A Markov process has three states that describe an environment for a shuttle service: state 1...

A Markov process has three states that describe an environment for a shuttle service: state 1 = driving in the city, state 2 = driving in the suburbs, state 3 = driving in the rural area. If the shuttle driver is driving in the city, there is a 50% chance the next passenger pickup assignment will also be in the city. If the shuttle driver is driving in the suburbs, there is a 40% chance the next passenger pickup assignment will be in the suburbs. If the shuttle driver is driving in the rural area, there is a 20% chance the next passenger pickup assignment will also be in the country. Anytime there is a switch from the driving environment, it is equally likely to be either of the two environments. If the shuttle driver starts in the city, what is the probability that the tenth passenger pickup will be in the rural area?

Solutions

Expert Solution

The states of the Markov process are state 1 = driving in the city, state 2 = driving in the suburbs, state 3 = driving in the rural area.

The transition probability from state 1 to state 1 is 0.5. The transition probability for other environment is equally likely. Thus, transition probability from state 1 to state 2 is 0.25 and transition probability from state 1 to state 3 is 0.25.

The transition probability from state 2 to state 2 is 0.4. The transition probability for other environment is equally likely. Thus, transition probability from state 2 to state 1 is 0.3 and transition probability from state 2 to state 3 is 0.3.

The transition probability from state 3 to state 3 is 0.2. The transition probability for other environment is equally likely. Thus, transition probability from state 3 to state 1 is 0.4 and transition probability from state 3 to state 2 is 0.4.

Thus, the transition probability matrix is,

To calculate the probability that the tenth passenger pickup will be in the rural area, we need to calculate the matrix P¹⁰.

If the shuttle driver starts in the city, what is the probability that the tenth passenger pickup will be in the rural area is the row 1 and column 3 in the matrix P¹⁰ .

Thus, the probability that the tenth passenger pickup will be in the rural area is 0.2542373.

orchestra answered 1 year ago

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