Probobility ffi Two Porqdoxes First Paradox: Under certa'in c'ircumstances, Aou haue your best chance of wi,nni,ng a tenn'is tournament i,f you play most of gour gam,es aga'inst the best possr,ble opponent, Alice and her two sisters, Betty and Carol, are avid tennis players. Bettv is the best of the three sisters, and Carol plays at the same level as Alice. Alice defeats Carol 50% of the time but only defeats Betty 40% of the time. Alice's mother offers to give her $100 if she can win two consecutive games when playing three alternating games against her two sisters. Since the games will alternate, Alice has two possibilities for the sequence of opponents. One possibility is to play the first game against Betty, followed by a game with Carol, and then another game with Betty, We will refer to this sequence as BCB, The other possible sequence is CBC. l-. Make a guess of the best sequence for Alice to choose-the 2. one having the majority of the games against the weaker oppouent or the one having the majority of the games against the stronger opponent. Calculate the probability of Alice getting the $100 reward if she chooses the sequence CBC. 3. Calculate 4. 5. the probability of Alice getting the $100 reward if she chooses the sequence BCB. Which sequence should Alice choose? How would you explain to someone who didn't know probability why the sequence that you chose is best? Second Paradox: The probabi,li,ty of a male appli,cant bei,ng ad,mi,tted to a grad,uate school can be hi,gher than the probabi,li,ty for a female appl,icønt, euen though for each d,epartment the probabi,li,ty of a female bei,ng ad,mi.tted i,s higher. (This apparent contradiction is known as Simpsonts paradox.) To simplify matters, consider a university with two professional graduate programs, medicine and law. Suppose that last year 1000 men and 1000 women applied, and the outcome was as shown in Table 1. TABTE 1 Men 6. Women Law 700 560 r40 400 340 60 Medicine 300 40 260 600 160 440 \Mhat is the probability that a male applicant was accepted to a professional program? female? 7. Wirich gender does the university appear to be favoring? 8. What is the probability that a male applicant was accepted to law school? female? 9. What is the plobability that a male applicant was accepted to medical school? female? L0. Which gender do the individual professional schools appear to be favoring? l-1. Without using probability, justify the apparent contradiction between the answers for part 7 and part 10. 233