In a large metropolitan area,10%of the commuters currently use the public transportation system, whereas the remaining90%commute via automobile. The city has recently revitalized and expanded its public transportation system. It is expected that 6 months from now40%of those who are now commuting to work via automobile will switch to public transportation, and60%will continue to commute via automobile. At the same time, it is expected that 20% of those now using public transportation will commute via automobile and 80% will continue to use public transportation.

(a) Construct the transition matrix for the Markov chain that describes the change in the mode of transportation used by these commuters.

PA

T= AT=PA

(b) Find the initial distribution vector for this Markov chain.

X₀₌ ?

?

(c) What percentage of the commuters are expected to use public transportation 6 months from now? (Round your answer to the nearest percent.) %