A quality assurance manager at a high-precision fabrication facility is analyzing the occurrence of component defects across different work shifts.
Event S S\,S is defined as a component being produced during the night shift.
Event M M\,M is defined as a component having a mechanical defect.
The results of a random audit are summarized in the following table of probabilities:
MM′S0.080.32S′0.040.56\begin{array}{|c|c|c|} \hline & M & M' \\ \hline S & 0.08 & 0.32 \\ \hline S' & 0.04 & 0.56 \\ \hline \end{array}SS′M0.080.04M′0.320.56A component is selected at random from the audit.
(i) Find P(S)P(S)P(S).
(ii) Find P((S∩M)′)P((S \cap M)')P((S∩M)′).
(iii) Find P(M∣S′)P(M | S')P(M∣S′).
Determine whether the events S S\,S and M M\,M are independent. Fully justify your answer.
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.