3 Jury Selection In many jurisdictions in the United States, to select a jury of twelve from a pool of potential jurors, the defense and prosecution will alternately strike (or excuse) one person from the pool until only twelve remain. The U.S. supreme court has ruled in Batson v. Kentucky and McCollum v. Georgia that prosecutors may not strike jurors based on race or gender. However, prosecutors are not required to explain why they chose to strike a particular person, so one must reason based on probabilities to determine whether a given sequence of strikes reflects bias. Suppose that twelve jurors are to be selected from a pool consisting of eight men and eight women by the procedure outlined above. Assume that the prose- cution and defense both strike jurors uniformly at random with no bias. Assume that the defense makes the first strike, the prosecution the second, and so forth. Let i if the prosecution's i'th strike excuses a man, P if the prosecution's i'th strike excuses a woman. Let i if the defense's i'th strike excuses a man, D 10 if the defense's i'th strike excuses a woman. { {: 2 Problem 3. (a) (2 points) Calculate P[P1 = 0, P2 = 0, D = 1, D2 = 1). Hint: You probably don't want to do this by counting simple events as in the last jury selection problem. You'll have an easier time if you use theorems like the multiplicative law (chain rule) instead. (b) (2 points) Calculate P[P1 = 0, P = 0, D = 0, D2 = 0). (c) (2 points) Calculate P[P1 = 0, P2 = 0). Suppose that twelve jurors are to be selected from a pool consisting of eight men and eight women by the procedure outlined above. Assume that the prose- cution and defense both strike jurors uniformly at random with no bias. Assume that the defense makes the first strike, the prosecution the second, and so forth. Let 1 if the prosecution's i'th strike excuses a man, P = O if the prosecution's i'th strike excuses a woman. Let f if the defense's i'th strike excuses a man, D lo if the defenses ith strike excuses a woman. { 2 Problem 3. (a) (2 points) Calculate P[P1 = 0, P2 = 0, D = 1, D2 = 1). Hint: You probably don't want to do this by counting simple events as in the last jury selection problem. You'll have an easier time if you use theorems like the multiplicative law (chain rule) instead. (b) (2 points) Calculate P[P1 = 0, P, = 0, D = 0, D2 = 0). (c) (2 points) Calculate P[P1 = 0, P = 0).