Conditional Probabilities From Life Table Entries

Why the Denominator Depends on the Conditioning Age

A life table records survivor counts at successive ages. To answer a question about a person who is already alive at age $x$, the relevant group is not the original cohort at birth: it is the members of that cohort who reached age $x$. The number of those survivors is the denominator for a conditional probability from age $x$ onward.

This is a direct application of conditional probability. The life table entries provide counts on a common cohort basis, so ratios of entries give probabilities conditional on having reached the age represented by the denominator. The method is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Let $\ell_x$ be the number of members of a life table cohort alive at exact age $x$. For $n>0$ and $\ell_x>0$, the conditional probability of surviving from age $x$ to age $x+n$, given survival to age $x$, is $$ {}_np_x=\frac{\ell_{x+n}}{\ell_x}. $$ The conditional probability of dying between ages $x$ and $x+n$, given survival to age $x$, is $$ {}_nq_x=\frac{\ell_x-\ell_{x+n}}{\ell_x} =1-\frac{\ell_{x+n}}{\ell_x}. $$

The entries $\ell_x$ and $\ell_{x+n}$ refer to survivors from the same original cohort. The difference $\ell_x-\ell_{x+n}$ counts how many members of that cohort die after reaching age $x$ and before reaching age $x+n$. Dividing this difference by $\ell_x$, rather than by the cohort size at birth, conditions the result on being alive at age $x$.

A life table may be presented with a chosen radix, such as 100,000, for its initial cohort. The radix sets the scale of the entries; it does not change the conditional probability. If every entry is multiplied by the same positive constant, that constant cancels from the ratio. Life table values may also be rounded, in which case probabilities calculated from them are approximations to the precision of the table.

How to Read an Interval Probability

The subscript in ${}_np_x$ identifies both the interval length $n$ and the age $x$ at which the conditioning begins. For example, ${}_5p_{60}$ concerns survival from age 60 to age 65, conditional on being alive at age 60. Its denominator is $\ell_{60}$, even if the table also gives entries at other ages.

For a general interval from age $a$ to age $b$, where $b>a$, use $\ell_a$ as the denominator. The probability of surviving the interval is $\ell_b/\ell_a$, and the probability of dying within it is $(\ell_a-\ell_b)/\ell_a$. If the condition instead says the person was alive at an earlier age, the desired event may include both reaching age $a$ and dying before age $b$. In that case, count the relevant deaths but divide by the survivors at the stated conditioning age.

Worked Example: Conditional Mortality Over Five Years

A hypothetical life table gives $\ell_{50}=80{,}000$ and $\ell_{55}=76{,}000$. Find the probability that a person alive at age 50 survives to age 55, and the probability that the person dies before age 55. Also find the expected number of deaths among 12,000 people who have reached age 50 and are subject to this table.

Method. Use age 50 as the conditioning age. The interval is five years, so the denominator for both conditional probabilities is $\ell_{50}=80{,}000$.

Survival probability. The numerator is the number of the age-50 cohort members who reach age 55: $$ {}_5p_{50}=\frac{\ell_{55}}{\ell_{50}} =\frac{76{,}000}{80{,}000}=0.95. $$ Thus the probability of surviving the full interval is $0.9500$.

Death probability. The number who die between ages 50 and 55 is $80{,}000-76{,}000=4{,}000$. Therefore, $$ {}_5q_{50}=\frac{4{,}000}{80{,}000}=0.05. $$ The conditional probability of death is $0.0500$. As a check, $0.9500+0.0500=1.0000$.

Expected deaths among 12,000 age-50 survivors. Applying the same conditional death probability to this group gives $$ 12{,}000(0.05)=600. $$ The expected number is 600 deaths. This is an expectation under the table’s mortality basis, not a guarantee that exactly 600 deaths will occur.

Common mistake avoided: Dividing 4,000 deaths by a hypothetical radix of 100,000 would answer a different question: the deaths as a proportion of the original cohort. The question conditions on reaching age 50, so the appropriate denominator is 80,000.

