## Friday, April 13, 2012

### PROBABILITIES OF DYING

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 px for (Probability of Living 1 Year)
Substitute qx 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 p46 = 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 “nqxthat is, q with subscripts of n preceding and x following. It is read “n q x”. An example would be “10q45 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) lx dx 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 dx column, beginning with d35 and ending with d38

To Illustrate: Using the following portion of a mortality table and a six-year setback for females, calculate how many of 100,000 females age 27 can be expected to die within ten years.

 Female Age (x) lx 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