Seventy students will take MATH/STAT 414 midterm 2 (independently due to random seats),and the amount of time (in minutes) needed by any student to finish the exam is Exponentially distributed, i.e.,T ii.i.d.Expo()for i= 1,2, . . . ,70, where Ti is the time needed for the i-th student to finish the exam. Tak believes that the average time needed for each student is 60 minutes. The exam starts at 9:05am.

(a) What is the probability that at least one of the students will finish the exam before10:05am?

Hint: How would you express the event that "no students will finish the exam before 10:05am"usingXi's? After answering this, use the independence of events.

(b)Surprisingly, Tak observed that one student, whose name will remain hidden for his or her own safety, finished the exam first at 9:35am (in 30 minutes!). Given this information (i.e.,after 30 minutes), what is the probability that at least one other student (among the remaining students) will finish the exam before 10:05am?

Hint: Memoryless property.