Why the Ratio Gives a Conditional Probability

The formula follows from the definition of conditional probability. Let $A$ be the event that a member of the original cohort reaches age $x+n$, and let $B$ be the event that the member reaches age $x$. Reaching age $x+n$ implies reaching age $x$, so $A$ is contained in $B$. The conditional probability of $A$ given $B$ is therefore $P(A)/P(B)$. In a life table, the corresponding cohort proportions are $\ell_{x+n}/\ell_0$ and $\ell_x/\ell_0$, where $\ell_0$ is the cohort size at the table’s starting age. Their ratio cancels the original cohort size: $$ P(A\mid B) =\frac{\ell_{x+n}/\ell_0}{\ell_x/\ell_0} =\frac{\ell_{x+n}}{\ell_x}. $$

The death formula follows because, among those alive at age $x$, the outcomes over the interval are survival to age $x+n$ or death before age $x+n$. These outcomes are mutually exclusive and exhaustive under the table’s model. Consequently, their conditional probabilities sum to one, and ${}_nq_x=1-{}_np_x$. Equivalently, the deaths in the interval number $\ell_x-\ell_{x+n}$, which is divided by the age-$x$ survivors.

These formulas require entries that refer to the same life table basis and cohort. They also require a positive number of survivors at the conditioning age. If $\ell_x=0$, conditioning on survival to age $x$ is not defined from these entries because there are no survivors at that age.

Changing the Conditioning Age

The numerator alone does not determine the probability. The denominator is selected by the wording of the condition: “given alive at age $a$” means divide by $\ell_a$. In particular, the probability of dying between ages $a$ and $b$, given survival to an earlier age $c$, is $(\ell_a-\ell_b)/\ell_c$, provided $c\leq a<b$. The numerator counts the members who experience the specified deaths; the denominator counts the people eligible under the stated condition.

Worked Example: Distinguishing Two Conditioning Ages

A hypothetical table has $\ell_{60}=100{,}000$, $\ell_{62}=98{,}000$, and $\ell_{67}=94{,}500$. Find (1) the probability of dying between ages 62 and 67, given survival to age 62, and (2) the probability of dying between ages 60 and 67, given survival to age 60.

Method. Identify the conditioning age separately for each part, then use the number of deaths in the specified interval as the numerator.

(1) Condition on reaching age 62. The deaths between ages 62 and 67 number $98{,}000-94{,}500=3{,}500$. The conditioning group has 98,000 members. Hence $$ P(\text{die from 62 to 67}\mid\text{alive at 62}) =\frac{3{,}500}{98{,}000} =0.035714\ldots \approx 0.0357. $$ The conditional death probability is $0.0357$, or $3.57\%$ to two decimal places as a percentage.

(2) Condition on reaching age 60. Deaths between ages 60 and 67 number $100{,}000-94{,}500=5{,}500$. The conditioning group has 100,000 members. Thus $$ P(\text{die from 60 to 67}\mid\text{alive at 60}) =\frac{5{,}500}{100{,}000}=0.0550. $$ The probability is $0.0550$. As a check, the two answers are not equal: the second includes deaths from ages 60 to 62 as well as deaths from ages 62 to 67.

Common mistake avoided: Using $100{,}000$ as the denominator in part (1) would give $3{,}500/100{,}000=0.0350$, which is the proportion of the age-60 cohort dying in the later interval. It is not the probability conditional on reaching age 62.

Combining Adjacent Intervals

Conditional probabilities can be checked across adjacent intervals. For $x<y<z$, the probability of surviving from $x$ to $z$ is the probability of surviving from $x$ to $y$, multiplied by the probability of surviving from $y$ to $z$ conditional on reaching $y$: $$ \frac{\ell_z}{\ell_x} =\frac{\ell_y}{\ell_x}\cdot\frac{\ell_z}{\ell_y}. $$

The intermediate entry $\ell_y$ cancels. This identity is useful both for calculating a multi-interval probability and for checking that the conditioning ages have been handled consistently. For an event of dying in a later interval while conditioning at an earlier age, first account for survival to the start of that later interval.

