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10. A darts player throws a dart at a standard board and the position of the dart is noted. Random variable X represents the distance (in m) of the dart from the board's center. a. Considering the statements below and the provided table of distances, construct a PDF for random variable X. . The player hits bullseyes (total of both types) on 4% of her throws, triples on 8% of her throws, and doubles on 5% of her throws. The player is equally likely to hit all spots within the two bullseyes. The player is equally likely to hit all spots within the triple ring. The player is equally likely to hit all spots within the double ring The player is equally likely to hit all spots outside of the scoring area. The PDF for X is continuous over [0, 170] and varies linearly between constants. The player has no angular control over her thrown darts; that is, she hits the twenty numbered wedges with equal probability. Dart Board Area Distance from Board Center Distance from Board Center to Nearest Point (mm) to Farthest Point (mm) Center Bullseye 0 7 Outer Bullseye 7 16 b/w Bullseye & Triple 16 106 Triple Ring 106 116 b/w Triple & Double 116 160 Double Ring 160 170 Non-scoring Area 170 225 * Dimensions are approximately correct, but metal wires have been neglected. b. What is the probability that a dart thrown by the player lands between 10 and 100 mm away from the center of the board? Bonus Questions for Darts Enthusiasts C. What is the probability that a thrown dart scores fewer than 5 points? d. What is the probability that a thrown dart scores greater than 37 points