Show that in general a b c a c b c and that a b c a c

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2) Show that in general ( A B ) c = A c B c and that ( A B ) c = A c B c (so the results of (f) and (g) and of (d) and (e) above were not mere coincidences). 3) Suppose that P ( A ) = 0 . 02, P ( B ) = 0 . 01 and P ( A B ) = 0 . 023. Calculate (a) P ( A B ), (b) P ( A c B c ), (c) P ( A B c ), (d) P ( A | B c ), (e) P ( A | B ). Problem 3.16 A large company hires most of its employees on the basis of two tests. The two tests have scores ranging from one to five. The following table summarizes the perfor- mance of 16,839 applicants during the last six years. From this table we learn, for example, that 3% of the applicants got a score of 2 on Test 1 and 2 on Test 2; and that 15% of the applicants got a score of 3 on Test 1 and 2 on Test 2. We also learn that, for example, 20% of the applicants got a score of 2 on Test 1 and that 25% of the applicants got a score of 2 on Test 2. A group of 1500 new applicants have been selected to take the tests. (a) What should the cutting scores be if between 140 and 180 applicants will be short–listed for a job interview? Assume that the company wishes to short–list people with the highest possible performances on the two tests.
3.3. EXERCISES 55 Table 3.5: Score 1 2 3 4 5 Total 1 0.07 0.03 0.00 0.00 0.00 0.10 2 0.15 0.03 0.02 0.00 0.00 0.20 3 0.08 0.15 0.09 0.02 0.01 0.35 4 0.10 0.04 0.08 0.01 0.02 0.25 5 0.00 0.00 0.06 0.02 0.02 0.10 Total 0.40 0.25 0.25 0.05 0.05 1.00 Table 3.6: Score Test 1 Test 2 1 0.10 0.40 2 0.20 0.25 3 0.35 0.25 4 0.25 0.05 5 0.10 0.05 (b) Same as (a) but assuming now that the company wishes to hire people with the highest possible performances on at least one of the two tests. (c) (Continued from (a)) A manager suggests that only applicants who obtain marks above a certain bottom line in one of the tests be given the other test. Noticing that giving and marking each test costs the company $55, recommend which test should be given first. Approximately how much will be saved on the basis of your advise? (d) Repeat (a)–(c) if the two tests performances are independent and the probabilities are given by Table 2.6. Problem 3.17 A computer company manufactures PC compatible computers in two plants, called Plant A and B in this exercise. These plants account for 35 % and 65 % of the production, respectively. The company records show that 3 % of the computers manufactured by Plant A must be repaired under the warranty. The corresponding percentage for plant B is 2.5 %. (a) What is the percentage of computers that are repaired under the warranty and come from Plant A? (b) What percentage of computers repaired under the warranty come from Plant A? From Plant B? Problem 3.18 Twenty per cent of the days in a certain area are rainy (there is some mea- surable precipitation during the day), one third of the days are sunny (no measurable pre- cipitation, more than 4 hours of sunshine) and fifteen per cent of the days are cold (daily average temperature for the day below 5 o C). 1 - Would you use the above information as an aid in

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