One-Year Mortality Rates qx And px

One-Year Mortality and Survival

A life table records how many members of a cohort are alive at different exact ages. To interpret the change between two adjacent entries, focus on the people who are alive at the start of the year. The one-year mortality probability describes the chance that a person in this group dies before reaching the next age; the one-year survival probability describes the chance that the person reaches that age.

The conditioning matters. These are probabilities for a life who has already reached age \(x\), not probabilities calculated using everyone who belonged to the cohort at its starting age. In actuarial notation, the one-year probabilities are often written \(q_x\) and \(p_x\), respectively.

Definition: For a life aged exactly \(x\), \(q_x\) is the probability of dying in the year from age \(x\) to age \(x+1\), conditional on being alive at age \(x\). The probability of surviving that year to age \(x+1\), conditional on being alive at age \(x\), is \(p_x\). In the notation for future lifetime, \(q_x={}_1q_x\) and \(p_x={}_1p_x\).

Let \(\ell_x\) be the number of lives in a life table who survive to exact age \(x\). The adjacent entry \(\ell_{x+1}\) counts those who survive to exact age \(x+1\). The number who die in the interval is therefore \(\ell_x-\ell_{x+1}\). When \(\ell_x>0\), the one-year probabilities follow by dividing the relevant count by the number alive at the start of the interval.

q_x=\frac{\ell_x-\ell_{x+1}}{\ell_x} \qquad\text{and}\qquad p_x=\frac{\ell_{x+1}}{\ell_x}

Both numerators refer to outcomes among the lives represented by \(\ell_x\). The probabilities are complements because death during the year and survival to the next exact age are mutually exclusive outcomes that together account for that starting group:

q_x+p_x =\frac{\ell_x-\ell_{x+1}}{\ell_x} +\frac{\ell_{x+1}}{\ell_x} =1.

Why the Life Table Ratios Give Conditional Probabilities

The ratios are not merely convenient calculations. They express the conditional probability directly. Imagine choosing one member at random from the \(\ell_x\) lives alive at age \(x\). Of those lives, \(\ell_x-\ell_{x+1}\) die before reaching age \(x+1\). Thus the fraction of the age-\(x\) group who die during that year is the probability of death conditional on being in that group. The fraction who reach age \(x+1\) gives the conditional probability of survival.

Identify the starting group.
Use \(\ell_x\), the number alive at exact age \(x\), as the denominator for both probabilities.

Count each outcome in that group.
The number dying in the interval is \(\ell_x-\ell_{x+1}\); the number surviving to age \(x+1\) is \(\ell_{x+1}\).

Divide by the starting group.
The resulting fractions are \(q_x\) and \(p_x\), conditional on survival to age \(x\).

This reasoning assumes the two entries belong to the same life table and refer to the same cohort as it moves from age \(x\) to age \(x+1\). A life table may represent a hypothetical cohort, so its entries need not be counts from an observed group of that size. Their ratios still give the table’s one-year probabilities. The entries should be nonnegative, and ordinarily \(\ell_{x+1}\leq \ell_x\); otherwise the implied death count or probability would be inconsistent with the usual life table interpretation.

Worked Example: Calculating Both Rates From Adjacent Entries

A hypothetical life table has \(\ell_{50}=80{,}000\) and \(\ell_{51}=79{,}200\). Find \(q_{50}\) and \(p_{50}\), and interpret them for a person who has reached age 50.

Method. Find the number dying between ages 50 and 51 by subtracting the adjacent survivor counts:

d_{50}=\ell_{50}-\ell_{51} =80{,}000-79{,}200=800.

The starting group contains \(80{,}000\) lives. Therefore,

q_{50}=\frac{800}{80{,}000}=0.0100, \qquad p_{50}=\frac{79{,}200}{80{,}000}=0.9900.

A life who has reached age 50 has a \(0.0100\), or 1.00%, probability of dying before age 51 according to this table, and a \(0.9900\), or 99.00%, probability of surviving to age 51. As a check, \(0.0100+0.9900=1.0000\). The calculation avoids treating the 800 deaths as a fraction of some earlier cohort size: the denominator is the number alive at age 50.

Finding One Probability From the Other

If one of the two one-year probabilities is known, the other follows by taking its complement. Similarly, if a life table gives \(\ell_x\) and one probability, the adjacent survivor count can be recovered by multiplying by the corresponding probability. The count should agree with the interpretation of the probability as a fraction of the age-\(x\) starting group.

Worked Example: Using a Given Mortality Probability

A hypothetical table has \(\ell_{62}=50{,}000\), and the one-year mortality probability at age 62 is \(q_{62}=0.0180\). Find \(p_{62}\), the number expected to survive to age 63 in the table, and the number dying during the year.

Method. First use the complementary-outcome relationship:

p_{62}=1-q_{62}=1-0.0180=0.9820.

The number surviving is the starting count multiplied by the survival probability:

\ell_{63}=\ell_{62}p_{62} =50{,}000(0.9820)=49{,}100.

The number dying is the difference between the starting and ending counts:

\ell_{62}-\ell_{63} =50{,}000-49{,}100=900.

