Astudent who is trying to write a paper for a course has a choice of two topics, a and b. if topic a is chosen, the student will order two books through interlibrary loan, whereas if topic b is chosen, the student will order four books. the student believes that a good paper necessitates receiving and using at least half the books ordered for either topic chosen. if the probability that a book ordered through interlibrary loan actually arrives in time is 0.9 and books arrive independently of one another, which topic should the student choose to maximize the probability of writing a good paper? (enter your answers to four decimal places.)

Say she chooses topic A. She needs at least one book to arrive on time. The probability that no books arrive on time is 0.1 x 0.1 = 0.01 so the probability that at least one book arrives on time is 1 - 0.01 = 0.99.
Say she chooses topic B. She needs at least two books to arrive on time. The probability of no books arriving on time is 0.1 x 0.1 x 0.1 x 0.1 = 0.0001. The probability of exactly one book arriving on time is 4 x 0.1 x 0.1 x 0.1 x 0.9 = 0.0036 (binomial probability distribution formula - the 4 comes from 4C1). So the probability of her not getting enough books is 0.0001 + 0.0036 = 0.0037 and the probability she does get the books she needs on time is 1 - 0.0037 = 0.9963.
So she should choose topic B.