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.
