Conditional Probability and Bayes
In probability notation, what is the difference between P(A|B) and P(B|A)?
In a probability tree diagram, if all leaf nodes represent mutually exclusive outcomes, what should be the sum of all leaf node probabilities?
If P(Rain) = 0.3 and P(Cloudy | Rain) = 0.8, what is P(Rain AND Cloudy)?
A security system has a 95% detection rate for intruders and a 3% false alarm rate for normal situations. If intruders attempt break-ins 1% of the time, what is the probability that an alarm indicates an actual intruder?
A weather prediction model correctly predicts rain 85% of the time when it will rain, and correctly predicts no rain 92% of the time when it won't rain. If it rains 25% of the days in a region, what is the probability that it will actually rain when the model predicts rain?
An online fraud detection system flags 6% of legitimate transactions and correctly identifies 94% of fraudulent transactions. If 0.5% of all transactions are fraudulent, what percentage of flagged transactions are actually fraudulent?
A pharmaceutical company tests a new drug using three different labs. Lab A conducts 40% of tests with 1% error rate, Lab B conducts 35% with 2% error rate, and Lab C conducts 25% with 4% error rate. If a test result contains an error, what is the probability it came from Lab C?
A diagnostic imaging technique has 88% sensitivity and 94% specificity for detecting a brain tumor. In a high-risk population where 8% have tumors, if 500 people are scanned and test positive, approximately how many actually have tumors?
In a factory quality control system, products pass through two inspection stages. Stage 1 catches 90% of defective items, and Stage 2 catches 80% of defective items that passed Stage 1. If 5% of products are initially defective, what percentage of products that pass both inspections are actually defective?