Probability of dying and survival by age

A life table gives the chance of dying within one year at each age. Most questions about retirement need more than one year: what is the chance of reaching 85, or of still being alive when the savings run out? This page, part of our guides to how the maths works, shows how one-year probabilities are chained into survival over any period, including part years.

The idea in plain English

To be alive in three years, you have to survive this year, then next year, then the year after. Each step has its own survival chance, read from the table at the age you will be then. Multiply the three chances together and you have the chance of surviving all three years.

The probability of dying within the period is everything else: one minus the probability of surviving it. “Probability of dying by age” tables quote exactly these two numbers, for one year or for a span of years.

Between birthdays, a table says nothing, so a calculator needs an assumption to answer a question like “what is the chance of being alive 10½ years from now?”. The usual one, and the one every calculator on this site uses, is that deaths within a year of age are spread evenly through that year.

The notation

Survival probability ₜpₓThe chance that someone aged x is still alive t years from now.How it works: Survival probability: the probability that someone aged Age xYour age today, in whole years.How it works: Life tables is still alive Time tA length of time from now, in years. It can include a fraction of a year.How it works: Survival probability years from now. The $t$ sits on the lower left; when $t = 1$ it is left out, giving One-year survival pₓThe chance that someone aged x survives one more year: 1 − qₓ.How it works: Life tables.
Death probability ₜqₓThe chance that someone aged x dies within t years: 1 − ₜpₓ.How it works: Survival probability: the probability that they die within $t$ years.
Whole years kA whole number of years from now.How it works: Survival probability: a whole number of years, and Fraction of a year sThe part of the next year, between 0 and 1, after the last whole year: t = k + s.How it works: Survival probability a part of a year, so that $t = k + s$ with $0 \le s \lt 1$ .
Death probability in year kThe one-year death probability at age x + k, the year being part-lived.How it works: Survival probability and One-year survival at age x + kThe chance that someone who has reached age x + k survives one more year: 1 − q at that age.How it works: Survival probability: the one-year death and survival probabilities at age $x+k$ , the age reached after $k$ years.
Deferred death probabilityThe chance that someone aged x survives k years and then dies in the year after: kpₓ times qₓ₊ₖ.How it works: Survival probability: the probability of surviving $k$ years and then dying in the following year. The bar after $k$ marks the deferral.

The formulas

Survival over whole years multiplies the one-year chances:

\begin{gathered} {}_{k}p_{x} = p_{x}\,p_{x+1} \cdots p_{x+k-1} = \prod_{j=0}^{k-1} \left(1 - q_{x+j}\right) \\ {}_{t}q_{x} = 1 - {}_{t}p_{x} \end{gathered}

Each factor is the chance of surviving one more year at the age reached by then.

For a part year, the Uniform distribution of deathsThe assumption that deaths within a year of age happen evenly through that year. It fills in survival between birthdays.How it works: Survival probability (UDD) assumption says that a fraction $s$ of the year’s deaths happen in the first fraction $s$ of the year, so ${}_{s}q_{x+k} = s\,q_{x+k}$ . That gives

\begin{gathered} {}_{k+s}p_{x} = {}_{k}p_{x}\,\left(1 - s\,q_{x+k}\right) \\ 0 \le s \lt 1 \end{gathered}

Survive k whole years, then the first part s of the next year.

and the chance of dying in a particular future year:

{}_{k|}q_{x} = {}_{k}p_{x}\;q_{x+k} = {}_{k}p_{x} - {}_{k+1}p_{x}

Alive after k years, then dies before k + 1.

Other assumptions exist for part years. A constant force of mortality within each year gives ${}_{s}p_{x} = (p_{x})^{s}$ . Over a single year the two differ only in the fourth or fifth decimal place at retirement ages. UDD has the practical advantage that survival is a straight line between birthdays, and it reproduces the SSA’s published life expectancies, which the engine’s tests check.

