MPH 560 Module 2 Probability in Public Health Example

Chance and Certainty: Probability Concepts for Screening and Disease Transmission

Student Name

Master of Public Health Program, Aspen University

MPH 560: Applied Biostatistics for Public Health

Instructor Name

Month Day, Year

Chance and Certainty: Probability Concepts for Screening and Disease Transmission

Public health decisions are made under uncertainty. Will a contact become infected? Does a positive screening test mean disease? Probability provides the language and rules for answering such questions. This paper reviews basic probability concepts and applies them to disease transmission and screening, using composite examples whose calculations can be checked step by step.

Basic Rules

A probability ranges from 0, impossible, to 1, certain. The complement rule states that the probability an event does not occur is 1 minus the probability it does. The addition rule gives the probability that either of two events occurs. The multiplication rule gives the probability that two independent events both occur as the product of their probabilities. Independence means that one event does not change the probability of the other.

Samples and Populations

Probability connects samples to populations. When a sample is drawn at random, probability theory describes how sample statistics vary around population values, which is the basis for all inference later in the course (Whitley & Ball, 2002). Non-random samples break this link, so conclusions may not generalize.

Transmission Example

Suppose each of five household contacts of a person with a respiratory infection independently has a 10% chance of becoming infected. The probability that a given contact escapes infection is 0.9, so the probability that all five escape is 0.9 multiplied by itself five times, 0.59. By the complement rule, the probability that at least one contact is infected is 1 minus 0.59, or about 0.41, a 41% chance.

The Binomial Distribution

Counting infected contacts in this way is a binomial problem: a set number of independent chances, each with the same probability of the event. With five contacts and a 10% probability each, the expected number infected is 5 times 0.1, or 0.5. The binomial model helps estimate how many secondary cases to expect and plan contact tracing resources.

Conditional Probability

Conditional probability is the probability of one event given that another has occurred. In screening, the key conditional probabilities are sensitivity, the probability of a positive test given disease, and specificity, the probability of a negative test given no disease (Altman & Bland, 1994a). But patients and clinicians usually want the reverse: the probability of disease given a positive test.

A Screening Example

Consider screening 10,000 people for a condition with 2% prevalence using a test with 90% sensitivity and 95% specificity. The table shows the expected results.

Interpreting Predictive Values

Of the 670 people who test positive, only 180 have the disease, so the probability of disease given a positive result, the positive predictive value, is 26.9%. Nearly three of every four positives are false. The negative predictive value is very high, 99.8%, because the condition is uncommon. Predictive values depend on prevalence as well as on the test (Altman & Bland, 1994b).

How Prevalence Changes Everything

Apply the identical test where one person in five is affected, as in a symptomatic clinic population, and roughly 82% of positives are true. Nothing about the test differs between the two settings; only the mix of people tested does. This is why screening low-risk populations produces many false positives and why confirmatory testing is essential.

Bayes' Theorem

The calculation above is an application of Bayes' theorem, which updates the probability of disease after a test result. It combines the prior probability, prevalence, with the test's sensitivity and specificity to produce the posterior probability. Clinicians use the same logic informally when they consider a patient's risk before interpreting a result. Writing the calculation as counts out of 10,000, as in the table, is usually easier than applying the formula directly and less prone to error.

Implications for Public Health

Probability reasoning helps design programs: targeting screening to higher-risk groups, planning follow-up capacity for positives, explaining results to patients honestly and estimating transmission for contact tracing. Misunderstanding probability can lead to overreaction to positive tests or false reassurance. Health departments that explain predictive values to the public before launching screening campaigns help people react calmly to positive results and follow through on confirmatory tests.

Independence in Practice

The transmission example assumed that each contact's risk was independent. In reality, household contacts share air, meals and surfaces, so if one becomes infected others may be more likely to as well. Violated independence can make simple probability calculations misleading, which is why outbreak models often account for clustering.

Probability in Risk Communication

Probabilities are often misunderstood. Natural frequencies, such as 180 of 670 people with a positive test, are easier to grasp than conditional percentages. Presenting screening results as counts out of 1,000 or 10,000 people helps patients and policymakers understand what a test result means.

Relative Frequency and Surveillance

In surveillance, probabilities are often estimated from observed frequencies. If 30 of 1,000 people tested positive last month, the estimated probability of a positive test is 3%. As more data accumulate, estimates become more stable, reflecting the law of large numbers.

Sensitivity and Specificity Trade-Offs

Changing a test's cutoff trades sensitivity against specificity. A lower threshold for a positive result catches more true cases but produces more false positives. Programs choose cutoffs by weighing the harm of missed cases against the cost and anxiety of false alarms, which depends on the condition and the follow-up available.

Probability and Risk Factors

Epidemiologists also use probabilities to express risk: the probability of developing a disease over a period among people with and without an exposure. Comparing these probabilities produces the risk ratios and risk differences that describe associations in cohort studies.

Probability Distributions for Continuous Data

For continuous measures such as blood pressure, the normal distribution describes many variables approximately. About 95% of values lie within 1.96 standard deviations of the mean, a property used throughout inference in later modules.

Conclusion

Probability rules allow public health professionals to reason about transmission and screening. In the composite examples, the chance that at least one of five contacts became infected was 41%, and a test with 90% sensitivity and 95% specificity yielded a positive predictive value of only 26.9% at 2% prevalence. Understanding conditional probability and the role of prevalence is essential for designing and communicating about screening programs.

References

Altman, D. G., & Bland, J. M. (1994a). Diagnostic tests 1: Sensitivity and specificity. BMJ, 308(6943), 1552. https://doi.org/10.1136/bmj.308.6943.1552

Altman, D. G., & Bland, J. M. (1994b). Diagnostic tests 2: Predictive values. BMJ, 309(6947), 102.1. https://doi.org/10.1136/bmj.309.6947.102

Whitley, E., & Ball, J. (2002). Statistics review 2: Samples and populations. Critical Care, 6(2), 143-148. https://doi.org/10.1186/cc1473