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6 Denzel and Cherie are friends who often go to the cinema together. On such visits there is a probability of 0.4 that Denzel will buy popcorn. The probability that Cherie will buy popcom is 0.7 if Denzel buys popcom and 0.35 if he does not. Cherie buys 0.7 Denzel buys 0.4 does not buy buys 0.35 does not buy does not buy (a) When Denzel and Cherie visit the cinema together: (i) find the probability that both buy popcorn; (2 marks) (ii) show that the probability that neither buys popcorn is 0.39; (2 marks) (iii) find the probability that exactly one of them buys popcorn. (2 marks) (b) Sohaib sometimes joins Denzel and Cherie on their cinema visits. On these occasions, the probability that Sohaib buys popcorn is 0.55 if both Denzel and Cherie buy popcorn and 0.25 if exactly one of Denzel and Cherie buys popcorn. Find the probability that when Denzel, Cherie and Sohaib visit the cinema together: (i) all three buy popcorn; (2 marks) (ii) Cherie and Sohaib buy popcorn but Denzel does not. (2 marks)20. A manufacturing firm employs three analytical plans for the design and development of a particular product. For cost reasons, all three are used at varying times. In fact, plans 1, 2, and 3 are used for 30%, 20%, and 50% of the products, respectively. The defect rate is different for the three procedures as follows: P(D P ) = 0.01, P(D|P,) = 0.03, P(D P,) = 0.02, where P(D P) is the probability of a defective product, given plan j. If a random product was observed and found to be defective, which plan was most likely used and thus responsible? 21. A company producing electric relays has three manufacturing plants producing 50, 30, and 20 percent, respectively, of its product. Suppose that the probabilities that a relay manufactured by these plants is defective are 0.02, 0.05, and 0.01, respectively. (a) If a relay is selected at random from the output of the company, what is the probability that it is defective? (b) If a relay selected at random is found to be defective, what is the probability that it was manufactured by plant 2?Problem 6. [28 points] Scores for a final exam are given by picking an integer uniformly at random from the set {50, 51,...,97, 98). The scores of all 128 students in the class are assigned in this manner. For parts (a), (b), (c) and (d) you may NOT assume that these scores are assigned independently. For parts (e), (f), (g) and (h) you MAY assume that these scores are assigned independently. Let $1, .... Sims be their scores. Let S = ()!_ S,) be the average score of the dass. (a) [3 points] For ie {1, ..., 128}, what is E[S,] ? (b) [2 points] Show that E(S] = 74. Make no independence assumptions. (c) [4 points] Prove that Pr($ 2 88] Make no independence assumptions
