Probability Theory for Health Economics

This lesson introduces the probability concepts needed for postgraduate work in health economics and health-service management. It is an independent study aid prepared in the context of MHEC 51023. It is not an official University of Kelaniya handout and does not represent institutional guidance or endorsement.

All examples below use hypothetical data. Their purpose is to show the calculation and interpretation, not to describe a real population, health programme, diagnostic test or supplier.

Learning objectives

After working through the lesson, you should be able to:

define a sample space and an event;
state the three Kolmogorov axioms correctly;
distinguish mutually exclusive events from independent events;
calculate complements, unions, intersections and conditional probabilities;
apply the general multiplication rule and Bayes’ theorem; and
distinguish sensitivity, specificity and positive predictive value when interpreting a diagnostic test.

Why probability matters in health economics

Health decisions are made under uncertainty. A treatment may benefit some patients and not others. A screening programme can produce true and false results. Costs and outcomes vary between people and over time. Probability supplies a consistent language for representing that uncertainty; statistics uses sample data to estimate population quantities and assess how uncertain those estimates are.

Descriptive statistics summarise observed data. Inferential statistics use a sample to learn about a larger population. Probability connects the two because it describes the sampling variation that remains even when a study has been designed and conducted properly.

Basic definitions

Probability

For an event A, P(A) denotes its probability. A probability lies between 0 and 1. Values closer to 1 indicate that the event is more likely under the specified model.

When a finite set of outcomes is equally likely:

P(A) = number of outcomes belonging to A / total number of outcomes.

This shortcut must not be used when the outcomes have unequal probabilities. For a fair six-sided die, for example, P(3) = 1/6; for a loaded die, the physical or empirical probability model is needed.

Sample space

The sample space, written Ω, contains every possible outcome of the experiment as defined. Examples include:

Experiment or measurement	Sample space
One binary test result	`{positive, negative}`
Recorded ABO blood group	`{A, B, AB, O}`
Number of emergency arrivals in one hour	`{0, 1, 2, ...}`
Waiting time in minutes	Any non-negative real value

The definition must match the measurement. A four-category recovery scale, for example, is not the same sample space as survival status at 30 days.

Events

An event is a subset of the sample space. If A is the event “the test result is positive”, then A contains the positive outcome. The empty event, ∅, contains no outcomes. The certain event, Ω, contains every outcome.

The Kolmogorov axioms

A probability measure satisfies three axioms:

Non-negativity: every probability is non-negative, so for every event A, P(A) ≥ 0.
Normalization (normalisation): P(Ω) = 1.
Countable additivity: for any countable collection of pairwise disjoint events A₁, A₂, ..., P(A₁ ∪ A₂ ∪ ...) = P(A₁) + P(A₂) + ....

Several familiar rules follow from these axioms. They are not replacement axioms:

P(∅) = 0;
0 ≤ P(A) ≤ 1;
P(Aᶜ) = 1 − P(A); and
P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

The legacy lesson incorrectly listed P(∅) = 0 and P(Ω) = 1 as separate second and third axioms and omitted countable additivity. The formulation above corrects that error.

Set operations on events

Complement

Aᶜ contains every outcome not in A. If the probability that a device has no more than two defects is 0.86, then the probability that it has more than two defects is 1 − 0.86 = 0.14.

Intersection

A ∩ B means that both events occur. In a patient dataset, if A means female and B means diagnosed with diabetes, A ∩ B identifies female patients diagnosed with diabetes.

Union and the addition rule

A ∪ B means that at least one of the events occurs. The general addition rule is:

P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Suppose 34% of a hypothetical workforce is in a lower salary category, 41% has 6–10 years of experience, and 10% belongs to both groups. Then:

P(lower salary or 6–10 years) = 0.34 + 0.41 − 0.10 = 0.65.

Mutually exclusive events

Events are mutually exclusive, or disjoint, when they cannot occur together: A ∩ B = ∅. For disjoint events, the addition rule becomes P(A ∪ B) = P(A) + P(B).

Mutual exclusivity and independence mean different things. Two mutually exclusive events with positive probability are dependent: learning that one occurred tells us that the other did not occur.

Independence and the multiplication rule

Two events are independent when observing one does not change the probability of the other. If A and B are independent:

P(A ∩ B) = P(A) × P(B).

Independence should be justified by the design or examined from data; it should not be assumed because two variables have different names.

For dependent events with P(A) > 0, use the general multiplication rule:

P(A ∩ B) = P(A) × P(B | A).

Suppose a hypothetical batch contains six acceptable and four unacceptable vials. Two are sampled without replacement. The probability that both sampled vials are acceptable is:

(6/10) × (5/9) = 30/90 = 1/3 ≈ 0.3333.

The second probability changes because the first acceptable vial is not returned to the batch.

Conditional probability

For P(B) > 0:

P(A | B) = P(A ∩ B) / P(B).

The order matters. P(disease | positive) is the probability of disease after observing a positive result. P(positive | disease) is the test’s sensitivity. They answer different questions.

