In a certain city, the Democratic, Republican, and Consumer parties always nominate candidates for mayor. The probability of winning in any election depends on the party in power and is given by the following transition matrix.
Using the given transition matrix and assuming the initial-probability vector is , find the probability vector for the fourth stage of the Markov chain. (This initial-probability vector indicates that a Consumer is certain to win the initial election.)

Question

In a certain city, the Democratic, Republican, and Consumer parties always nominate candidates for mayor. The probability of winning in any election depends on the party in power and is given by the following transition matrix. ​  ​
Using the given transition matrix and assuming the initial-probability vector is  , find the probability vector for the fourth stage of the Markov chain. (This initial-probability vector indicates that a Consumer is certain to win the initial election.)
​

Quizplus · Accepted Answer

To find the probability vector for the fourth stage of the Markov chain, we need to multiply the initial-probability vector by the transition matrix four times.

We have:

P0 = [1 0 0]

P1 = P0 x transition matrix = [0.4 0.45 0.15]

P2 = P1 x transition matrix = [0.4425 0.396 0.1615]

P3 = P2 x transition matrix = [0.43804 0.3978 0.16416]

P4 = P3 x transition matrix = [0.431516 0.403452 0.165032]

Therefore, the probability vector for the fourth stage of the Markov chain is [0.431516 0.403452 0.165032].

The correct answer is C.

In a Certain City, the Democratic, Republican, and Consumer Parties