Transition Probability Matrices

Two products A and B currently share the market with shares 50% and 50% each respectively. Each week some brand switching takes place. Of those who bought A the previous week, 60% buy it again whereas 40% switch over to B. Of those who bought B the previous week, 80% buy it again where as 20% switch over to A. Find their shares after one week and after two weeks. If the price war continues, when is the equilibrium reached?

The subscription department of a magazine sends out a letter to a large mailing list inviting subscriptions for the magazine. Some of the people receiving this letter already subscribe to the magazine while others do not. From this mailing list, 45% of those who already subscribe will subscribe again while 30% of those who do not now subscribe will subscribe. On the last letter, it was found that 40% of those receiving it ordered a subscription. What percent of those receiving the current letter can be expected to order a subscription?

A new transit system has just gone into operation in Chennai. Of those who use the transit system this year, 30% will switch over to using metro train next year and 70% will continue to use the transit system. Of those who use metro train this year, 70% will continue to use metro train next year and 30% will switch over to the transit system. Suppose the population of Chennai city remains constant and that 60% of the commuters use the transit system and 40% of the commuters use metro train next year.

What percent of commuters will be using the transit system year after the next year?

A new transit system has just gone into operation in Chennai. Of those who use the transit system this year, 30% will switch over to using metro train next year and 70% will continue to use the transit system. Of those who use metro train this year, 70% will continue to use metro train next year and 30% will switch over to the transit system. Suppose the population of Chennai city remains constant and that 60% of the commuters use the transit system and 40% of the commuters use metro train next year.

What percent of commuters will be using the transit system in the long run?

Two types of soaps A and B are in the market. Their present market shares are 15% for A and 85% for B. Of those who bought A the previous year, 65% continue to buy it again while 35% switch over to B. Of those who bought B the previous year, 55% buy it again and 45% switch over to A. Find their market shares after one year and when is the equilibrium reached?

Choose the correct alternative:

If \[{\begin{matrix} & \begin{matrix}A&&B\end{matrix} \\ T = \begin{matrix}A\\B\end{matrix} & \begin{pmatrix}0.4&0.6\\0.2&0.8\end{pmatrix}\\ \end{matrix}}\] is a transition probability matrix, then at equilibriuium A is equal to

Choose the correct alternative:

If \[{\begin{matrix} & \begin{matrix}A&&B\end{matrix} \\ T = \begin{matrix}A\\B\end{matrix} & \begin{pmatrix}0.7&0.3\\0.6&x\end{pmatrix}\\ \end{matrix}}\] is a transition probability matrix, then the value of x is

Choose the correct alternative:

In a transition probability matrix, all the entries are greater than or equal to