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 0.8. If it is congested, he will take route B the next day with probability 0.7.
On the other hand, if he takes route B one day and it is not congested, he will take route B again the next day with probability 0.9. Similarly, if route B is congested, he will take route A the next day with probability 0.6.
a. Construct the transition matrix for this problem. (Hint: There are four 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?
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