# The Pamphlet Collection of Sir Robert Stout: Volume 8

# I. Probability and the Average Duration of Life

### I. Probability and the Average Duration of Life.

1. If *a* be the number of ways in which an event can happen, and *b* the number of ways in which it can fail, and all these ways are equally possible, the probability of the event happening is , and the probability of its failing is

2. The probability of the continuance of life of one person is very uncertain, and differs under different circumstances in persons of the same age. It is greater in healthy than in unhealthy persons; in persons living in a healthy climate or locality than in persons living in an unhealthy one; in persons living in the country than in persons living in a large town or city; in persons following healthy or safe occupations than in persons following unhealthy or dangerous ones; in females than in males; and in married persons than in single persons. For general purposes it is sufficient to find the average probability out of a large number of unselected persons of the same age. This average probability is found by a Table of Mortality showing, from experience, out of a large number of unselected persons of the same age, the number dying in each succeeding year of age, and, consequently, the number living at each successive age. It is necessary to assume, that out of any other equal number of persons of the same age and class the number living at each age would be nearly the same, if they live under similar circumstances.

page 2(a) | The Friendly Societies' Table, constructed by Mr. Ansell, from returns obtained by the Society for the Diffusion of Useful Knowledge from Friendly Societies in England for 1823-27, embracing the experience of 24,323 years of life.^{*} |

(b) | The Friendly Societies' Table, constructed by Mr. Finlaison, from the returns of Friendly Societies in England and Wales for 1836-40, sent to the Registrar, embracing the experience of 1,147,143 years of life.^{†} |

(c) | The Odd Fellows', Manchester Unity Table, constructed by Mr. Ratcliffe, from returns of the Lodges composing the Unity for 1846-48, embracing the experience of 621,561 years of life.^{‡} |

(d) | The Friendly Societies' Table, constructed by Mr. Finlaison, from the returns of Friendly Societies in England and Wales for 1846-50, sent to the Registrar, embracing the experience of 793,759 years of life.^{§} |

(e) | The Odd Fellows', Manchester Unity Table, constructed by Mr. Ratcliffe, from returns of the Lodges composing the Unity for 1856-60, embracing the experience of 1,006,272 years of life.^{¶} |

(f) | The Odd Fellows', Manchester Unity Table, constructed by Mr. Ratcliffe, from returns of the Lodges composing the Unity for 1866-70 (see Table I., page 5), embracing the experience of 1,321,048 years of life, being the largest experience hitherto collected of Friendly Societies in the United Kingdom.^{‖} |

4. A comparative view of these tables would be useful, hut would be foreign to the object of this Treatise.

^{*} *A Treatise on Friendly Societies*. By Charles Ansell, Esq., F.R.S., 1835, p. 119.

^{†} *Contributions to Vital Statistics*. By F. G. P. Neison, F.L.S., 1845, p. 28.

^{‡} *Observations on the Rate of Mortality and Sickness existing amongst Friendly Societies*. By Henry Ratcliffe, 1850, p. 24.

^{§} *Return of the Friendly Societies' Sickness and Mortality*. Mr. Alexander Glen Finlaison's Report. Ordered by the House of Commons to be printed, 16th August, 1853, p. 3.

^{¶} *Observations on the Hate of Mortality and Sicckess existing amongst Friendly Societies*. By Henry Ratcliffe, 1862, p. 25.

^{‖} *Supplementary Report, July 1st. 1872*, of the Independent Order of Odd Fellows, Manchester Unity Friendly Society, p. 20.

5. Let *l _{x}* denote the number of persons living at the age

*x*according to any mortality table,

*the number living at the age*

^{n}l_{x}*n*years older than

*x*, and

*-*the number living at the age

^{n}l_{x}*n*years younger than

*x*; then, by Art. I, the probability of a person whose age is

*x*living n years is

Example 1. Probability of a person aged 18 living 7 years, or attaining the age of 25, according to the Manchester Unity Table of Mortality for 1866-70.

6. Of *l _{x}* persons living at the age

*x, l*is the number dying between the ages

_{x}—^{n}l_{x}*x*and

*x*+

*n*, or within

*n*years; then, by Art. 1, the probability of a person whose age is

*x*dying within n years is

Example 2. Probability of a person aged 18 dying within 7years, or before attaining the age of 25, according to the Manchester Unity Table of Mortality for 1866-70.

7. The ratio of the number of persons dying within a year to the number living at the beginning of the year is called the *yearly rate of mortality*. Thus, of 100,000 persons living at the age 18, 560 die within a year. Therefore 560÷100000×100=.5600, is the yearly mortality per cent., according to the Manchester Unity Table of Mortality for 1866-70.^{*}

8. The average number of years which persons of the same age, one with another, live according to the given mortality table, is called the *expectation, or average duration, of life*.

^{*} The number dying in each year of age being expressed in the Manchester Unity Table of Mortality by a whole number, decimal omitted, the correct number, including decimal, may be found by multiplying the mortality per cent. for each age by the number living at each age, and then dividing by 100. Thus the number dying in the 20th year of age is 5817 x 99440/100=578.442480, and so on. The decimal must be supplied as in column *d* of Table I. in calculating the rate of mortality per cent, for each age.

Let *ex* denote the expectation, or average duration, of life of a person of the age *x, z* the number of years from age *x* to the oldest age attained by any life according to the table, and let us suppose that the persons dying in each year die at equal intervals therein. Then of *l _{x}* persons living at the age

*x, l*—

_{x}^{1}

*l*die in the first year, living among them ½(

_{x}*l*—

_{x}^{1}

*l*) years, while those who attain the age

_{x}*x*+

*1*live 1

*lx*whole years; therefore the

*l*persons live in the first year, ½(

_{x}*l*

_{x}^{1}

*l*)+

_{x}^{1}

*l*=½(

_{x}*l*+

_{x}^{1}

*l*) years. Similarly

_{x}*l*persons live through life ½(

_{x}*l*+

_{x}^{1}

*l*)+½(

_{x}^{1}

*l*+

_{x}^{2}

*l*)+...+½(

_{x}^{z−1}

*l*+

_{x}^{z}

*l*)=½

_{x}^{z}

*l*=½

_{x}*l*+

_{x}^{1}

*l*+...+

_{x}^{z}

*l*) years, which divided by

_{x}*lx*, gives

This demonstrates the form of the rule.

The rule is, *divide the sum of the numbers living at all the older ages by the number living at the given age, and then add half a year to the quotient*.

The number in column L of Table I., opposite each age, is the sum of the numbers living at all the older ages; thus

L_{x}=^{1}*l _{x}*+

^{2}

*l*+

_{x}^{3}

*l*... to the last tabular age;

_{x}L* _{x}*=

^{1}

*l*+

_{x}^{1}L

*;*

_{x}By means of column L we are now enabled to compute the average duration of life by simple division.

Example 3. Average duration of life of a person aged 18, according to the Manchester Unity Table of Mortality for 1866-70.

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