Worked example: a man aged 65

Using the SSA 2023 period table for men:

Year 1: $q_{65}$ = 0.016455, so $p_{65}$ = 0.983545.
Year 2: $q_{66}$ = 0.017574, so $p_{66}$ = 0.982426.
Surviving both: ${}_{2}p_{65}$ = 0.983545 × 0.982426 = 0.966260.

Carrying on for all 20 years to age 85 gives ${}_{20}p_{65}$ = 0.4493: a 45% chance, about 45 in 100, that he reaches 85. The probability of dying before 85 is ${}_{20}q_{65}$ = 0.5507, or 55%.

For a part year, take 10.5 years. After 10 whole years, ${}_{10}p_{65}$ = 0.794057. The death probability in the next year, at age 75, is $q_{75}$ = 0.033802. The first 0.5 of that year under UDD gives

${}_{10.5}p_{65}$ = 0.794057 × (1 − 0.5 × 0.033802) = 0.780637.

Finally, the chance, seen from today, that he dies at age 80 (after reaching 80 but before 81) is ${}_{15|}q_{65} = {}_{15}p_{65}\,q_{80}$ = 0.642164 × 0.055633 = 0.035725, about 3.6%.

Try it: watch survival build up

Pick an age and a number of years, or add one year at a time. Each row multiplies the running total by that year’s survival chance; a part year at the end uses the UDD step.

Build up survival year by year

Table

MaleFemale

Age now

Years aheadPart years allowed, e.g. 10.5

Alive at 75, ${}_{t}p_{x}$: 0.7940679.4%
Dies before 75, ${}_{t}q_{x}$: 0.2059420.6%

Year $k$	Age	$q_{x+k}$	Multiply by	Alive at end
0	65	0.016455	0.983545	0.98355
1	66	0.017574	0.982426	0.96626
2	67	0.018735	0.981265	0.94816
3	68	0.019981	0.980019	0.92921
4	69	0.021366	0.978634	0.90936
5	70	0.022903	0.977097	0.88853
6	71	0.024615	0.975385	0.86666
7	72	0.026504	0.973496	0.84369
8	73	0.028648	0.971352	0.81952
9	74	0.031071	0.968929	0.79406

Source: US Social Security Administration, 2023 period life table (Period Life Table, 2023, as used in the 2026 Trustees Report). Do My Numbers calculations.

Where survival probabilities lead

Add the survival probabilities over every future year and you get the average remaining lifetime: life expectancy. Combine two people’s survival and you get the chance that both, or at least one, are alive: joint and last survivor. Weight each future payment by the chance of being alive to receive it and you get the value of a life annuity.

Like the table they come from, these are population averages for one calendar year’s death rates. They describe the table, not any one person’s future.

Where this is used

Questions

How do you calculate the probability of dying from a life table?

For one year, read qₓ at the person's age. For several years, multiply the one-year survival chances (1 − qₓ) for each year in turn to get the chance of surviving, then subtract that from 1. The result is the probability of dying within the period, written tqₓ.

What is the probability of dying in a given year of age?

It is qₓ, the life table's one-year death probability at that age, for someone who has just reached it. Seen from today, the chance of dying in a later year k is the chance of surviving to it, kpₓ, times the death probability at that age, qₓ₊ₖ.

What does uniform distribution of deaths mean?

It is the assumption that, within any single year of age, deaths happen evenly through the year. It lets a calculator find survival to a part year, such as 10.5 years from now, from a table that only gives whole years. Other assumptions exist, but for one year at a time they give almost the same answer.

Is survival probability the same as life expectancy?

No. Survival probability is the chance of being alive at a given future age. Life expectancy is the average number of years still to live, which adds up the survival probabilities over every future year. Two people with the same life expectancy can have different chances of reaching 90.

Sources

Social Security Administration, Actuarial Life Table (period life table)

Survival probability: the chance of reaching an age