The probability that one of the events will happen is the
total of the probabilities of each individual event happening.
(Probability of
living 1 year) + (Probability of Dying Within 1Year) = (Probability of Either
Living or Dying That Year)
Symbols can be substituted for each of the above
expressions, as follows:
Substitute p_{x}
for (Probability of Living 1 Year)
Substitute q_{x}
for (Probability of Dying within 1 Year)
Substitute 1 (certainty) for (Probability of Either Living
or Dying That Year)
Consequently, the equation is
To Illustrate:
Given that p_{46} = 0.995138,
how many persons age 46 can be expected to die before reaching age 47 out of a
group of 1,000,000?
Solution:
Basic equation
Which means that out of 1,000,000 persons
age 46, we can expect 4,862 to die before reaching age 47.
PROBABILITIES OF DYING
WITHIN n YEARS:
The probability that a person age x will die within n years or will die before reaching age (x + n) is represented by
the symbol “_{n}q_{x}” that
is, q with subscripts of n preceding and x following. It is read “n q x”.
An example would be “_{10}q_{45}” which
is read “10 q 45”. It means the probability that a person age 45 will
die within the next 10 years, that is, that the person will die before reaching
age 55.
The probability that a person age x will
die within n years (_{n} q _{x}) is found by dividing the
difference between the number living at ages x and (x + n) by the number living
at age x. this is expressed in equation form as:
The numerator equals the number of people
who die between ages x and (x + n) because the number living at age x is
reduced by all those who die in the interval in order to arrive at the number
still living at age (x + n).
To Illustrate:
Using the following portion of a mortality table, calculate the probability
that a person age 35 will die within the next four years.
Age (x)

l_{x}

d_{x}

35

9827

13

36

9814

13

37

9801

15

38

9786

16

39

9770

16

40

9754

17

Solution:
Basic equation
Substitute the values from table
The number in the numerator, 57, is the
number of people who die between ages 35 and 39. It is equal to the total of
the numbers in the d_{x} column, beginning with d_{35} and ending
with
d_{38 }
To Illustrate:
Using the following portion of a mortality table and a sixyear setback for
females, calculate how many of 100,000 females age 27 can be expected to die
within ten years.
Female Age (x)

l_{x}

21

9647694

22

9630039

23

9612127

24

9593960

25

9575636

26

9557155

27

9538423

28

9519442

29

9500118

30

9480358

31

9460165

Solution
Six years must be subtracted from the ages before using the
table. This means using the table as if calculating the probability of a female
age 21 reaching age 31. The number of years involved, n, is 10
Basic equation:
Substitute the values from the table
No comments:
Post a Comment