On Any Particular Day an Individual Can Take One of Two

question 34

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?

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Definitions:

Union

In mathematics, particularly in set theory, it is the set containing all elements from all sets being considered.

Events A And B

Two occurrences within a probability space that are considered for their individual likelihood as well as their combined or conditional probabilities.

Sample Space

Sample space is the set of all possible outcomes or results of a probabilistic experiment.

Outcomes

The possible results that may come from a given action, experiment, or situation.