XXVIII-1. To find, by Table VI., the probability that a given life will survive a certain number of years, or attain a given age. Divide the number living at the increased age by the number living at the given age; the result will be the required probability.

Required the probability that a person now aged twenty will attain the age of fifty, according to the Northampton table of mortality?

The number living at 50, the increased age, by Table VI.,=2857
The number living at 20, the given age, by Table VI., . =5132'
Therefore 2857-5132 (IV.)=.5567, the required probability.

What is the probability of a person now aged forty-five surviving one year?

The number living at 46, the increased age, by Table VI.,=3170
The number living at 45, the present age, by Table VI., . =3248'
Therefore 3170-3248 (IV.) .9760, the required probability.

A person aged sixty requires to know what probability there is of his attaining the age of sixty-two?

Ans., .9195, the probability.

Ans., .8325, the probability.
Ans., .9670, the probability.
Ans., .5479, the probability.
Ans., .9370, the probability.

2. To find the probability of a given life failing to attain a

given age.

Divide the difference of the number living at the present and increased ages by the number living at the present age; the result will be the probability required.

Required the probability that a person now aged twenty shall not live to attain the age of fifty, according to the Northampton table of mortality? 5132, the number living at 20, the present age;

2857, the number living at 50, the increased age;

Therefore 5132-2857=2275, the difference between the numbers living;
And 2275÷5132 (IV.)=.44329, the required probability of failure.

The probability of failure is the difference between the probability of attaining the given age and unity;
Thus the probability of a life of 20 attaining the age of 50 was found, by question first, to be .5567;
Therefore 1.0000.5567.4433, the required probability of failure nearly.

XXIX. To find the expectation of life at any given age; or the number of years which lives of any given age, taken one with another, have an equal chance of attaining.

Divide the sum of all the living beyond the given age by the number living at the given age; the result, increased by half unity or .5, will be the expectation required.

Required the expectation of a life aged ninety-three, according to the Northampton table of mortality?

9+4+1=14, the number, by Table VI., of all the living beyond the age of 93; And 16, the number, by Table VI., living at the given age of 93;

Therefore 14-16=.87, which, increased by .5, 1.37, the expectation required.

Required the expectation of a life aged ninety-four?

of 94;

4+1=5, the number, by Table VI., of all the living beyond the age And 9, the number, by Table VI., living at the given age of 94; Therefore 59= .55, which, increased by .5, 1.05, the expectation required.

=

Required the expectation of a life aged ninety-five?

1 is the number, by Table VI., of all the living beyond the age of 95;
And 4, the number, by Table VI., living at the given age of 95;
Therefore 1÷4=.25, which, increased by .5, .75, the expectation required.

=

Required the expectation of a life aged ninety-six ?

O is the number, by Table VI., at the age of 96;

=

Therefore 0, increased by half unity or .5, .5, the expectation required.

What expectation of life has an infant which is just one year old? 278898, the number, by Table VI., of all the living beyond the age of 1; And 8650, the number, by Table VI., living at the given age of 1;

Therefore 278898÷8650=32.24, which, incr. by .5, -32.74, the expectation required.

Required the expectation of a life aged two years?

271615, the number, by Table VI., of all the living beyond the age of 2;

And 7283, the number, by Table VI., living at the given age of 2;

Therefore 271615-7283-37.29, which, incr. by .5, 37.79, the expectation required.

Required the expectation of a life aged three years?

264834, the number, by Table VI., of all the living beyond the age of 3 ; And 6781, the number, by Table VI., living at the given age of 3; Therefore 264834-678139.05, which, incr. by .5,=39.55, the expectation required.

Continue these operations until the following Table VII., of the expectations of life at every age, be entirely constructed, according to the Northampton table of mortality.

Showing the expectations of life at every age, deduced from the observations made at Northampton.

XXX.-To find the present value, according to any table of mortality, of an annuity of £1, during the whole term of any given life, at any given rate per cent.

Multiply the pres. val., as found by Tab. III., of £1 for one year, at the given rate per ct.,byThe probability (XXVIII. 1.) of the given life surviving one year; and by—

The present value, increased by unity, of an annuity of £1, on a life one year older than the given life:

The result will be the pres. value of £1 per ann., during the whole term of the given life.

What is the present value of an annuity of £1 during the whole term of a life aged ninety, according to the Northampton Table of mortality, at the several rates of 3, 4, and 5 per cent. interest per annum?

