Question

At one liberal arts​ college, students are classified as humanities​ majors, science​ majors, or undecided. The chances are 36​% that a humanities major will change to a science major from one year to the​ next, and 17​% that a humanities major will change to undecided. A science major will change to humanities with probability 0.31​, and to undecided with probability 0.34. An undecided will switch to humanities or science with probabilities of 0.31 and 0.23​, respectively. Find the​ long-range prediction for the fraction of students in each of these three majors. The​ long-range prediction is humanities, ​science, undecided.

Answer 1

We are given the probabilities here as:
P( H --> S) = 0.36,
P( H --> U) = 0.17,
Therefore P(H --> H) = 1 - 0.36 - 0.17 = 0.47

Also,
P(S --> H) = 0.31,
P( S --> U) = 0.34,
Therefore, P(S --> S) = 1 - 0.31 - 0.34 = 0.35

P(U --> H) = 0.31,
P(U --> S) = 0.23
P(U --> U) = 1 - 0.31 - 0.23 = 0.46

Using the above probabilities, we obtain the transition probability matrix here as:

Let the long range prediction probabilities be X, Y and Z respectively here.

From First column, we have:
X = 0.47X + 0.31Y + 0.31Z
0.53X = 0.31Y + 0.31Z
Y + Z = (0.53/0.31)X
But Y + Z = 1 - X
Therefore,
1 - X = (0.53/0.31)X
X = 0.3690

Y + Z = (0.53/0.31)X = 0.6310

From second column, we have:
Y = 0.36X+ 0.35Y + 0.23Z
0.65Y = 0.36*0.3690 + 0.23Z
0.65Y = 0.1328 + 0.23Z
0.65Y = 0.1328 + 0.23*(0.6310 - Y)
Y = 0.27797 / (0.65 + 0.23) = 0.3159

Z = 0.6310 - Y = 0.6310 - 0.3159 = 0.3151

therefore the long range steady state probabilities here are given as:
( 0.3690, 0.3159, 0.3151)