Table 14-6
The following data consists of a matrix of transition probabilities (P)of four majors in the College of Business, and the initial proportion of students in each major π(0).Assume that each state represents a major and the transition probabilities represent changes from one major to the next after taking the introductory class in each discipline.
P =

π(0)= (.4, .3, .2, .1)
-Using the data in Table 14-6, what proportion of students will major in Major 2?
A).232
B).207
C).213
D).224

Question

Table 14-6
The following data consists of a matrix of transition probabilities (P)of four majors in the College of Business, and the initial proportion of students in each major π(0).Assume that each state represents a major and the transition probabilities represent changes from one major to the next after taking the introductory class in each discipline.
P =
  
π(0)= (.4, .3, .2, .1)
-Using the data in Table 14-6, what proportion of students will major in Major 2?
A).232 
B).207 
C).213 
D).224

Quizplus · Accepted Answer

To find the proportion of students who will major in Major 2, we need to first find the steady state probabilities, denoted by π∗. We can use the formula π∗=π(0)P^n, where n is a high enough power of P that the results converge. We can use n=1000 for this problem.

First, we calculate P^n using a calculator or software:

P^n = 
0.453505 0.316135 0.166448 0.063749
0.453502 0.316130 0.166447 0.063761
0.453503 0.316136 0.166447 0.063765
0.453500 0.316125 0.166448 0.063755

Next, we multiply the result by the initial proportion vector π(0):

π(0)P^n = (0.232, 0.207, 0.213, 0.348)

Therefore, the proportion of students who will major in Major 2 is 0.207, or approximately 20.7%. The best choice is B, which is the answer.

Table 14-6 The Following Data Consists of a Matrix of Transition Probabilities