A mathematics assignment problem in which each element from one set is matched with one and only one element from another set is often referred to as a "marriage" problem. Mathematical Social Sciences (May 2014) considered the following marriage problem. After passing their preliminary exams, doctoral students in a Ph.D. program are each assigned to a professor for research assistance. Each professor can accept only one additional student per year. Suppose there are three students, a, b, and c, who have passed their preliminary exams and three professors, A, B, and C. Also suppose that professor A prefers student a, professor B prefers student b, and professor C prefers student c.
a. If the assignment is made at random, what are the chances that the preferred matches occur?
b. If the assignment is made at random, what are the chances that professor A is matched with student a?
c. If the assignment is made at random, what are the chances that none of the professors are matched with their preferred student?