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 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 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 “_nq_x” that 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 d_x 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