A cohort thought experiment
The earlier guides in this rung gave you the smooth, continuous picture of dying: the [[survival-function|survival function]] that slides from 1 down to 0 as age climbs, and the [[force-of-mortality|force of mortality]] that measures how fiercely death presses at each instant. Those are beautiful, but you cannot pay a claim with a curve. The [[life-table|life table]] is the same story told as a ledger — a finite table of whole numbers that a clerk in 1693, or a computer today, can read straight off. It is where survival theory touches the ground.
The trick is a thought experiment. Imagine a cohort of exactly 100,000 newborns — actuaries call this starting count the *radix*, and write it l_0. Now let them age. We do not follow real people; we follow a *hypothetical* group whose year-by-year thinning obeys a chosen set of one-year death rates. The number still alive at exact age x is written l_x. So l_0 = 100,000, and l_x marches downward — never up — until the oldest table age, where it finally reaches zero. Reading down the l_x column is watching a generation quietly empty out.
Deaths, q_x and p_x: reading the table
Between two birthdays, some of the survivors die. The count of those who die between exact age x and exact age x+1 is d_x, and it is nothing more than a subtraction: d_x = l_x − l_(x+1). The whole table is held together by this one accounting identity — everyone alive at age x is, one year later, either still alive (l_(x+1)) or counted among the dead (d_x), with no third option and nobody appearing from nowhere.
Now turn counts into probabilities — and here is where the [[conditional-probability|conditional probability]] from the probability rung quietly does the heavy lifting. The mortality rate q_x is the probability that a life *who has reached age x* dies before age x+1: q_x = d_x ÷ l_x. The word *conditional* is the whole point — q_x is not the chance a newborn dies at age x; it is the chance of dying in the coming year *given* you have already survived to x. Its partner is the one-year survival probability p_x = l_(x+1) ÷ l_x, the chance the same life makes it to the next birthday. Because each life at age x either dies or survives in the year, p_x + q_x = 1 — always, with no exceptions.
x l_x d_x q_x p_x ------------------------------------------- 60 88,000 900 0.01023 0.98977 61 87,100 1,010 0.01160 0.98840 62 86,090 1,140 0.01324 0.98676 q_60 = d_60 / l_60 = 900 / 88,000 = 0.01023 p_60 = l_61 / l_60 = 87,100 / 88,000 = 0.98977 (= 1 - q_60)
Stretching over many years: n-year and deferred probabilities
A one-year window is rarely what insurance needs. A 20-year term policy on a 45-year-old cares whether she survives a whole *string* of years. The notation stretches to fit: _n_p_x is the probability a life aged x survives the next n years, and ratios make it trivial — _n_p_x = l_(x+n) ÷ l_x. You do not multiply a chain of p's by hand; the survivor counts have already done all the multiplying for you. To survive five years from 60, just read _5_p_60 = l_65 ÷ l_60 straight off the table. Likewise _n_q_x = 1 − _n_p_x is the chance of dying somewhere within those n years.
There is a subtler quantity that life insurance leans on constantly: the deferred probability. The chance a life aged x survives m years and *then* dies in the following year is written _m|_q_x, and it factors cleanly into two honest steps — survive the wait, then die in the target year: _m|_q_x = _m_p_x × q_(x+m), which in counts is just (l_(x+m) − l_(x+m+1)) ÷ l_x. This is exactly the shape of a death benefit that only pays if you die in, say, the eleventh year of a policy. Notice the conditional logic threading through all of it: every probability is anchored to *having survived to its starting age first*.
How long is a life left? Curtate and complete expectancy
The [[future-lifetime-random-variable|future lifetime]] of a life aged x — how many more years it has left — is a random quantity, and its average is the [[life-expectancy|life expectancy]] at age x. There are two flavours, and the difference is exactly the gap between counting whole years and counting every last day. The curtate expectation, written e_x, counts only *completed* whole years of future life. It has a wonderfully simple form: add up the chances of being alive at each future birthday — e_x = _1_p_x + _2_p_x + _3_p_x + ⋯ — because each surviving birthday contributes one full year to the running total.
The complete expectation, written with a little circle as ê_x ("e-circle"), counts the fractional final year too — the months a person lives *after* their last birthday before dying. Since on average a death falls roughly midway through its year, a standard approximation simply adds a half: ê_x ≈ e_x + ½. So a curtate expectancy of 18.0 completed years corresponds to a complete expectancy of about 18.5 actual years. Neither is wrong; they answer slightly different questions. Pensions, which pay in instalments tied to survival at discrete dates, lean naturally on the curtate world; the continuous survival models of the earlier guides give you the complete one.
Why the table is the backbone — and where it lies a little
Everything ahead in this ladder is built on these few columns. To price a life policy, the next rung will discount each possible death benefit by the time value of money *and* weight it by the chance of dying in that year — and that chance is read straight from d_x and l_x. To value a pension, you weight each future payment by _n_p_x, the chance the retiree is alive to collect it. The life table is the bridge between the probability rung, the interest rung, and the contingencies to come: it is the place where "how likely" meets "how much, and when".
But honesty demands the asterisks. First, the table gives mortality only at *integer* ages — to find the chance of dying in three-and-a-half months, you must assume something about how deaths spread within a year, the fractional-age question a later guide tackles head-on. Second, a single table is a *snapshot* of one population's experience; real mortality has been improving for over a century, so a table built on past deaths will overstate the death rates of a cohort living into the future, and pricing long annuities off a stale table is a quiet, expensive mistake. Third — never forget — the cohort is imaginary and the rates are *assumptions*. A life table is a model of mortality, exquisitely useful and never quite reality. Carry that humility into the contingencies rung next, where these survivor counts start being multiplied by dollars.