Question

College instructors who adopt this book are (we hope!) twice as likely to continue to use the book the following year as they are to drop it, whereas nonusers are four times as likely to remain nonusers the following year as they are to adopt this book.

(a) Determine the probability that a nonuser will be a user in two years.


(b) In the long term, what proportion of college instructors will be users of this book?

Answer 1

(a)

Let the problem be modeled as Markov chanin with states Users (U) and NonUsers (N)

The transition probability from state U to state U is 2/3 (twice as likely to continue to use the book )

The transition probability from state U to state N is 1 - 2/3 = 1/3

The transition probability from state N to state N is 4/5 (four times as likely to remain nonusers )

The transition probability from state U to state N is 1 - 4/5 = 1/5

The transition probability matrix is,

Probability that a nonuser will be a user in two years = P(X2 = U | X0 = N)

= P(X2 = U, X1 = U, X0 = N) + P(X2 = U, X1 = N, X0 = N)

= (1/5) * (2/3) + (4/5) * (1/5)

= 22/75 = 0.2933333

(b)

Let be the long term proportion of distribution of users and non-users.

Then a + b = 1

and

which gives,

(2/3)a + (1/5)b = a

(1/5)b = (1/3)a

b = (5/3)a

a + b = 1

=> a + (5/3)a = 1

=> (8/3)a = 1

=> a = 3/8

In the long term, the proportion of college instructors will be users of this book = 3/8 = 0.375