Conditional Probability and Bayes
What does P(A|B) represent in probability notation?
In a probability tree diagram, how do you calculate the probability of reaching a specific leaf node?
If P(Disease) = 0.01 and P(Test Positive | Disease) = 0.95, what is P(Disease AND Test Positive)?
A medical test has a sensitivity of 90% (correctly identifies 90% of sick patients) and a specificity of 85% (correctly identifies 85% of healthy patients). If 2% of the population has the disease, what is the probability that a randomly selected person tests positive?
Using the same medical test from the previous question (sensitivity 90%, specificity 85%, disease prevalence 2%), what is the probability that a person who tests positive actually has the disease?
In an email spam detection system, 40% of emails are spam. The spam filter correctly identifies 95% of spam emails and incorrectly flags 8% of legitimate emails as spam. If an email is flagged as spam, what is the probability it is actually spam?
A rare genetic condition affects 1 in 10,000 people. A genetic test for this condition has 99.9% sensitivity and 99.5% specificity. What is the probability that someone who tests positive actually has the condition?
A company uses three machines (A, B, C) to produce items. Machine A produces 50% of items with 2% defective rate, Machine B produces 30% with 3% defective rate, and Machine C produces 20% with 5% defective rate. If a randomly selected item is defective, what is the probability it came from Machine B?
In a clinical trial, a new diagnostic test is being evaluated. The test has 92% sensitivity and 88% specificity for a condition that affects 15% of the tested population. If 1000 people are tested, approximately how many people who test negative actually have the condition (false negatives)?