Question

In a study of the domestic market share of the three major automobile manufacturers A, B, and C in a certain country, it was found 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 10% would buy and car manufactured by C. Of the customers who bought a car manufactured by B, 90% would again buy a car manufactured by B, whereas 5% each would buy cars manufactured by A and C. Finally, 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 10% would buy a car manufactured by B. Assuming that these sentiments reflect the buying habits of customers in the future model years, determine the market share that will be held by each manufacturer in the long run.

Answer 1

According to given conditions

if buy A then again A-> 0.7500 B-> 0.1500 C---> 0.1000

if buy B then again A-> 0.0500 B---> 0.9000 C-->0.0500

if buy C then again   A--> 0.0500 B-> 0.1000 C--> 0.8500

Writing in matrix form

P = 0.7500 0.1500 0.1000
0.0500 0.9000 0.0500
0.0500 0.1000 0.8500

now P*P =  0.5750 0.2575 0.1675
0.0850 0.8225 0.0925
0.0850 0.1825 0.7325

using calculator in long run you can multiply A*A*A again and again until you get constant matrix

Using calculator  

A^100 =  0.1667 0.5417 0.2917
0.1667 0.5417 0.2917
0.1667 0.5417 0.2917

that is A share is 16.67 % , B share is 54.17% and C share is 29.17%