The Chicago Bulls basketball team has hired you as a consultant to help them analyze their player statistics and gather sports analytics for them. The following statistics show their top 9 players regular season performance so far this year. The listed statistics track: field goal peercentage (FG%), 3-point field goals percentage (3FG%), free throw percentage (FT%), total assists (AST), total rebounds (RBD), total steals (STL), and total blocks (BLK). Player Height (in.) FG% 3FG% FT% AST RBD STL BLK Patrick Williams 79 48.3% 39.1% 72.8% 99 327 64 46 Zach LaVine 77 50.7% 41.9% 84.9% 282 289 46 27 Coby White 77 41.6% 35.9% 90.1% 328 284 38 15 Nikola Vucevic 83 47.1% 38.8% 87.0% 102 300 23 20 Lauri Markkanen 84 48.0% 40.2% 82.6% 45 268 26 15 Garrett Temple 77 41.5% 33.5% 80.0% 124 160 43 30 Thaddeus Young 80 55.9% 26.7% 62.8% 291 422 74 40 Thomas Satoransky 79 51.4% 35.6% 84.8% 271 142 40 14 Denzel Valentine 76 37.3% 33.1% 94.1% 105 197 30 7 Totals 1647 2389 384 214 Figure 1: Top 9 Chicago Bulls player stats 1. (35 pts) The Chicago Bulls want to see if they can use player statistics to help determine which position each player should have on the team. (a) Centers are tall players that stay under the basket and often get more rebounds and blocks than other players. Which players are in the set C = {players at least 78 inches tall, with at least 200 rebounds, and at least 20 blocks}. (b) Point guards and shooting guards stay on the outside of the court both offensively and defensively, which means they generally have more assists, steals, and a better 3-point field goal percentage than others. Which players are in the set G = {players with at least 35 steals, at least 200 assists, and at least a 35% 3-point field goal percentage}.(c) Forwards play a diverse roll on the team and do lots of different things. They are usually not the smallest player on the team and they don't often have the highest number of assists, which are both indicators of a guard player. Which players are in the set F = {players at least 77 inches tall, with at most 280 assists. }. (d) Are any of the sets C, G, and F mutually exclusive with another? If so which pairs of these sets are mutually exclusive? Answer in complete sentences. PM (e) Based on the sets you defined in parts (a-c), fill in the following Venn diagram (Figure 2) to represent how many players would be good for each position: Center, Guard, or Forward. Fill in all regions with the number of players in that region of the diagram. (Make sure all players are accounted for!) S Ga |Q Search F Figure 2: Empty Venn diagram for problem 1.e (f) Use the sets C, G, and F along with set operations to describe the set of players who can be either Centers or Forwards, but not Guards. What is the cardinality of this set? (g) In a complete sentence describe the set Gn (C U F)' in terms of what positions the players in this set play. Who are the players in this set? Problem Points Deliverables la Write "C = { }" containing the players in set C. 1b Write "G = { }" containing the players in set G. 1c Write "F = { }" containing the players in set F. 1d 4 Answer the questions in complete sentences. le 11 Fill in the Venn Diagram with player counts in each region. You may redraw the diagram in your project work if you prefer! If 4 Use set operations to describe the requested set. The cardinality can be specified with a complete sentence or mathematical notation. 1g 4 Describe the requested set in a complete sentence. Write "Gn (CUF) = { }" containing the players in the set. Table 1: Rubric for problem 12. (20 pts) Suppose the players are sorted into the following sets based on the position that they play (these sets may have different players than the ones you made in problem 1): C = {Center Players} = { Nikola Vucevic, Lauri Markkanen }, G = {Guard Players} = {Zack LaVine, Coby White, Thomas Satornasky, Denzel Valentine}, F = {Forward Players} = {Patrick Williams, Garrett Temple, Thaddeus Young}. (a) If the Bull's want a starting lineup with 1 center, 2 guards, and 2 forwards, how many unique 5 player lineups can they make? 2 (b) If they instead want a starting lineup with 1 center and 4 other players who are either guards or forwards, how many unique 5 player lineups can they make? (c) If they don't care about how many players they have of each position, how many unique 5 player lineups can they make from all 9 players? (d) If the Bull's want to give 4 players a pay raise, each of a different amount, how many ways can they assign these raises to their players? Problem Points Deliverables 2a 5 Answer in a complete sentence and show your work! 2b Answer in a complete sentence and show your work! 2c OT OT CT Answer in a complete sentence and show your work! 2d Answer in a complete sentence and show your work!3. (35 pts) The Bull's have an upcoming series where they will play 5 games in a row against the Golden State Warriors. Immediately after the series against the Warriors, the Bull's will play a 5 game series against the Atlanta Hawks. We are interested in how many games the Bulls win against each team. We write the possible outcomes as an ordered pair (x,y) where x is the number of games won against the Warriors and y is the number of games won against the Hawks. The following table shows the possible outcomes in the sample space. Games won vs Hawks 1 2 3 4 5 1 (1,1) (1,2) (1,3) (1,4) (1,5) 2 (2,1) (2, 2) (2, 3) (2,4) (2, 5) Games won vs Warriors 3 (3, 1) (3, 2) (3, 3) (3,4) (3, 5) 4 (4, 1) (4, 2) (4,3) (4,4) (4, 5) 5 (5,1) (5,2) (5,3) (5,4) (5, 5) Figure 3: Outcomes in the sample space of problem 3. (a) What are the outcomes in the event A = {the Bulls win at most 4 total games}? Write the set A in set builder notation (A = {(x, y) | conditions on x and y that make (x, y) part of the set }). (b) What are the outcomes in the event B = { the Bulls win an even number of total games}? Write the set B in set builder notation (B = {(x, y) | conditions on x and y that make (x, y) part of the set }). (c) What are the outcomes in the event C = {the Bulls win an odd number of games against the Warriors and an even number of games against the Hawks }? Write the set C in set builder notation (C = {(x, y) | conditions on x and y that make (x, y) part of the set }). (d)(d) If every outcome in the sample space is equally likely, what is the probability of any one particular outcome occurring? Answer in a complete sentence. (e) Assuming that every outcome in the sample space is equally likely, what are P(C) and P(C')? Write the set C' in set builder notation (C' = {(x, y)| conditions on x and y that make (x, y) part of the set }). Co (f) Let D = {(x, y)| x + y > 6} and E = {(x, y)| x = 3}. Assuming that every outcome in the sample space is equally likely, find P(D| E). (g) Let F = {(x, y) | x + y is odd} and G = {(x, y)| y 2 3}. Assuming that every outcome in the sample space is equally likely, find P(FIG). Problem |Points |Deliverables 3a Write "A = { }" containing the outcomes in A. Write A in set-builder notation. 3b Write "B = { }" containing the outcomes in B. Write B in set-builder notation. 3c Write "C = { }" containing the outcomes in C. Write C in set-builder notation. 3d OT OT CT Answer in a complete sentence. 3e Find P(C) and P(C'). Write C' in set-builder notation. 3f Find P(D E). 3g 5 Find P(FIG). Table 3: Rubric for problem 3