Worked Example: Death in a Later Interval Given Earlier Survival

A hypothetical table gives $\ell_{50}=100{,}000$, $\ell_{53}=98{,}500$, $\ell_{57}=97{,}000$, and $\ell_{60}=95{,}800$. For a person alive at age 53, find the probability of surviving to age 57 and then dying before age 60. Verify the result by counting the relevant deaths directly.

Method. The conditioning age is 53, but the deaths of interest occur from age 57 to age 60. Calculate the probability of reaching 57 given survival to 53, then multiply by the probability of dying from 57 to 60 given survival to 57.

Survive from 53 to 57. $$ P(\text{alive at 57}\mid\text{alive at 53}) =\frac{97{,}000}{98{,}500}. $$

Die from 57 to 60, given survival to 57. There are $97{,}000-95{,}800=1{,}200$ such deaths, so $$ P(\text{die from 57 to 60}\mid\text{alive at 57}) =\frac{1{,}200}{97{,}000}. $$ Multiplying the conditional probabilities gives $$ \frac{97{,}000}{98{,}500}\cdot\frac{1{,}200}{97{,}000} =\frac{1{,}200}{98{,}500} =0.0121827\ldots \approx 0.0122. $$

Direct check. Among the 98,500 people alive at age 53, the 1,200 who die between ages 57 and 60 are exactly those who survive to 57 and then die in the stated interval. Thus the direct ratio is also $1{,}200/98{,}500=0.0122$. The probability is not $1{,}200/97{,}000$, because that latter ratio conditions on survival to age 57 rather than age 53.

Common mistake avoided: The later-interval death probability alone does not include the chance of reaching the start of that interval. When conditioning at age 53, the probability must include survival from 53 to 57.

Common Denominator Traps

Before calculating, translate the condition into a specific age. That age determines the denominator. Then translate the event into a count. These two steps prevent the most frequent errors.

Using the radix by habit. A radix is the initial cohort size, not necessarily the group specified by a conditional question.
Using the endpoint as the denominator. For deaths from age $a$ to age $b$, divide by the number alive at $a$, not by the number alive at $b$.
Confusing “during the interval” with “given alive at its start.” The count of deaths is $\ell_a-\ell_b$; the conditional probability is that count divided by $\ell_a$.
Ignoring an earlier condition. If a question conditions at age $c<a$, then deaths in the later interval $a$ to $b$ are divided by $\ell_c$, not by $\ell_a$.
Forgetting the complement check. For the same starting age and interval, survival and death probabilities must add to one.

Key takeaway: For a probability conditional on being alive at age $x$, use $\ell_x$ as the denominator. Survival to $x+n$ has numerator $\ell_{x+n}$; death before $x+n$ has numerator $\ell_x-\ell_{x+n}$. Match the denominator to the age named in the condition.

Check Your Understanding

For each problem, identify the conditioning age before forming the ratio. Probabilities in the answers are rounded to four decimal places.

A hypothetical table gives $\ell_{40}=72{,}000$ and $\ell_{44}=68{,}400$. Find the probabilities that a person alive at age 40 survives to age 44 and dies before age 44. (Answer: survival $0.9500$; death $0.0500$.)
A table gives $\ell_{45}=90{,}000$ and $\ell_{48}=87{,}300$. Find the probability of dying between ages 45 and 48, given survival to age 45. (Answer: $0.0300$.)
A hypothetical table has $\ell_{60}=50{,}000$, $\ell_{63}=48{,}500$, and $\ell_{65}=47{,}600$. Find (a) the probability of dying from 63 to 65 given survival to 63, and (b) the probability of surviving to 63 and then dying before 65, given survival to 60. (Answer: (a) $0.0186$; (b) $0.0180$.)
A table gives $\ell_{70}=40{,}000$, $\ell_{73}=38{,}000$, and $\ell_{78}=36{,}100$. Find the probability of surviving from age 73 to age 78, given survival to age 73, and its complementary death probability. (Answer: survival $0.9500$; death $0.0500$.)
A hypothetical table gives $\ell_{55}=100{,}000$, $\ell_{58}=98{,}000$, and $\ell_{62}=95{,}000$. For a person alive at age 55, find the probability of surviving to 58 and then dying before 62. (Answer: $0.0300$.)

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