Suppose that there was a cancer diagnostic test that was 95...

Suppose that there was a cancer diagnostic test that was 95 percent accurate both on those that do and those that do not have the disease. If $.4$ percent of the population have cancer, compute the probability that a tested person has cancer, given that his or her test result indicates so.

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Key Concepts

Base Rate (Prevalence)

Base rate, or prevalence, is the proportion of individuals in a population who have a particular condition before any testing is done. Recognizing the base rate is important since even tests with high sensitivity and specificity can yield counterintuitive results when the disease is rare.

Sensitivity and Specificity

Sensitivity and specificity are critical measures of a diagnostic test's performance. Sensitivity refers to the test's ability to correctly identify those with the disease, while specificity pertains to its ability to correctly identify those without the disease. Both metrics are essential for understanding the reliability of test outcomes.

Conditional Probability

Conditional probability examines the chance of an event occurring given that another event has already occurred. In the context of test results, it is used to calculate the probability that a person actually has a disease given that they have tested positive.

Bayes' Theorem

Bayes' Theorem is a fundamental concept in probability that enables one to update the probability of a hypothesis as more evidence or information becomes available. In diagnostic testing, it is used to determine the likelihood of a disease given a positive test result by incorporating both the test accuracy and the prevalence of the disease.

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