On Any Particular Day an Individual Can Take One of Two

Essay

On any particular day an individual can take one of two routes to work. Route A has a 25% chance of being congested, whereas route B has a 40% chance of being congested.
The probability of the individual taking a particular route depends on his previous day's experience. If one day he takes route A and it is not congested, he will take route A again the next day with probability .8. If it is congested, he will take route B the next day with probability .7.
On the other hand, if on a day he takes route B and it is not congested, he will take route B again the next day with probability .9. Similarly if route B is congested, he will take route A the next day with probability .6.
a.Construct the transition matrix for this problem. (HINT: There are 4 states corresponding to the route taken and the congestion. The transition probabilities are products of the independent probabilities of congestion and next day choice.)
b.What is the long-run proportion of time that route A is taken?