Question Type 2: Using Bayes' theorem Exercises

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?

20% of incoming emails are spam. An email 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, calculate the probability that it is actually spam.

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?

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

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 vaccine study is conducted where $95\%$ of people receive the vaccine. Given vaccination, the probability of a false positive test is $1\%$. Without vaccination, the probability of a false negative is $2\%$.

If a person tests negative, find the probability that they were vaccinated.

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?

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?

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?

Consider the following vaccine study data: $P(V)=0.95$, $P(-\mid V)=0.99$, and $P(-\mid V')=0.02$, where $+$ denotes a positive test result and $-$ denotes a negative test result.

Find the probability that a person was vaccinated given a positive test result.

A disease affects $5\%$ of the population. A diagnostic test has a false positive rate of $2\%$ and a false negative rate of $3\%$.

If a person tests negative, what is the probability they do not have the disease?

A factory produces items where $2\%$ are defective. An inspection machine flags $80\%$ of defective items. It also falsely flags $1\%$ of non-defective items.

Calculate the probability that an item is actually defective given that it has been flagged.