Probability of Death Calculator

Enter mortality inputs

Choose the input style that matches your data source. The calculator converts it to annual probability, survival, and cumulative death risk.

Input mode

Accumulation model

Time horizon (years)

Annual probability q

Use values from 0 to below 1.

Observed deaths

Used with deaths and population mode.

Population exposed

Population must exceed zero.

Hazard rate μ

Use when risk is modeled continuously.

Cohort size

Expected deaths are based on this cohort.

Decimal places

Example data table

These sample rows show how different annual inputs change multi-year mortality estimates for a cohort of 50,000.

Scenario	Annual q	Horizon	Model	Cumulative Death Probability	Expected Deaths
Low annual risk	0.0010	10 years	Discrete	0.009955	497.75
Deaths ÷ population	0.0012	10 years	Discrete	0.011934	596.70
Hazard-based	0.009950	5 years	Continuous	0.048771	2,438.55
Higher annual risk	0.0200	15 years	Discrete	0.261430	13,071.50

Formula used

Annual probability from counts:

q = deaths ÷ population

Annual probability from hazard:

q = 1 − e^(−μ)

Continuous hazard from annual probability:

μ = −ln(1 − q)

Discrete survival over t years:

S(t) = (1 − q)^t

Continuous survival over t years:

S(t) = e^(−μt)

Cumulative death probability:

P(death by t) = 1 − S(t)

Expected deaths in cohort N:

Expected deaths = N × P(death by t)

50% risk horizon for discrete annual probability:

t₅₀ = ln(0.5) ÷ ln(1 − q)

How to use this calculator

Choose the input mode that matches your data source.
Enter annual probability, deaths and population, or hazard rate.
Select a discrete or continuous accumulation model.
Set the time horizon in years and cohort size.
Choose decimal precision, then press the calculate button.
Review death probability, survival, expected deaths, and projections.
Use the CSV or PDF buttons to export results.

Frequently asked questions

1. What does this calculator estimate?

It estimates the probability of death over a chosen time horizon using annual probability, observed deaths divided by population, or a constant hazard rate.

2. What is the difference between discrete and continuous models?

The discrete model assumes risk compounds by separate periods, such as yearly steps. The continuous model assumes a constant hazard acting at every moment.

3. When should I use deaths and population mode?

Use it when you know observed deaths and the exposed population. The calculator converts that ratio into an annual probability before projecting risk.

4. Why are expected deaths based on a cohort size?

A probability alone is unitless. Cohort size translates that probability into an expected count, helping you estimate deaths and survivors in a group.

5. What does the 50% risk horizon mean?

It is the number of years required for cumulative death probability to reach 50 percent, assuming the annual risk remains unchanged over time.

6. Does this model account for changing age or health status?

No. It assumes the same annual probability or hazard throughout the chosen horizon. Changing risk profiles require life tables or time-varying models.

7. Can I use decimal time horizons?

Yes. Decimal years are accepted. The projection table lists full years and adds a final partial row when the horizon includes a fraction.

8. What makes this a maths calculator rather than a medical tool?

It performs probability and survival calculations from supplied rates. It does not diagnose conditions, predict personal outcomes, or replace professional assessment.