Conditional Probability

Conditional Probability for Branches and Transitions in Health Economic Models

Conditional probability answers a narrower question than an ordinary probability, namely how likely an event is once something else is known to have happened, so the calculation counts only the cases where that other event occurred. It is written $P(A \mid B)$ and equals the probability that both events happen divided by the probability of $B$. Most probabilities in health economic models are of this kind. A branch at a chance node applies only to people who reached that node, and a transition probability in a Markov model applies only to people in a given health state at the start of a cycle. The reversal of a conditional probability with new evidence has its own page on Bayes theorem.

The definition and why the denominator matters

Grinstead and Snell, in their introductory probability textbook, frame conditional probability as the revised probability of an event once another event is known to have occurred. Outcomes outside the conditioning event drop out, and those inside it keep their relative weights. The result is the formula below.

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$

where $A$ is the event of interest, $B$ is the conditioning event, $P(A \cap B)$ is the probability that both occur and $P(B)$ is the probability of the conditioning event, which must be greater than zero.

The conditioning event sets the denominator, so it decides which population the number describes. Grinstead and Snell illustrate this with a life table in which 89.835% of a female birth cohort live to age 60 and 57.062% live to age 80. For a woman who has already reached 60, the probability of reaching 80 is 57,062 divided by 89,835, or 0.6352, higher than the 0.57062 that applies at birth.

Swapping the two events changes the question. In diagnostic test evaluation, sensitivity is the probability of a positive test among people with the condition, $P(T^{+} \mid D^{+})$, while the positive predictive value is the probability of the condition among people who test positive, $P(D^{+} \mid T^{+})$. Rautenberg and colleagues, in a primer on decision trees for diagnostics, note that unlike sensitivity and specificity, predictive values depend on prevalence, so the prevalence used must match the setting the predictive values describe.

Three rules that follow from the definition

Rearranging the definition gives the multiplication rule, which builds a joint probability from a sequence of conditional steps. Summing such products over a complete set of mutually exclusive cases gives the law of total probability. Grinstead and Snell combine the two to derive Bayes' formula.

$$P(A \cap B) = P(B)\,P(A \mid B)$$

where the symbols are as defined above, so the joint probability is the probability of reaching $B$ multiplied by the probability of $A$ among those who reach it.

$$P(A) = \sum_{i=1}^{m} P(H_i)\,P(A \mid H_i)$$

where $H_1, \dots, H_m$ are mutually exclusive events that together cover every possible case, $P(H_i)$ is the probability of case $i$ and $P(A \mid H_i)$ is the probability of $A$ within that case.

Two events are independent when knowing that one has occurred leaves the probability of the other unchanged, so that $P(A \mid B) = P(A)$. Grinstead and Snell show that this holds exactly when $P(A \cap B) = P(A)\,P(B)$. In a model, independence is an assumption about the disease and the data. Multiplying two marginal probabilities to obtain a joint probability is valid only when that assumption holds.

Branch probabilities in decision trees

A decision tree is a multiplication rule drawn as a diagram. Grinstead and Snell point out that the weight on a second-level branch of a probability tree is the conditional probability of that branch given the first-level outcome. The probability of reaching the end of a pathway is therefore the product of the branch probabilities along it, and the probabilities leaving any chance node must sum to one because they cover mutually exclusive and exhaustive outcomes for the people at that node.

The same population can be split in either order. Rautenberg and colleagues describe a disease-based tree, which branches first on true disease status and then on the test result using sensitivity and specificity, and a test-based tree, which branches first on the test result and then on disease status using predictive values. Built correctly, the two give the same model results, because each is a valid factorisation of the same joint probabilities. For sequential testing they add that correlation between tests and conditional dependence are relevant.