Checking from the mortality probability gives \(50{,}000(0.0180)=900\), the same death count. The common mistake avoided here is to use \(q_{62}\) as the survival proportion: \(0.0180\) applies to deaths, while \(0.9820\) applies to survivors.

Interpreting Rates for a Group

The life table ratio can also be applied to a different group of lives aged \(x\), provided the same one-year mortality assumption is being used. Multiplying the group size by \(q_x\) gives the expected number who die during the year; multiplying by \(p_x\) gives the expected number who survive. These are expected counts, not a guarantee of the exact number of outcomes in an actual group.

For example, if a hypothetical group of 2,400 lives aged \(x\) is subject to \(q_x=0.0080\), the expected number of deaths is \(2{,}400(0.0080)=19.2\), and the expected number surviving is \(2{,}400(0.9920)=2{,}380.8\). Fractional expected counts are possible; they do not mean that a fraction of a person experiences an outcome.

Worked Example: Applying a Life Table Rate to an Age-Specific Group

A hypothetical table gives \(\ell_{40}=100{,}000\) and \(\ell_{41}=99{,}200\). A separate group contains 3,000 people, all aged exactly 40. Using the table’s one-year mortality probability, calculate the expected number of deaths and survivors in this group.

Method. First calculate the table’s age-40 mortality probability from its adjacent entries:

q_{40} =\frac{100{,}000-99{,}200}{100{,}000} =\frac{800}{100{,}000} =0.0080.

The complementary survival probability is

p_{40}=1-0.0080=0.9920.

Apply each probability to the 3,000 people who are alive at age 40:

E[\text{deaths}]=3{,}000(0.0080)=24, \qquad E[\text{survivors}]=3{,}000(0.9920)=2{,}976.

The expected counts add to the original group size: \(24+2{,}976=3{,}000\). They are also in the same ratio as the table’s one-year probabilities. The common mistake avoided is multiplying by a probability for a person of a different starting age; the given entries determine \(q_{40}\), so this rate applies to people aged 40, not to an unspecified age group.

Common Checks and Exam Traps

A reliable calculation starts by identifying the age at the beginning of the interval. For rates at age \(x\), the denominator is \(\ell_x\), and the next entry is \(\ell_{x+1}\). Reversing the entries or dividing by the ending count changes the conditional group and does not give \(q_x\) or \(p_x\).

For instance, with \(\ell_x=25{,}000\) and \(\ell_{x+1}=24{,}500\), the correct mortality probability is

q_x=\frac{25{,}000-24{,}500}{25{,}000}=0.0200.

The incorrect calculation \(500/24{,}500\) divides deaths by the number surviving to the end of the interval. That is not the fraction of the starting age-\(x\) group who die. The correct survival probability is \(24{,}500/25{,}000=0.9800\), and the two correct probabilities sum to 1.

Use the correct interval. \(q_x\) refers to the year beginning at age \(x\), so use \(\ell_x\) and \(\ell_{x+1}\).
Use the correct denominator. Both probabilities are conditional on being alive at age \(x\), so divide by \(\ell_x\).
Do not confuse probability with force. \(q_x\) is a probability over a one-year interval; it is not itself the instantaneous force of mortality.
Check the complements. If calculated \(q_x\) and \(p_x\) do not add to 1, recheck the subtraction, denominator, and age labels.

Key takeaway: From adjacent life table entries, \(q_x=(\ell_x-\ell_{x+1})/\ell_x\) and \(p_x=\ell_{x+1}/\ell_x\). Each is a conditional probability for a life already alive at age \(x\), and together they sum to 1.

Check Your Understanding

For each problem, use the age-\(x\) entry as the starting group and check that the death and survival probabilities are complements.

A hypothetical table gives \(\ell_{66}=24{,}000\) and \(\ell_{67}=23{,}760\). Find \(q_{66}\) and \(p_{66}\). (Answer: \(q_{66}=0.0100\), \(p_{66}=0.9900\).)
At age 80, a hypothetical table has \(\ell_{80}=40{,}000\) and \(q_{80}=0.0275\). Find \(p_{80}\), \(\ell_{81}\), and the number dying in the interval. (Answer: \(p_{80}=0.9725\), \(\ell_{81}=38{,}900\), deaths \(=1{,}100\).)
A hypothetical life table has \(\ell_{52}=75{,}000\) and \(\ell_{53}=74{,}550\). For a separate group of 3,000 people aged exactly 52, find the table’s \(q_{52}\) and the expected number of deaths and survivors in the group. (Answer: \(q_{52}=0.0060\), expected deaths \(=18\), expected survivors \(=2{,}982\).)
A table has \(\ell_x=18{,}400\) and \(\ell_{x+1}=18{,}124\). Find both one-year probabilities and the expected number of deaths among 800 people aged \(x\) subject to this table rate. (Answer: \(q_x=0.0150\), \(p_x=0.9850\), expected deaths \(=12\).)
A hypothetical table gives \(p_x=0.9600\) and \(\ell_{x+1}=18{,}432\). Find \(q_x\) and \(\ell_x\), then calculate the expected number of deaths among 1,500 people aged \(x\). (Answer: \(q_x=0.0400\), \(\ell_x=19{,}200\), expected deaths \(=60\).)

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