Actuarial Mathematics for Modelling · Assurance and annuity functions involving two lives

Joint Life and Last Survivor Status: Definitions and Survival Probabilities

Updated 11 October 2026 · Fact-checked

The joint life status (xy) survives while both lives are alive, so it fails at the first death. The last survivor status (x̄ȳ) survives while at least one life is alive, so it fails at the second death. For independent lives: tpxy = tpx × tpy, and tpx̄ȳ = tpx + tpy − tpxy.

Understand Joint Life and Last Survivor Status

A status is a condition that is either alive or failed at each time. In single-life work the status is one person. With two lives, you combine two people into one status and ask when that status fails.

The joint life status (xy) exists as long as both (x) and (y) are alive. It fails at the first death. Its future lifetime is T_xy = min(T_x, T_y). Think of a chain that breaks when either link breaks.

The last survivor status (x̄ȳ) exists as long as at least one of (x) and (y) is alive. It fails at the second death. Its future lifetime is T_x̄ȳ = max(T_x, T_y). Think of two ropes holding a weight: it falls only when both break.

The two statuses are linked by a simple identity. Exactly one of two things holds: either T_xy is the smaller and T_x̄ȳ the larger, or they are the same two values swapped. So T_xy + T_x̄ȳ = T_x + T_y, and the survival probabilities satisfy tpxy + tpx̄ȳ = tpx + tpy. This holds for any two lives, independent or not.

The product rule tpxy = tpx × tpy needs independence of T_x and T_y. Exam questions usually state this. If it is not stated, check the wording before using it.

Key rules to remember

Joint life survival (independent lives): tpxy = tpx × tpy; Both must survive t years. Needs independence.
Joint life death probability: tqxy = 1 − tpxy = 1 − tpx × tpy; At least one dies within t years.
Last survivor survival: tpx̄ȳ = tpx + tpy − tpxy; At least one survives t years. Valid for any two lives, since it is the addition rule for events.
Last survivor death probability (independent lives): tqx̄ȳ = tqx × tqy; Both die within t years. Needs independence.
Complement form: tpx̄ȳ = 1 − tqx × tqy; Quick route for independent lives.
Status lifetimes: T_xy = min(T_x, T_y); T_x̄ȳ = max(T_x, T_y); Gives T_xy + T_x̄ȳ = T_x + T_y.
Exactly one survives (independent lives): tpx × tqy + tqx × tpy; Equals tpx̄ȳ − tpxy.

How to solve Joint Life and Last Survivor Status questions

Use this routine for any question asking for a probability involving two lives.

1Write the event in words: both alive, at least one alive, both dead, exactly one alive, or at least one dead.
2Match the event to a status: both alive is (xy); at least one alive is (x̄ȳ).
3Check whether independence is stated. If so, you may multiply single-life probabilities.
4List the single-life values you need: tpx, tpy, tqx, tqy. Compute the missing ones from the life table or given formula.
5Choose the shortest formula: tpxy = tpx × tpy for joint life; tqx̄ȳ = tqx × tqy for last survivor death.
6Calculate with at least four decimal places and keep the working visible.
7Sanity check: tpxy ≤ tpx, tpy ≤ tpx̄ȳ, and all probabilities lie between 0 and 1.

Quickest way: Translate to AND / OR, then use complements

When to use it: Use for any MCQ or short part where lives are independent and single-life probabilities are given.

Joint life = AND of survivals: multiply tpx and tpy.
Last survivor death = AND of deaths: multiply tqx and tqy.
Last survivor survival = 1 − (tqx × tqy). This avoids the three-term formula.
Exactly one alive = tpx × tqy + tqx × tpy.
Check: tpxy + tpx̄ȳ should equal tpx + tpy.

