Question

The registrar of a law school has compiled the following statistics on the progress of the school's students working toward the LLB degree: Of the first-year students in a particular year, 80% successfully complete their course of studies and move on to the second year, whereas 20% drop out of the program; of the second-year students in a particular year, 92% go on to the third year, whereas 8% drop out of the program; of the third-year students in a particular year, 98% go on to graduate at the end of the year, whereas 2% drop out of the program.

(a) Construct the transition matrix associated with the Markov process. (Label your matrix using this order: Drop out, Graduate, First-Year, Second-Year, Third-Year)

(b) Find the steady-state matrix. (Round your answers to four decimal places.)

(c) Determine the probability that a beginning law student enrolled in the program will go on to graduate. (Round your answer to four decimal places.)

Answer 1

a) We are given 5 transition states here: Drop out, Graduate, First-Year, Second-Year, Third-Year.

Regarding the transition probabilities we are given here that:
P( first year --> Second year) = 0.8,
P( first year --> drop out) = 0.2,
P( second year --> third year) = 0.92,
P( second year --> drop out) = 0.08,
P( third year --> graduate) = 0.98,
P( third year --> drop out) = 0.02,

Therefore the probability transition matrix here is given as:

b) Let the steady state probabilities for the 5 states be D, G, F, S and T respectively.

From third column, we have:
F = 0
From second column, we have:
G = G + 0.98T
Therefore T = 0
From fourth column, we have:
S = 0.8F = 0
From last column, we have:
T = 0.92S = 0
From first column, we have:
D = D + 0.2F + 0.08S + 0.02T

Also, we know here that:
D + G = 1

Now let everybody be first yearite initially,
Then P(Drop out) = 0.2 + 0.8*0.08 + 0.8*0.92*0.02 = 0.27872

Therefore 0.27872 is the required probability of drop out here, while 1 - 0.27872 = 0.72128 is the required probability of graduate here.

Therefore the matrix here is given as:

c) This is already computed in the above part as: 0.72128

Therefore 0.72128 is the required probability here.