5. A company is considering using Markov theory to analyse consumers switching between three different brands of hand cream. An analysis of consumer behaviour data has produced the transition matrix shown below for the probability of switching every six months between brands. The current market shares for the three brands are 64%, 27% and 9% for brands A, B and C respectively. From brand A B C Long-run (%, 2dp) To brand A B 0.81 0.15 0.01 0.56 0.47 0.51 Here, for example, there is a probability of 0.51 that a customer switches from brand C to brand B. (a) Copy the following table into your answer book and fill in the long-run prediction for the market shares for these brands. (15 marks) C 0.04 0.43 0.02 Brand A Brand B Brand C After six months (%, 2dp) Long-run (%, 2dp) kb) Due to their low market share brand C has decided to exit the market immediately. Consumers who are currently purchasing brand C will now choose either to immediately purchase brand A or to immediately purchase brand B (with equal probability). However, after this immediate purchase they will behave as normal consumers of brands A and B. Explaining clearly any assumptions you make copy the following table into your answer book and fill in the predicted market shares both after six months, and in the long-run, for brands A and B. Brand A Brand B (10 marks)