POINTS For full credit, you must show ALL your work and EXPLAIN ALL your answers! Point values given are those I will use IF I choose to grade that problem. 1. Although Astonishingly Above Average Academically, Always Acing Any Assign- ment, As An Athlete Ariel's Accomplishments Are Actually Astounding, An Artful Agility Arousing Awe And Amazement Among Attendees At Any Athletic Arena. Despite her special skills, school rules stipulate that poor Ariel may play only one varsity sport per season, so each year she must choose which sports she will play. Her favorite president, Teddy Roosevelt, was famous for his motto, "Speak softly and carry a Big Stick," so Ariel has a tender spot in her heart for Field Hockey, Ice Hockey, and Softball. As she is uncertain, however, whether she likes this motto because she likes Teddy, or whether she likes Teddy because she likes carrying a Big Stick, she decides at least to consider other sports where she is less well-armed. This year Ariel considers the sports tabled below (letters are suggestions to use in your Tree). In the Fall, Ariel is 3 times as likely to choose Field Hockey as Football. In each successive season, if she DID carry a Big Stick in the previous season, then she is only twice as likely to choose a Big Stick in the next season, but if she did NOT carry a Big Stick, then she is 4 times as likely to choose a Big Stick in the next season. FALL WINTER SPRING F: Football G: Gymnastics R: Rugby H: Field Hockey H: Ice Hockey S: Softball a. Please Make a Tree Diagram Showing All of Ariel's Possible Sport Choices by Season, Including Probabilities/Conditional Probabilities of Each Choice. (15) b. What is the Probability That Ariel Carries a Big Stick: i. In Each Season? ii. In Exactly 2 Seasons? iii. In At Least 1 Season? iv. In the Spring Season? v. In Exactly Two Consecutive Seasons? c. Over her 4-year Varsity Career, What is the Probability That Ariel: i. Plays Rugby Exactly Twice? ii. Carries a Big Stick in All Three Seasons 3 Or More Years? (15) (10) (10) (10) (10) iii. Carries No Big Stick in All Three Seasons At Least One Year? d. Let A = {Ariel Plays Field Hockey} and B = {Ariel Carries A Big Stick In Win- ter AND Spring}: Are Events A and B Independent? PROVE your answer! (10)