5 In this question, give probabilities correct to 4 decimal places.
Alpha and Delta are two companies which compete for the ownership of insurance bonds. Boyles and Cayleys are companies which trade in these bonds. When a new bond becomes available, it is first acquired by either Boyles or Cayleys. After a certain amount of trading it is eventually owned by either Alpha or Delta. Change of ownership always takes place overnight, so that on any particular day the bond is owned by one of the four companies. The trading process is modelled as a Markov chain with four states, as follows.
On the first day, the bond is owned by Boyles or Cayleys, with probabilities \(0.4,0.6\) respectively.
If the bond is owned by Boyles, then on the next day it could be owned by Alpha, Boyles or Cayleys, with probabilities \(0.07,0.8,0.13\) respectively.
If the bond is owned by Cayleys, then on the next day it could be owned by Boyles, Cayleys or Delta, with probabilities \(0.15,0.75,0.1\) respectively.
If the bond is owned by Alpha or Delta, then no further trading takes place, so on the next day it is owned by the same company.
- Write down the transition matrix \(\mathbf { P }\).
- Explain what is meant by an absorbing state of a Markov chain. Identify any absorbing states in this situation.
- Find the probability that the bond is owned by Boyles on the 10th day.
- Find the probability that on the 14th day the bond is owned by the same company as on the 10th day.
- Find the day on which the probability that the bond is owned by Alpha or Delta exceeds 0.9 for the first time.
- Find the limit of \(\mathbf { P } ^ { n }\) as \(n\) tends to infinity.
- Find the probability that the bond is eventually owned by Alpha.
The probabilities that Boyles and Cayleys own the bond on the first day are changed (but all the transition probabilities remain the same as before). The bond is now equally likely to be owned by Alpha or Delta at the end of the trading process.
- Find the new probabilities for the ownership of the bond on the first day.