Question Type 2: Using Bayes' theorem Bootcamps

Given a disease prevalence of 1%, a test has sensitivity 99% and specificity 95%. What is the probability that a person has the disease given a positive test result?

A disease affects 5% of the population. A diagnostic test has a false positive rate of 2% and false negative rate of 3%. If a person tests negative, what is the probability they do not have the disease?

In a population, 5% carry a virus. A test correctly identifies infected individuals with probability 0.90 and correctly clears uninfected individuals with probability 0.98. If someone tests negative, what is the probability they are uninfected?

A factory’s inspection machine flags 2% of items as defective. If an item is defective, the machine catches it with probability 0.80; if an item is good, it falsely flags it 1% of the time. What is the probability an item is actually defective when flagged?

An HIV screening test has sensitivity 99.7% and specificity 99.8%. Prevalence of HIV in the tested population is 0.05%. If a person tests negative, what is the probability they are HIV–free?

An email service labels 20% of incoming mail as spam initially. Its filter correctly tags spam 95% of the time and incorrectly flags genuine mail as spam 1% of the time. If an email is flagged as spam, what is the probability it is actually spam?

Two suppliers produce parts: Supplier A 60% of output, defect rate 3%; Supplier B 40%, defect rate 5%. If a randomly chosen part is found defective, what is the probability it came from Supplier A?

Using the same vaccine study data ($P(V)=0.95$, $P(-\mid V)=0.99$, $P(-\mid \neg V)=0.02$), what is the probability a person was vaccinated given a positive test result?

In allergy testing, 10% of patients have an allergy. The test has 85% sensitivity and 90% specificity. If a patient tests negative, what is the probability they truly have no allergy?

A vaccine study: 95% of people receive the vaccine. Given vaccination, probability of a false positive test is 1%. Without vaccination, probability of a false negative is 2%. If a person tests negative, what is the probability they were vaccinated?

At airport security, 0.5% of passengers carry prohibited items. The scanner detects them with 98% probability and gives false alarms 2% of the time. If an alarm sounds, what is the probability the passenger is carrying a prohibited item?

A rare condition has prevalence 0.1%. A test has 99.9% sensitivity and 99.5% specificity. If a person tests positive, what is the probability they truly have the condition?