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 .9 and books arrive independently of one another, which topic should the student choose to maximize the probability of writing a good paper? What if the arrival probability is only .5 instead of .9?

Solution : Step 1 : It is given that the student have two choices to select a topic A or 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. Student believe that in order to make a good note at least half of the books ordered for either topic should receive on correct time.If the probability that a book ordered through interlibrary loan actually arrives in time is .9. We have to find that which topic should the student choose to maximize the probability of writing a good paper. How this will change if the arrival probability is 0.5 Step 2 : 1) Lets check the each cases. Case 1 If the student chose choice A the he/she need one of the ordered book on time to write a good paper. Here probability that a book arrive on the correct time is = 0.90 And probability that a book not arrive in the correct time is = 0.10 In this case probability that no book arrive on correct time is= 0.10*0.10 = .01 So Probability that at least one of the book on the correct time is= 1-.01 =.99 Case 2 If the student select the 2nd choice B Then he/she needs at least 2 books at a time to write a paper So in this case probability that no book arrive on correct time is =0.1*.1*.1*.1 =.0001 Probability that at least one of the book on the correct time is = 4*0.1*0.1*.0.1*0.9(since the events are independent) = .0036 So it is clear that choosing topic A is the correct option to write a good paper on correct time.