ONLY NEED #4 PLEASE

12 4 14 12 9 4 Darts, like any game of skill, involves some degree of randomness. In this project, we will try to analyze the chances 5 20 1 18 of earning various point totals in individual dart tosses. A typical 9 dartboard is displayed in Figure 1. We may consider two random variables in the position of each 13 dart toss (see Figure 2): 11 6 1. The angle, 0, from the horizontal of the line (in red) on which the dart lands. 8 10 16 15 2. The position along this line, X, centered at the middle of the bullseye. 7 2. 19 3 17 If the location of the dart is the black dot in Figure 2, we see that X and define the location of the dart, with a positive X Figure 1 above the horizontal and a negative X below it. If a player aims a dart at the bullseye in the center of the 5 20 1 18 dartboard and tosses it, of course he/she may or may not hit the target with some probability, but we can say more if we know something about the pattern in the randomness. (14 In particular, a highly skilled player when aiming at the 111 bullseye, will hit the center of the bullseye on average, but with 6+ decreasing probability, the dart will land further away from 8 10 the target, so a normal distribution centered at 0 is reasonable for X. A highly skilled player also shows no particular angular 16 15 tendency, so we may model e as a uniformly distributed random variable that is independent of X. 19 3 17 Common rules of darts have the following scoring scheme': Figure 2 Where dart lands Point value Inner Bullseye 50 Outer Bullseye 25 Single Region (black/white area within Double ring) Score given on outside ring (1-20) Triple Ring (inner red/green ring) Triple the score given on the outside ring Double Ring (outer red/green ring) Double the score given on the outside ring Outside the Double Ring 0 13 7 Questions (Assume all throws are aimed at the center of the bullseye.) 1. What is the range of the random variable e? 2. For a skilled player, what is the mean of the normal random variable X? 3. Should the normal random variable X have a high or low o for the most skilled players? 4. Find the probability of the dart landing in each region for a player with o = 50 mm and for a player with 0 = 100 mm given the following specifications for the dartboard. Each region corresponds to the given ranges of (absolute) distances from the center of the inner bullseye. Region Distance is more than... Distance is less than... Inner Bullseye 0 mm 6.35 mm Outer Bullseye 6.35 mm 15.9 mm Inner Single Region 15.9 mm 95.3 mm Triple Ring 95.3 mm 104.8 mm Outer Single Region 104.8 mm 158.8 mm Double Ring 158.8 mm 168.3 mm Beyond 168.3 mm 5. What is the probability of landing in a single scoring region (shown in yellow in Figure 3) for each player? 5 20 1 18 6. What is the probability of landing in the wedge of the 9 dartboard marked with 20 if o = 50 mm (i.e., in the yellow 14 region or beyond)? (See Figure 4.) 13 11 6 7. What is the probability of scoring 7 points on one dart throw if o = 50 mm? 8 10 8. What is the probability of scoring 8 points on one dart 16 15 throw if o = 50 mm? 2 9. What is the probability of scoring 6 points on one dart throw if o = 50 mm? Figure 3 10. What are the highest and lowest possible scores from a single dart throw? What is the probability of each if 12 51 o = 50 mm? 9 12 4 1 19 3 17 18 4 12 12 4 14 12 9 4 Darts, like any game of skill, involves some degree of randomness. In this project, we will try to analyze the chances 5 20 1 18 of earning various point totals in individual dart tosses. A typical 9 dartboard is displayed in Figure 1. We may consider two random variables in the position of each 13 dart toss (see Figure 2): 11 6 1. The angle, 0, from the horizontal of the line (in red) on which the dart lands. 8 10 16 15 2. The position along this line, X, centered at the middle of the bullseye. 7 2. 19 3 17 If the location of the dart is the black dot in Figure 2, we see that X and define the location of the dart, with a positive X Figure 1 above the horizontal and a negative X below it. If a player aims a dart at the bullseye in the center of the 5 20 1 18 dartboard and tosses it, of course he/she may or may not hit the target with some probability, but we can say more if we know something about the pattern in the randomness. (14 In particular, a highly skilled player when aiming at the 111 bullseye, will hit the center of the bullseye on average, but with 6+ decreasing probability, the dart will land further away from 8 10 the target, so a normal distribution centered at 0 is reasonable for X. A highly skilled player also shows no particular angular 16 15 tendency, so we may model e as a uniformly distributed random variable that is independent of X. 19 3 17 Common rules of darts have the following scoring scheme': Figure 2 Where dart lands Point value Inner Bullseye 50 Outer Bullseye 25 Single Region (black/white area within Double ring) Score given on outside ring (1-20) Triple Ring (inner red/green ring) Triple the score given on the outside ring Double Ring (outer red/green ring) Double the score given on the outside ring Outside the Double Ring 0 13 7 Questions (Assume all throws are aimed at the center of the bullseye.) 1. What is the range of the random variable e? 2. For a skilled player, what is the mean of the normal random variable X? 3. Should the normal random variable X have a high or low o for the most skilled players? 4. Find the probability of the dart landing in each region for a player with o = 50 mm and for a player with 0 = 100 mm given the following specifications for the dartboard. Each region corresponds to the given ranges of (absolute) distances from the center of the inner bullseye. Region Distance is more than... Distance is less than... Inner Bullseye 0 mm 6.35 mm Outer Bullseye 6.35 mm 15.9 mm Inner Single Region 15.9 mm 95.3 mm Triple Ring 95.3 mm 104.8 mm Outer Single Region 104.8 mm 158.8 mm Double Ring 158.8 mm 168.3 mm Beyond 168.3 mm 5. What is the probability of landing in a single scoring region (shown in yellow in Figure 3) for each player? 5 20 1 18 6. What is the probability of landing in the wedge of the 9 dartboard marked with 20 if o = 50 mm (i.e., in the yellow 14 region or beyond)? (See Figure 4.) 13 11 6 7. What is the probability of scoring 7 points on one dart throw if o = 50 mm? 8 10 8. What is the probability of scoring 8 points on one dart 16 15 throw if o = 50 mm? 2 9. What is the probability of scoring 6 points on one dart throw if o = 50 mm? Figure 3 10. What are the highest and lowest possible scores from a single dart throw? What is the probability of each if 12 51 o = 50 mm? 9 12 4 1 19 3 17 18 4 12