At one liberal arts college, students are classified as humanities majors, science majors, or undecided. The chances are 40% that a humanities major will change to a science major from one year to the...

Extracted text: At one liberal arts college, students are classified as humanities majors, science majors, or undecided. The chances are 40% that a humanities major will change to a science major from one year to the next, and 35% that a humanitles major will change to undecided. A science major will change to humanities with probability 0.35, and to undecided with probabillty 0.25. An undecided will switch to humanities or science with probabilities of 0.40 and 0.20, respectively. Complete parts (a) and (b) below. (a) Find the long-range prediction for the fraction of students in each of these three majors. First, find the transition matrix P. Let the first state be that a student is a humanities major, the second that a student is a science major, and the third state that a student is undecided. (Type an integer or decimal for each matrix element.) Find the long-range prediction for the fraction of students in each of these three majors. The long-range prediction is humanities, science, and undecided. (Type integers simplified fractions.) (b) For a regular 2-by-2 matrix having column sums of 1, the equilibrium vector is Make a conjecture, and describe how this conjecture, if true, would allow you to predict the answer to part (a) with very little computation. A regular 3-by-3 matrix having column sums of 1 has an equilibrium vector of (Type an integer or simplified fraction for each matrix element.) How can this conjecture be used to predict the answer to part (a)? The column sums of the transition matrix in part (a) equal to 1. The conjecture V be used to find the long-range predictions for the fraction of students in each major. Enter your answer in each of the answer boxes.