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Three companies Benfel Ltd, Cromwell Ltd and Samtec Ltd were recently launched in the Kenyan market. The companies have operated in the Kenyan market for a few months now. At the beginning of the month of August 2019, a survey on monthly brand switching behavior was conducted on a representative sample among customers of the three companies. The survey revealed that Benfel Ltd retains 70% of its customers and loses 20% to Cromwell Ltd and 10% to Samtec Ltd. Similarly, Cromwell Ltd retains 60% of its customers and loses 30% to Benfel Ltd and 10% to Samtec Ltd. Furthermore, Samtec Ltd retains 50% while it loses 30% to Cromwell Ltd and 20% to Benfel Ltd. It has been determined that there will be 3,500, 4,300 and 2,200 customers shopping with Benfel Ltd, Cromwell Ltd and Samtec Ltd respectively at the start of the month of September 2019. Required:
(a) Formulate the transition matrix for this system
b) Evaluate the distribution of the sample surveyed as at the beginning month of August 2019
c) Compare the long run state of the market amongst the three companies
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