7. (Q.67, 2.5 pts) Engineering courses often have assigned seating to facilitate the Socratic method. Suppose that there are 100 first-year engineering students, and each takes the same two courses: Calculus and Physics. Both are held in the same lecture hall (which has 100 seats), and the seating is uniformly random and independent for the two courses. Find a simple but accurate approximation to the probability that no one has the same seat for both courses. Answer: e - Analysis: Student j has the same seat in two courses sounds like a rare event, and we have 100 rare events. The total number of a large number of rare events to happen has Poisson distribution with parameter 1 = the expected value of the total number of rare events that occur.1) (0.5 pts) Define rare events Aj: student j has the same seat in the two courses. What is the probability of Aj? 2) (1 pt) Let r.v. X be the # of rare events that occur. X = I(A1) +... + I(A10o) Compute EX 3) (0.5 pts) What's the distribution of X? Provide the distribution name and the parameter 4) (0.5 pts) Compute the probability that no one has the same seat for both courses. 8. (Q.70, 2 pts) Ten million people enter a certain lottery. For each person, the chance of winning is one in ten million, independently. Find a simple, good approximation for the PMF of the number of people who win the lottery. Answer: X ~Pois(1) Analysis: a person wining a lottery is a rare event. The total number of people wining the lottery has a Poisson distribution with 1 = the expected value of the total number of rare events that occur. 1) (0.5 pts) Define rare events Aj: person j wins the lottery. What is the probability of Aj? 2) (1 pt) Let r.v. X be the # of rare events that occur. X = I(A1) +... + I(A10o)-Compute EH 3) {0.5 pts) What's the distribution of X? Provide the distribution name and the parameter