Transition probabilities, the Markov property and survival data

Sonnenberg and Beck, in their practical guide to Markov models in medical decision making, define a transition probability as the net probability of moving from one state to another during a single cycle. That probability is conditional on the state occupied at the start of the cycle. The Markov property adds that the process has no memory for earlier cycles, so the next transition depends only on the current state. The Decision Support Unit's Technical Support Document 19 (TSD 19) illustrates the point with a three-state cancer model needing a probability of death conditional on being progression-free and another conditional on having progressed.

That structure is what separates a state-transition model from a partitioned survival model. TSD 19 explains that in a state-transition model mortality has a structural link to earlier progression events, whereas mortality in a partitioned survival model is determined only by time-to-death data and is not explicitly linked to progression. Conditional probabilities also organise competing exits from a state. Where the cause of death matters, the ISPOR-SMDM task force report on state-transition modelling gives one unbiased approach, which is to model the probability of death first, followed by a conditional distribution of cause-specific deaths. The report also recommends converting probabilities between time units through rates (covered under rate-to-probability conversion).

Survival data supply many of these probabilities. Clark and colleagues, in a tutorial on survival analysis, describe the Kaplan-Meier estimator as the product of a sequence of conditional survival probabilities, and the hazard as the instantaneous event rate for someone who has already survived to time $t$. Because surviving beyond a later time implies surviving beyond an earlier one, the definition of conditional probability gives the probability of an event in an interval among people alive at its start.

$$P(\text{event in } (t, t+u] \mid T > t) = 1 - \frac{S(t+u)}{S(t)}$$

where $T$ is the time to the event, $S(t)$ is the survival function giving the probability of remaining event-free beyond time $t$, and $u$ is the cycle length.

National life tables apply the same idea to background mortality. The Office for National Statistics defines the life table quantity $q_x$ as the probability that a person aged exactly $x$ will die before reaching age $x + 1$, which is a probability conditional on having survived to age $x$. Sonnenberg and Beck describe the most convenient approach as a table of age-specific mortality rates, with the appropriate value looked up for each cycle as the cohort ages.

Worked example with an adverse event, an admission and a survival curve (illustrative)

The numbers are illustrative and describe no real treatment. In part A, a new treatment causes an adverse event in 10% of patients. Among patients with the adverse event, 30% are admitted to hospital, and among patients without it, 2% are admitted. Each admission costs GBP 3,000.

The four pathway probabilities sum to one, which checks that the branches at each node are complete. The law of total probability gives the overall admission probability.

$$P(\text{adm}) = 0.10 \times 0.30 + 0.90 \times 0.02 = 0.048$$

where $P(\text{adm})$ is the probability that a treated patient is admitted, 0.10 and 0.90 are the probabilities of having and not having the adverse event, and 0.30 and 0.02 are the admission probabilities conditional on each.

The expected admission cost is 0.048 × GBP 3,000 = GBP 144 per patient, of which 0.030 × GBP 3,000 = GBP 90 follows an adverse event. Reversing the conditioning, the share of admissions that follow an adverse event is 0.030 / 0.048 = 0.625. Two shortcuts give wrong answers. Treating the events as independent and multiplying the marginal probabilities gives 0.10 × 0.048 = 0.0048 for the joint probability of an adverse event and an admission, less than a sixth of the correct 0.030. Applying the 30% admission probability to the whole cohort would give 0.30 × GBP 3,000 = GBP 900 per patient instead of GBP 144.

In part B, a fitted survival curve gives $S(2) = 0.80$ and $S(3) = 0.70$ at two and three years. Twelve months is the cycle length.

$$1 - \frac{S(3)}{S(2)} = 1 - \frac{0.70}{0.80} = 0.125$$

where $S(2)$ and $S(3)$ are the illustrative probabilities of surviving beyond two and three years. The annual transition probability for people alive at the start of year three is 0.125. The difference $S(2) - S(3) = 0.10$ is the probability, seen from time zero, of dying during year three. Using 0.10 as the year-three transition probability would apply it to the 0.80 still alive and remove only 0.08 of the original cohort, understating deaths in that cycle by a fifth.

