On Any Particular Day,an Individual Can Take One of Two

question 21

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 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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Definitions:

Metabolic Tolerance

The body's adjustment to a substance through the increase of metabolic processes, reducing the substance's effectiveness over time.

Cellular Tolerance

A cellular adaptation in which the response to a substance decreases after repeated exposure, often seen with drugs or toxins.

Hippocampus

A region of the brain crucial for the formation of new memories and spatial navigation.

Amygdala

An almond-shaped section of nervous tissue located in the temporal (side) lobe of the brain; involved in emotions such as fear and pleasure.