5 Each day that Adam is at work he carries out one of three tasks A, B or C. Each task takes a whole day. Adam chooses the task to carry out on each day according to the following set of three rules.

1. If, on any given day, he has worked on task A then the next day he will choose task A with probability 0.75 , and tasks B and C with equal probability.
2. If, on any given day, he has worked on task B then the next day he will choose task B or task C with equal probability but will never choose task A .
3. If, on any given day, he has worked on task C then the next day he will choose task A with probability \(p\) and tasks B and C with equal probability.
   1. Write down the transition matrix.
   2. Over a long period Adam carries out the tasks \(\mathrm { A } , \mathrm { B }\) and C with equal frequency. Find the value of \(p\).
   3. On day 1 Adam chooses task A . Find the probability that he also chooses task A on day 5 .
   Adam decides to change rule 3 as follows.
   If, on any given day, he has worked on task C then the next day he will choose tasks \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) with probabilities \(0.4,0.3,0.3\) respectively.
4. On day 1 Adam chooses task A. Find the probability that he chooses the same task on day 7 as he did on day 4 .
5. On a particular day, Adam chooses task A. Find the expected number of consecutive further days on which he will choose A. Adam changes all three rules again as follows.
   • If he works on A one day then on the next day he chooses C .
   • If he works on B one day then on the next day he chooses A or C each with probability 0.5.
   • If he works on C one day then on the next day he chooses A or B each with probability 0.5 .
   • Find the long term probabilities for each task.