1, and all these values of x to be
equally probable a priori, then the probability of his having told

the truth and the event having taken place is / — i-r. — ttt r,

r J px+(l'—p)(l—x)

taken from x=0 to x=l, which integral is

cM'-s^Ty-

9

If p = -^, this probability is '81601. Generally, Up > £, the

assertion that the event has taken place (on this hypothesis of
veracity) rather diminishes the probability that the event has taken
place; if p= J, the assertion does not alter the probability; if p < \,
the assertion rather increases it.

9-1601

9 1 + Se

If P=^> «= -81601, let jh=10; then tlf^

r 10 m+x

12

which is the probability that the individual will tell the truth in
another case. If the individual had told ten truths, the chance of

his telling the truth in another case would have been — •

All values of x between and 1 were supposed equally pos-
sible : if they are not, let <j>x be the probability of the possibility
of any value of x ; then the probability of an individual telling the
ffxdx

truth will be

'px+(l — x) (1— p)
grals being taken from x=0 to <e=1.

divided by f\$xdx, these inte-

Table formed from the Burials in All Saints' Parish, Northampton,
from 1735 to 1780. (Seepage 198, line 26.)

Age.

Actual

Number of

Burials.

By the

Observations.

As altered by
Dr. Price.

Under . . 2

Between 2 and 5

S „ 10

„ 10 „ 20

„ 20 „ 30

30 „ 40

„ 40 „ SO

. ,, SO „ 60

60 „ 70

• „ 70 „ 80

„ 80 „ 90

„ 90 „ 100

Total . . .

1,529
362
201
189
373
329
365
384
378
358
199
22

37981
899J
499*
469|
926|
817|
906}
954
939J
889|
494|
Mi-

4367
1034
574
543
747
750
778
819
806
763
423
46

4,689

ll 650

11650

1855.] and on some Questions in the Theory of Chances.

207

Table I. — Males.

Table II. — Females.

Expectation

of Life,

by the Author's

method.

Expectation

Age.

of Life, by
usual method.

29-75345

2813

1

36-69541)

35-76

2

40-21306

39-42

3

42-54615

41-97

4

43-83928

43-33

5

44-09357

43-20

10

42-75204

41-92

15

38-95786

38-05

20

35-82561

34-86

25

32-99367

32-00

30

30-27118

29-25

35

27-04332

25-97

40

2404172

22-92

45

21-34876

20-20

50

18-81444

17-64

55

16-34857

15-14

60

13-63392

12-36

65

12-05917

10-79

70

9-41263

8-05

75

8-43636

7-00

80

6-99009

5-43

85

5-90384

4-25

90

4-32000

2-50

95

2-14285

1-00

Table III.—

-Males.

Age.

Value of
Annuity by
the Author's

method.

Value of

Annuity by

usual method.

13-96256

1

17-35468

2

19-17322

3

20-38907

4

21-15972

5

21-42118

21-283

10

21-55443

21-512

15

2038198

20-283

20

19-42818

19285

25

18-55566

18-399

30

17-67138

17-492

35

16-36473

16134

40

15-05567

14-667

45

13-81590

13-493

50

12-59164

12-316

55

11-28375

10-866

60

9-63491

9-140

65

8-77709

8-220

70

6-94786

6-260

75

6-17140

5-291

80

5-11141

85

4-30505

Expectation

Expectation *

Age.

by the Author's
method.

of Life, by
usual method.

34-55535

33-27

1

40-04475

39-54

2

43-65276

43-25

3

45-87700

45-68

4

47-23533

4711

5

47-99860

47-44

10

45-69310

45-17

15

41-96030

41-36

20

38-76308

38-10

25

35-49329

34-78

30

32-79.565

32-27

35

3000384

29-26

40

27-13292

26-37

45

24-29072

23-50

50

21-43212

20-62

55

18-35900

17-52

60

15-09954

14-20

65

12-83834

11-94

70

9-78378

8-81

75

8-12794

7-14

80

6-30434

5-20

85

5-97402

4-81

90

4-55263

3-46

95

2-07692

1-71

Table IV. — Females.

Age.

Value of
Annuity by
the Author's

method.

Value of

Annuity by

usual method.

15-75290

1

18-42550

2

20-14850

3

21-32367

4

22-11858

5

22-64306

22-624

10

22-41169

22-439

15

21-25267

21-235

20

20-36435

20-323

25

19-33571

19-265

30

18-65687

18-583

35

17-62920

17534

40

16-56779

16-366

45

15-42043

15-282

50

14-14872

13-983

55

12-58232

12-376

60

1067266

10-410

65

9-37655

9080

70

7-26287

6-889

75

6-02689

5-338

80

4-65536

85

4-34792

Note. — We do not reprint all the tables originally given in this paper. They are of
value now only as exhibiting the results of the methods laid down by the Author, and
for that purpose those here quoted will suffice. — Ed. A. M.

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