Probability Case Study Questions for Class 12 – Printable PDF Format

Case Study Description: A public health department is evaluating a new, non-invasive diagnostic test for a very rare genetic disorder, let’s call it Disease $D$. The prevalence of Disease $D$ in the general population is estimated to be $0.5\%$ (or one in two hundred people). The test is known to be highly accurate. Specifically, the probability that the test correctly identifies a person \textbf{with} the disease (a true positive result) is $99\%$. However, no test is perfect. The probability that the test correctly identifies a person \textbf{without} the disease (a true negative result) is $98\%$. This means there is a chance of a false positive (testing positive when healthy) and a false negative (testing negative when sick). A person is selected at random from the population and is administered this test. The public health officials are particularly interested in the probability that a person who tests positive actually has the disease, as this impacts resource allocation and patient counseling. This scenario highlights how \textbf{Conditional Probability} and the concept of \textbf{Bayes’ Theorem} are crucial for interpreting medical test results, especially for rare diseases, where the prior probability is very low. Even a highly accurate test can yield surprising results when applied to the general population. Let $D$ be the event that a person has the disease, and $T$ be the event that the test result is positive. We are given $P(D)=0.005$, $P(T|D)=0.99$, and $P(T’|D’)=0.98$.