Probability

Probability: Concept Architecture

Orientation and learning roadmap

The page begins with events and the rules that keep their numerical assignments coherent. It then shows how conditioning changes the reference group, works through a diagnostic-testing example, and explains how probabilities enter health decision models without being confused with uncertainty about an estimate.

Define the event, population, and time period

A numerical probability is interpretable only when the event and its conditions are stated. For example, “death” is not a sufficiently specified model input, whereas death within one year among patients starting a defined treatment in a stated population is closer to one. The population, starting state, follow-up duration, competing events, and data source determine whether a reported probability applies to the decision at hand.

A probability model assigns values to events in a sample space of possible outcomes. Its basic rules require every event probability to lie from zero to one, the probability of the whole sample space to equal one, and probabilities of mutually exclusive events to add. These rules make calculations internally consistent; they do not guarantee that an estimated input is empirically credible.

For an event $A$ and its complement $A^c$:

$$0\leq P(A)\leq1,\qquad P(A^c)=1-P(A).$$

The probability can be written as a proportion or percentage, but the scale must be explicit. A value of 0.20 is 20%, not 0.20%, and does not say whether the event happens exactly once, at least once, or by a particular time unless the event has been defined that way.

Combine and condition events correctly

“And” refers to the intersection of events, while “or” refers to their union. Two outcomes can sometimes occur together, so adding their probabilities without accounting for their overlap double counts that overlap. Conditional probability changes the denominator to the group for whom the condition is true.

$$P(A\cup B)=P(A)+P(B)-P(A\cap B),$$

$$P(A\mid B)=\frac{P(A\cap B)}{P(B)}\quad\text{when }P(B)>0.$$

If $A$ and $B$ are mutually exclusive, their intersection has probability zero and the addition rule simplifies. Independence is different: it means $P(A\mid B)=P(A)$ when $P(B)>0$, or equivalently $P(A\cap B)=P(A)P(B)$. Mutually exclusive nonzero-probability events cannot be independent, since learning that one occurred rules out the other.

A worked diagnostic-testing example

A positive test does not have the same probability as the disease among people with a positive result. One is a test result conditional on disease status; the other reverses the condition and depends on the disease prevalence in the tested population. Bayes' rule and the total-probability rule keep those denominators explicit.

Suppose 10% of a population has a condition, test sensitivity is 80%, and specificity is 90%. Among 1,000 people, the illustrative expected counts are 100 with the condition and 900 without it; 80 of the first group test positive and 90 of the second group test positive. The resulting positive predictive value is:

$$P(\text{condition}\mid\text{positive})=\frac{0.80(0.10)}{0.80(0.10)+0.10(0.90)}=\frac{80}{170}\approx0.471.$$

Thus about 47.1% of positive tests come from people with the condition under these illustrative inputs. The 80% sensitivity is $P(\text{positive}\mid\text{condition})$, which answers a different question. These are expected counts, not observed trial data, and the result would change with prevalence or test performance in the actual population.

Use probabilities in health models

Decision trees assign branch probabilities to possible outcomes from the same chance node. Exhaustive, mutually exclusive branches should sum to one; otherwise the expected value from the branches will be misweighted. A state-transition model similarly uses cycle-specific probabilities for movement among defined health states, with each row of a transition matrix summing to one when the states exhaust the next-cycle possibilities.

A probability for a specified event and time interval is distinct from a rate or hazard that describes event occurrence per unit of time. Under a constant hazard $h$ over an interval of length $t$, the event probability is $1-\exp(-ht)$, subject to the assumed hazard process and competing-event structure. Substituting a rate directly for a cycle probability without checking units and assumptions can yield invalid model inputs.

Empirical proportions can estimate probabilities, but their applicability depends on the sampling frame, missing data, event definitions, and follow-up. In a decision model, uncertainty about an estimated probability may be represented with an appropriate parameter distribution for probabilistic sensitivity analysis. That distribution reflects uncertainty about the input; a single draw from it is not the same as the event probability itself.

Check the model's numerical and clinical meaning

A probability can be mathematically valid yet clinically implausible for the proposed population. Validation therefore includes both arithmetic constraints and a comparison with relevant evidence and expert understanding. Decisions should be examined across plausible uncertainty in influential probabilities rather than reported from a single unqualified estimate.

1. State the event and conditioning set. Identify the population, starting state, time horizon, comparator, and any risk factors or prior observations that define the probability.
2. Check the numerical rules. Keep values between zero and one and confirm that exhaustive, mutually exclusive model branches sum to one.
3. Match evidence to use. Examine source population, endpoint definition, follow-up, and transportability to the intended decision.
4. Preserve dependence. Avoid multiplying probabilities as if events were independent when shared causes or conditional relationships matter.
5. Propagate input uncertainty. Test plausible ranges and correlations, and report whether the preferred decision changes.

The appropriate probability is conditional on a well-defined question, not a universal property of a treatment or diagnosis. When evidence supports several plausible estimates, the model should show how those alternatives affect outcomes and decisions.

Evidence used for this explanation

The MIT materials establish the probability axioms, event rules, conditioning, and Bayes' rule used in the calculations. ISPOR-SMDM guidance connects parameter estimation and uncertainty analysis to transparent health decision modeling.

• MIT OpenCourseWare. Introduction to Probability: fundamentals, lectures on probability models, axioms, conditioning, and Bayes' rule.
• Briggs AH et al. Model Parameter Estimation and Uncertainty Analysis, ISPOR-SMDM Modeling Good Research Practices Task Force 6. Medical Decision Making (2012).
• Siebert U et al. State-Transition Modeling, ISPOR-SMDM Modeling Good Research Practices Task Force 3. Value in Health (2012).