Probability

Concept

Theoretically, Probability is the mathematical measure of the likelihood that an event will occur. Probability theory provides the formal framework for quantifying uncertainty and forms the foundation of statistical inference, stochastic processes, decision analysis and health economic modelling. In health economics, probabilities are used to represent clinical events such as disease progression, treatment response, adverse events, mortality and healthcare utilisation, allowing uncertainty to be incorporated into decision models.

Mathematically, probability assigns a value between 0 and 1 to an event within a defined sample space, satisfying the axioms established by Kolmogorov. Probabilities may be estimated from observed frequencies, statistical models or expert judgement and are manipulated using recognised probability rules, including conditional probability, joint probability and Bayes' theorem. These mathematical properties underpin transition probabilities in Markov models, event probabilities in decision trees and probabilistic sensitivity analyses.

In practice, probabilities are estimated from randomised controlled trials, observational studies, registries, epidemiological databases, systematic reviews and meta-analyses. They populate health economic models by determining the likelihood of transitions between health states or the occurrence of clinical events during each model cycle. Parameter uncertainty surrounding probabilities is represented using appropriate probability distributions, most commonly the beta distribution.

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Purpose

Used to quantify uncertainty, estimate the likelihood of clinical and economic events, and provide fundamental inputs for statistical analyses and health economic decision models.

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Mathematical Formulae

Primary Formula

P(A) = Number of favourable outcomes / Total number of possible outcomes

for equally likely outcomes.

Supporting Formulae

Complement rule:

P(A?) = 1 ? P(A)

Addition rule:

P(A ? B) = P(A) + P(B) ? P(A � B)

Conditional probability:

P(A | B) = P(A � B) / P(B)

Multiplication rule:

P(A � B) = P(B) P(A | B)

Related Mathematical Methods

• Probability theory
• Conditional probability
• Joint probability
• Bayes' theorem
• Markov models
• Decision trees
• Monte Carlo simulation
• Probabilistic sensitivity analysis

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Example

A clinical trial reports that 48 of 200 patients experience a serious adverse event.

The estimated probability is

P(Adverse Event) = 48/200 = 0.24

This probability is entered into a decision tree to estimate expected treatment costs and health outcomes. During probabilistic sensitivity analysis, the parameter is represented using a beta distribution to reflect sampling uncertainty.

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Excel Implementation

Function	Example Formula	Health Economics Application
COUNTIF	=COUNTIF(B2:B201,""Yes"")/COUNTA(B2:B201)	Estimate the probability of an observed event
COUNTIFS	=COUNTIFS(A2:A201,""Treatment"",B2:B201,""Yes"")/COUNTIF(A2:A201,""Treatment"")	Estimate subgroup-specific probabilities
AVERAGE	=AVERAGE(C2:C201)	Estimate a probability from binary outcomes coded as 0 and 1
BETA.DIST	=BETA.DIST(A2,12,38,FALSE)	Evaluate a beta probability distribution for probabilistic sensitivity analysis
RAND	=RAND()	Generate random probabilities for simulation

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VBA (Optional)

Automate estimation of event probabilities from clinical datasets and populate health economic models with updated probability parameters.

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Sources

• Kolmogorov AN. Foundations of the Theory of Probability. Chelsea Publishing; 1956.
• Ross SM. Introduction to Probability Models. 12th ed. Academic Press; 2019.
• Briggs AH, Claxton K, Sculpher MJ. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford University Press; 2015.
• NICE. Health Technology Evaluation Manual.