Actuarial Survival and Death Probabilities¶

Definition¶

For a life aged $x$ with future lifetime $T_x$ , define ${}_tp_x=\mathbb{P}(T_x>t)$ and ${}_tq_x=\mathbb{P}(T_x\le t)$ for $t\ge0$ $0$ . The deferred death probability is

{}_{u|t}q_x=\mathbb{P}(u<T_x\le u+t) ={}_up_x-{}_{u+t}p_x.

Here $u\ge0$ is the deferment and $t\ge0$ the subsequent coverage period. An omitted left-hand duration means one year: $p_x={}_1p_x$ $_{1} p_{x}$ , $q_x={}_1q_x$ $_{1} q_{x}$ .

Interpretation¶

These symbols encode the age, waiting period and coverage period of a survival event. Reading those three parts prevents confusion between dying within a period and surviving until it begins.

Timeline for a life aged 40 showing survival to time 5 with probability 0.98, survival to time 10 with probability 0.95, and death between times 5 and 10 with probability 0.03.

The figure uses the Examples values for a life aged 40. The blue arrows are the survival events $T_{40}>5$ and $T_{40}>10$ $10$ ; the red interval is the deferred death event $5<T_{40}\le10$ $10$ , open at 5 and closed at 10. Surviving past time 5 means either surviving past time 10 or dying in $(5,10]$ , so ${}_{5|5}q_{40}={}_5p_{40}-{}_{10}p_{40}=0.98-0.95=0.03$ $0.03$ , a probability measured from age 40.

The multiplication rule for conditional probability explains survival over successive periods: first survive to duration $t$ , then survive another $u$ years given attained age $x+t$ $t$ . This gives ${}_{t+u}p_x={}_tp_x{}_up_{x+t}$ whenever the first survival probability is positive.

Key Results¶

${}_tp_x+{}_tq_x=1$ $1$ .
${}_{t+u}p_x={}_tp_x\,{}_up_{x+t}$ $_{t} p_{x}_{u} p_{x + t}$ .
${}_{u|t}q_x={}_up_x\,{}_tq_{x+u}$ when survival to $x+u$ has positive probability.

Examples¶

If ${}_5p_{40}=0.98$ and ${}_{10}p_{40}=0.95$ $0.95$ , then ${}_{5|5}q_{40}=0.03$ $0.03$ . This is an unconditional probability at age 40, not the conditional five-year death probability at age 45.

Prerequisites¶

Sources¶

Life Insurance · Lecture 1

Aliases: international actuarial notation, deferred mortality probability, survival probability, death probability, deferred death probability, tpx, tqx, probability of death, probability of survival Keywords: probability of dying between two ages, u|t q x deferred mortality, t+u p x = t p x times u p x+t, p x plus q x equals one

Last updated: 22 October 2026 19:04