. The three most popular options on a certain type of new car are a built-in GPS (A), a sunroof (B), and an automatic transmission (C). If 40% of all purchasers request A, 55% request B, 70% request C, 63% request A or B, 77% request A or C, 80% request B or C, and 85% request A or B or C, determine the probabilities of the following events. int: "A or B" is the event that at least one of the two options is requested; try drawing a Venn gram and labeling all regions.] a) The next purchaser will request at least one of the three options. b) The next purchaser will select none of the three options. c) The next purchaser will request only an automatic transmission and not either of the other two options. d) The next purchaser will select exactly one of these three options. . An educational consulting firm is trying to decide whether high school students who have never used a handheld calculator can solve a certain type of problem more easily with a calculator that uses reverse Polish logic or one that does not use this logic. A sample of 25 students is selected and allowed to practice on both calculators. Then each student is asked to work on one problem on the reverse Polish calculator and a similar problem on the other. Let p = P(X), where S indicates that a student worked the problem more quickly using reverse Polish logic than without, and let X = number of S's a) If p = 0.5, what is P(7 X 18)? b) If p = 0.8, what is P(7 X 18)? . The Reviews editor for a certain scientific journal decides whether the review for any particular book should be short (1-2 pages), medium (3-4 pages), or long (5-6 pages). Data on recent reviews indicate that 60% of them are short, 30% are medium, and the other 10% are long. Reviews are submitted in either Word or LaTeX. For short reviews, 80% are in Word, whereas 50% of medium reviews are in Word and 30% of long reviews are in Word. Suppose a recent review is randomly selected. a) What is the probability that the selected review was submitted in Word format? b) If the selected review was submitted in Word format, what are the posterior probabilities of it being short, medium, or long? (Use the Bayes' rule as appropriate)