Counting rules

If one stage has m possible outcomes and a second stage has n possible outcomes for each first-stage outcome, the combined process has m × n outcomes. A design with three dose levels, two delivery methods and four durations therefore contains 3 × 2 × 4 = 24 combinations before any exclusions or constraints are applied.

Counting combinations is not the same as assigning probabilities. Equal probabilities require an additional modelling assumption.

Bayes’ theorem

Bayes’ theorem reverses a conditional probability when P(E) > 0:

P(H | E) = P(E | H) × P(H) / P(E).

Here P(H) is the prior probability, P(E | H) is the likelihood of the observed evidence under the hypothesis, and P(H | E) is the posterior probability after observing the evidence.

When the mutually exclusive hypotheses H₁, ..., Hₖ cover all possibilities:

P(Hⱼ | E) = P(E | Hⱼ)P(Hⱼ) / Σᵢ P(E | Hᵢ)P(Hᵢ).

The denominator is the total probability of the evidence across all hypotheses.

Diagnostic-test interpretation

Use precise terms:

Quantity	Definition
Sensitivity	`P(test positive
Specificity	`P(test negative
False-positive rate	`1 − specificity`
Positive predictive value (PPV)	`P(condition present
Negative predictive value (NPV)	`P(condition absent

Sensitivity and specificity describe performance conditional on condition status in the evaluated setting. PPV and NPV also depend on the condition prevalence, or more generally the pre-test probability, in the population where the test is used. A single percentage described as overall accuracy does not identify sensitivity, specificity or predictive value and should not replace these terms.

Consider a hypothetical condition with 1% prevalence, a test with 99% sensitivity and 95% specificity. The false-positive rate is 5%.

P(positive) = (0.99 × 0.01) + (0.05 × 0.99) = 0.0594.

PPV = P(condition | positive) = 0.0099 / 0.0594 = 0.1667.

The positive predictive value is therefore about 16.7% under these assumptions. This does not mean that the test has 16.7% sensitivity or that it performs the same way in every population. It shows how a low pre-test probability can produce many false positives relative to true positives.

Bayes example: supply quality

Suppose three suppliers provide 35%, 40% and 25% of a hospital’s lamps, with hypothetical defect rates of 2%, 3% and 5%, respectively.

P(defect) = (0.35 × 0.02) + (0.40 × 0.03) + (0.25 × 0.05) = 0.0315.

P(supplier C | defect) = (0.25 × 0.05) / 0.0315 = 0.3968.

About 39.7% of defective lamps would come from supplier C under this model. The calculation can inform an audit question, but supplier decisions would also require sample sizes, uncertainty intervals, severity, costs and contractual evidence.

Bayes example: differential diagnosis

In a simplified hypothetical fever population, suppose 70% of cases are influenza and 30% are measles. Assume that rash occurs in 8% of influenza cases and 95% of measles cases.

P(rash) = (0.70 × 0.08) + (0.30 × 0.95) = 0.341.

P(measles | rash) = (0.30 × 0.95) / 0.341 = 0.8358.

The posterior probability is about 83.6% within this two-diagnosis model and its assumed rates. Real diagnosis cannot be reduced to this illustration: other causes, patient history, examination, test performance and local epidemiology matter.

Practice questions

In a hypothetical population, 45% have private insurance, 30% have public insurance and 10% have both. What proportion has at least one of the two forms of insurance?
A treatment succeeds independently with probability 0.70 for each of three patients. Under that independence assumption, what is the probability that all three experience success?
A hypothetical condition has 2% prevalence. A screening test has 90% sensitivity and 85% specificity. What is the PPV?

Answer check

0.45 + 0.30 − 0.10 = 0.65, or 65%.
0.70³ = 0.343, or 34.3%.
(0.02 × 0.90) / [(0.02 × 0.90) + (0.98 × 0.15)] = 0.1091, or about 10.9%.

Limitations and appropriate use

The examples are simplified and hypothetical. They do not establish empirical rates for Sri Lanka or any other population.
Independence, equal likelihood and distributional form are modelling assumptions. A calculation is valid only when its assumptions match the problem.
Diagnostic-test interpretation requires a suitable reference standard, representative participants and uncertainty estimates. The worked examples omit those design questions so that the arithmetic remains visible.
Probability supports health-economic analysis but does not replace clinical judgement, epidemiological evidence, equity analysis or economic evaluation.

Academic integrity and AI disclosure

Use the worked examples to check your reasoning after attempting each problem. Do not submit the wording or calculations as your own assessed work without following the relevant academic-integrity rules and citing sources where required.

AI assistance was used during migration to extract the legacy lesson, restructure the material and check the arithmetic. The probability definitions, formulas and numerical results were then reviewed against the sources below. The author remains responsible for the published version.

Sources

Illowsky, B. and Dean, S. (2023) Introductory Statistics 2e. OpenStax, especially Two Basic Rules of Probability and the probability formula review.
Pennsylvania State University, Eberly College of Science (n.d.) STAT 414: Introduction to Probability Theory.
US Food and Drug Administration (2007) Statistical Guidance on Reporting Results from Studies Evaluating Diagnostic Tests.
Briggs, A., Claxton, K. and Sculpher, M. (2006) Decision Modelling for Health Economic Evaluation. Oxford University Press.