A high-precision manufacturing plant uses two independent-looking monitoring systems, Alpha and Beta, to detect micro-faults during production. Let AAA be the event that system Alpha flags a defect and BBB be the event that system Beta flags a defect. During a routine audit, the following probabilities were established:
P(A)=0.25,P(A∩B)=0.05andP(A′∩B′)=0.68 P(A) = 0.25, \quad P(A \cap B) = 0.05 \quad \text{and} \quad P(A' \cap B') = 0.68 P(A)=0.25,P(A∩B)=0.05andP(A′∩B′)=0.68Calculate the probability that for a single production unit:
At least one of the monitoring systems flags a defect.
System Beta flags a defect.
Given that system Alpha flags a defect, determine the probability that system Beta also flags it.
A quality control engineer suspects that the triggering of system Beta is linked to the performance of system Alpha. Determine whether or not events AAA and BBB are statistically independent.
Comment on the engineer's suspicion in light of your answer to part (d).
200 exam-style questions on OCR (MEI) A Level Maths 2.3 Probability, covering 2.3.1 Calculate the probability of an event, 2.3.2 Complementary events, 2.3.3 Expected frequency of an event, 2.3.4 Diagrams to calculate probabilities, 2.3.5 Mutually exclusive and independent events, 2.3.6 Add probabilities for mutually exclusive events, 2.3.7 Multiply probabilities for independent events, 2.3.8 Mutually exclusive and independent events (notation) (A-level only), 2.3.9 Venn diagrams for probabilities (A-level only), 2.3.10 Conditional probabilities (A-level only), and 2.3.11 Independence via conditional probability (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.