Common mistakes in Joint Life and Last Survivor Status

Treating the last survivor status as failing at the first death.
The words 'joint' and 'last' are mixed up.
Fix: Joint life (xy) fails at the first death. Last survivor (x̄ȳ) fails at the second death.
Using tpx̄ȳ = tpx × tpy.
Students apply the product rule to every two-life status.
Fix: The product gives the joint life only. For last survivor use tpx + tpy − tpxy.
Writing tqx̄ȳ = tqx + tqy.
Confusing 'both die' with 'either dies'.
Fix: Both dying is an AND event, so multiply: tqx × tqy for independent lives. Either dying is tqxy = 1 − tpx × tpy.
Using the product rule without checking independence.
It becomes automatic from practice questions.
Fix: Read the question for 'independent' or an assumption statement. Mention the assumption in your answer.
Forgetting to subtract tpxy in the last survivor formula.
Students add tpx and tpy and forget the overlap is counted twice.
Fix: Check the answer is at most 1. Or use 1 − tqx × tqy instead.
Mixing up ages, so tpx uses the wrong life.
Both lives look alike in the notation.
Fix: Write the age and the term next to each value, for example 10p60 and 10p55, before multiplying.

Worked examples

Example 1

Two independent lives (x) and (y) have 10px = 0.90 and 10py = 0.80. Calculate the probability that, after 10 years, (a) both are alive, (b) at least one is alive, (c) exactly one is alive.

Show the solution

(a) Joint life: 10pxy = 0.90 × 0.80 = 0.72.
(b) Last survivor: 10px̄ȳ = 0.90 + 0.80 − 0.72 = 0.98.
Check with the complement: 10qx = 0.10 and 10qy = 0.20, so 1 − 0.10 × 0.20 = 1 − 0.02 = 0.98. This matches.
(c) Exactly one alive = 10px̄ȳ − 10pxy = 0.98 − 0.72 = 0.26.
Check directly: 0.90 × 0.20 + 0.10 × 0.80 = 0.18 + 0.08 = 0.26.

Answer: (a) 0.72; (b) 0.98; (c) 0.26

Example 2

For independent lives (x) and (y), 5qx = 0.04 and 5qy = 0.10. Calculate 5qxy, 5qx̄ȳ and verify that 5pxy + 5px̄ȳ = 5px + 5py.

Show the solution

Survival probabilities: 5px = 0.96 and 5py = 0.90.
Joint life: 5pxy = 0.96 × 0.90 = 0.864, so 5qxy = 1 − 0.864 = 0.136.
Last survivor death: 5qx̄ȳ = 0.04 × 0.10 = 0.004.
So 5px̄ȳ = 1 − 0.004 = 0.996.
Verify: 5pxy + 5px̄ȳ = 0.864 + 0.996 = 1.860.
And 5px + 5py = 0.96 + 0.90 = 1.86. The identity holds.

Answer: 5qxy = 0.136; 5qx̄ȳ = 0.004; the identity is verified (both sides equal 1.86).

Exam tips

Write the status in words first. Marks are often lost by picking the wrong status, not by arithmetic.
State 'assuming independence' in written answers. Examiners give credit for stating assumptions.
Use the identity tpxy + tpx̄ȳ = tpx + tpy as a quick check, and to find one status when the other is given.
In MCQs, use the complement route 1 − tqx × tqy for last survivor survival. It is the fastest and has fewest errors.
Show age, term and notation in every line of working, for example 10p60, so the marker can follow it.

Practice questions from Assurance and annuity functions involving two lives

Joint Life and Last Survivor Status: frequently asked questions

What is the difference between joint life and last survivor status?

A joint life status (xy) is alive only while both lives are alive, so it ends at the first death. A last survivor status (x̄ȳ) is alive while at least one life is alive, so it ends at the second death.

How do I calculate tpx̄ȳ for independent lives?

Use tpx̄ȳ = tpx + tpy − tpx × tpy. Or compute 1 − tqx × tqy. Both give the same value.

Do I need independence for tpx̄ȳ = tpx + tpy − tpxy?

No. This identity holds for any two lives, because it is the addition rule for the events. Independence is only needed to write tpxy as tpx × tpy.

Is T_xy the minimum or the maximum of T_x and T_y?

T_xy is the minimum, since the joint status fails at the first death. T_x̄ȳ is the maximum, since the last survivor status fails at the second death.