In a large metropolitan area, 10% of the commuters currently use the public tran

Question

In a large metropolitan area, 10% of the commuters currently use the public transportation system, whereas the remaining 90% commute via automobile. The city has recently revitalized and expanded its public transportation system. It is expected that 6 months from now 10% of those who are now commuting to work via automobile will switch to public transportation, and 90% 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. P (b) Find the initial distribution vector for this Markov chain. (c) What percentage of the commuters are expected to use public transportation 6 months from now? (Round your answer to the nearest percent.)

Explanation / Answer

Out of every100 people, 20 now use public transport and 80 use cars.....

And of the 80 that use cars, 30% of these will switch to public transportation....thus.....the public transportation system will increase by 80 * .30 = 24 people out of every 100

However, it will also lose 10% of its current ridership to cars = 20 X .10   = 2 people out of every 100

So, the percentage of people using public transportation in the long run will be

[20 + 24 - 2] / 100 = 42%

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