1. (Permutation) In how many different ways can 4 married couples coming to a dinner be seated at a round table of eight seats (two seating arrangements are equivalent if everyone has the same immediate left and right neighbors). (1) if there are no restrictions? (2) if each married couple is seated together? (3) if the members of each sex are seated together? (4) Assume the eight people are seated randomly. Compute the probability P("each married couple is seated together or the members of each sex are seated together").

2.(Bayes formula, No pts) In the setup of an assembly line, the system is either correctly or incorrectly adjusted. The probability of a correct adjustment is 0.95. When correctly adjusted, the system operates with a 2% defective rate. However, if incorrectly adjusted, it has a 50% defective rate. Assume that defects occur independently of each other.

(1) (10 pts) Complete the tree diagram below by filling in the relevant probabilities and conditional probabilities, where C (resp., I) represents the event that the system is correctly (resp., incorrectly) adjusted, and D (resp., N) represents the event that the next produced item is defective (resp., non-defective);

(2) (5 pts) After the system starts a production run, what is the probability that a defect is observed? (Hint: D = (C D) (I D), Total Probability)

(3) (5 pts) Suppose that one item selected by an inspector is found to be defective. What is the probability that the system is incorrectly adjusted? (Hint: P(I | D), Bayes Theorem)