Conditioning, association and causal effects

Many model inputs come from comparing outcomes between groups, and those comparisons are conditional probabilities. Hernán and Robins, in their textbook on causal inference, define association as a different risk in two disjoint subsets of the population defined by the treatment actually received, and causation as a different risk in the same whole population under two different treatments. A conditional risk such as $P(Y = 1 \mid A = 1)$ describes the people who happened to be treated; it is a valid estimate of the risk under treatment only if treated and untreated people are otherwise comparable. Hernán and Robins give a hypothetical case in which aspirin halves five-year mortality yet the treated show a 50% higher risk, because people at high cardiovascular risk are preferentially prescribed it.

Conditioning can also create associations that have no causal basis. Hernán and Robins show that two independent causes of a common effect generally become associated once the analysis is restricted to people who have that effect. Selecting a model's source population on an outcome-related characteristic can therefore bias conditional estimates. The ISPOR-SMDM state-transition report suggests causal graphs to state causal assumptions explicitly when observational results feed a model.

Common errors with conditional probabilities in models

Most errors come from losing track of the denominator. The list below covers the ones that recur in model inputs and in reading published estimates.

• Transposing the conditional. Sensitivity is used where positive predictive value is needed, or the reverse. The two have different denominators and can differ widely.
• Assuming independence by default. Multiplying marginal probabilities to get a joint probability, as in part A above, ignores the dependence that the tree exists to represent.
• Using cumulative or unconditional probabilities as per-cycle transitions. A cycle's transition probability applies to people still in the state, as part B shows.
• Moving predictive values between settings. Predictive values carry the prevalence of the study population and need recalculating for a screening population with lower prevalence.
• Ignoring dependence between sequential tests. A second test's accuracy among people who tested positive on the first may differ from its accuracy in the whole population.
• Mixing time windows or populations. A conditional probability measured over five years, or in a selected hospital cohort, does not fit a one-year cycle or a general population without adjustment.
• Small denominators. A conditional probability estimated from few people in the conditioning group is unstable, and uncertainty in it belongs in the probabilistic sensitivity analysis.

Sources

• Clark TG, Bradburn MJ, Love SB, Altman DG. Survival analysis part I: basic concepts and first analyses. British Journal of Cancer. 2003;89(2):232-238. doi:10.1038/sj.bjc.6601118
• Grinstead CM, Snell JL. Grinstead and Snell's Introduction to Probability. 2nd ed. American Mathematical Society; CHANCE Project version dated 4 July 2006. Section 4.1. https://math.dartmouth.edu/~prob/prob/prob.pdf
• Hernán MA, Robins JM. Causal Inference: What If. Boca Raton: Chapman and Hall/CRC; 2020 (online edition dated 19 August 2026). Sections 1.5 and 6.3. https://miguelhernan.org/whatifbook
• Office for National Statistics. National life tables QMI. Last revised 10 December 2025. https://www.ons.gov.uk/peoplepopulationandcommunity/birthsdeathsandmarriages/lifeexpectancies/methodologies/nationallifetablesqmi
• Rautenberg T, Gerritsen A, Downes M. Health economic decision tree models of diagnostics for dummies: a pictorial primer. Diagnostics. 2020;10(3):158. doi:10.3390/diagnostics10030158
• Siebert U, Alagoz O, Bayoumi AM, Jahn B, Owens DK, Cohen DJ, Kuntz KM. State-transition modeling: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-3. Value in Health. 2012;15(6):812-820. doi:10.1016/j.jval.2012.06.014
• Sonnenberg FA, Beck JR. Markov models in medical decision making: a practical guide. Medical Decision Making. 1993;13(4):322-338. doi:10.1177/0272989X9301300409
• Woods B, Sideris E, Palmer S, Latimer N, Soares M. NICE DSU Technical Support Document 19: Partitioned survival analysis for decision modelling in health care: a critical review. Sheffield: Decision Support Unit; 2017. https://sheffield.ac.uk/nice-dsu/tsds/partitioned-survival-analysis