Question

A student 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.)

Answer 1

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.