Answer to Question #144078 in Statistics and Probability for Fridah

Question

n a study of the domestic market share of three major automobile manufacturers A, B and C in a certain country, it was found out that of the customers who bought a car manufactured by A, 75% would again buy a car manufactured by A, 15% would buy a car manufactured by B and the rest would buy a car manufactured by C. Of the customers who bought a car manufactured by B, 90% would again buy a car manufactured by B, 5% would buy a car manufactured by A and the rest would buy a car manufactured by C. Of the customers who bought a car manufactured by C, 85% would again buy a car manufactured by C, 5% would buy a car manufactured by A and the rest would buy a car manufactured by B.
Required
The long run market share of the manufacturers (14 Marks)

Answer 1

$Solution$

The above is an example of $3\times3$ Markov chain.

To calculate the long run market share of the manufacturers, we need to create first $3\times3$ transition matrix for this problem. The columns denote the conditional probability of buying a car manufactured by three companies and rows represents the respective manufacturer of the car

Hence the matrix $T$ will be equal to

We can define $T^{(n)}$ as an n step transition matrix, now we can find $T^{(2)}$

Now let us assume the current share in the market for different manufacturers as $s_0 = [0.56,\ 0.24,\ 0.20]$ , then the companies share after two years

$s_2 = s_0 * T_2\\ \implies[0.3612,\ 0.37885,\ 0.25995]$

Hence percentage share will be $A = 36.12\%,\ B = 37.885\%$ and $C = 25.995\%$