Assume the prevalence of a certain disease in the general...

Assume the prevalence of a certain disease in the general population is $1 \%$. Assume there exists a quite reliable test for the disease, say, the test on a diseased/healthy person is positive/negative with $99 \%$ probability. If the test is positive, what is the chance of having the disease? Many medical doctors say about $99 \%$. Show with Bayes' rule (2.19) that the true answer is scanty $50 \%$.

Key Concepts

Bayes’ Rule

Bayes’ Rule is a fundamental theorem in probability theory that provides a way to update the probability of a hypothesis based on new evidence. It is central to inferential statistics and decision-making processes in various fields, including medicine, where it's used to compute the likelihood of a disease given a test result.

Prior Probability

The prior probability is the initial estimation of the likelihood of an event, such as having a disease, before new evidence is taken into account. In medical testing, the prevalence of the condition in the general population serves as the prior probability, influencing subsequent updates as per Bayes' Rule.

Conditional Probability

Conditional probability measures the probability of an event occurring given that another event has already occurred. In diagnostic testing, it determines the likelihood of obtaining a positive test result if a patient does, or does not, actually have the disease, helping to interpret the reliability of test outcomes.

False Positive Paradox

The false positive paradox occurs when a diagnostic test, despite having high sensitivity and specificity, results in a high probability of false positives due to a very low prevalence (base rate) of the disease. This paradox demonstrates the importance of considering base rates when evaluating test results and prevents overestimating the probability of having the disease following a positive result.

Transcript

Please give Ace some feedback