Consider a senior Statistics concentrator with a packedextracurricular schedule, taking five classes, and writing athesis. Each time she takes an exam, she either scores very well (aleast two standard deviations above the mean) or does not. Herperformance on any given exam depends on whether she is operatingon a reasonable amount of sleep the night before (more than 7hours), relatively little sleep(between 4-7 hours, inclusive), orpractically no sleep (less than 4 hours). When she has hadpractically no sleep, she scores very well about 30% of the time.When she has had relatively little sleep, she scores very well 40%of the time. When she has had a reasonable amount of sleep, shescores very well 42% of the time. Over the course of a semester,she has a reasonable amount of sleep 50% of nights and practicallyno sleep 30% of nights. What is her overall probability of scoringvery well on an exam? What is the probability she had practicallyno sleep the night before an exam where she scored very well?Suppose that one day she has three exams scheduled. What is theprobability that she scores very well on exactly two of the exams,under the assumption that her performance on each exam isindependent of her performance on another exam? What is theprobability that she had practically no sleep the night prior to aday when she scored very well on exactly two out of threescams?