No values of annuities being as yet given, we must evidently commence with the eldest life, 96, in Table VI., and calculate the values of every intermediate life until we arrive at the value of the given life, 90.

The present value of £1 for one year, as found by Table III.,

At 3 per cent.=£.970874;

At 4 per cent.=£.961538;

At 5 per cent.=£.952381.

The probability (XXVIII. 1.) of a life of 96 surviving one year;
Therefore, during a life of 96,

The present value of an annuity, at any rate per cent.,=0.

The probability (XXVIII. 1.) of a life of 95 surviving one year=;

The pres. val., incr. by unity, of an ann. of £1, on a life one year older than 95,=1+0; Therefore, during a life of 95,

(£.970874 × 1) × 1.0=£.242, present value of an annuity of £1, at 3 per cent.; (£.961538 × 1) × 1.0=£.240, present value of an annuity of £1, at 4 per cent.; (£.952381 ) × 1.0=£.238, present value of an annuity of £1, at 5 per cent.

The probability (XXVIII. 1.) of a life of 94 surviving one year=;

The pres. value, incr. by unity, of an ann. of £1, on a life one year older than 94, At the several rates of 3, 4, and 5 per cent., -1.242, 1.240, and 1.238, respectively;

Therefore, during a life of 94,

(£.970874 × 4)× 1.242=£.536, present value of an annuity of £1, at 3 per cent.; (£.961538 × 4) × 1.240=£.530, present value of an annuity of £1, at 4 per cent.; (£.952381 × 3) × 1.238=£.524, present value of an annuity of £1, at 5 per cent.

The probability (XXVIII. 1.) of a life of 93 surviving one year=1%;
The pres. value, incr. by unity, of an ann. of £1, on a life one year older than 93,
At the several rates of 3, 4, and 5 per cent., -1.536, 1.530, and 1.524, respectively;
Therefore, during a life of 93,

(£.970874 X) X 1.536=£.839, present value of an annuity of £1, at 3 per cent.; (£.961538 X) X 1.530= £.827, present value of an annuity of £1, at 4 per cent.; (£.952381 × √) × 1.524=£.816, present value of an annuity of £1, at 5 per cent.

The probability (XXVIII. 1.) of a life of 92 surviving one year=1;
The pres. value, incr. by unity, of an ann. of £1, on a life one year older than 92,
At the several rates of 3, 4, and 5 per cent.,=1.839, 1.827, and 1.816, respectively;
Therefore, during a life of 92,

=

(£.970874 × 16) × 1.839 £1.190, present value of an annuity of £1, at 3 per cent.; (£.961538 × 1) × 1.827=£1.171, present value of an annuity of £1, at 4 per cent.; (£.952381 x ) x 1.816=£1.153, present value of an annuity of £1, at 5 per cent.

The probability (XXVIII. 1.) of a life of 91 surviving one year=;

The pres. value, incr. by unity, of an ann. of £1, on a life one year older than 91, At the several rates of 3, 4, and 5 per cent.,=2.190, 2.171, and 2.153, respectively; Therefore, during a life of 91,

(£.970874 x) x 2.190=£1.501, present value of an annuity of £1, at 3 per cent.; (£.961538 × 4)× 2.171=£1.474, present value of an annuity of £1, at 4 per cent.; (£.952381 × 4)× 2.153 £1.447, present value of an annuity of £1, at 5 per cent.

The probability (XXVIII. 1.) of a life of 90 surviving one year;
The pres. value, incr. by unity, of an ann. of £1, on a life one year older than 90,
At the several rates of 3, 4, and 5 per cent.,=2.501, 2.474, and 2.447, respectively;

Therefore, during a life of 90,

(£.970874 × 1) x 2.501-£1.794, present value of an annuity of £1, at 3 per cent.; (£.961538) × 2.474 £1.758, present value of an annuity of £1, at 4 per cent.; (£.952381 × 34) × 2.447=£1.723, present value of an annuity of £1, at 5 per cent.

Making the present value of an annuity of £1, during a life of ninety, at the several rates of 3, 4, and 5 per cent.,=£1.794, £1.758, and £1.723, respectively.

Continue these operations until the following Table VIII. be completely constructed, at the several rates of 3, 4, and 5 per cent. per

annum.