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1. If the transition matrix for a certain Markov chain is P = and the initial-state matrix is S₀=, find S₁.

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   4.   ?    None of the answers given; click to see the solution.

2. If the transition matrix for a certain Markov chain is P = and the initial-state matrix is
   S₀= , find S₄.

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   4.   ?    None of the answers given; click to see the solution.

3. A Markov Chain has three states, 1, 2 and 3. The probability of going from state 1 to state 2 in one trial is .2 and the probability of gong from state 1 to state 3 in one trial is .3. The probability of going from state 2 to state 1 in one trial is .1 and the probability of going from state 2 to state 3 in one trial is .6. The probability of going from state 3 to state 1 in one trial is .4 and the probability of going from state 3 to state 2 in one trial is .4. Write the transition matrix for this Markov chain.

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   4.   ?    None of the answers given; click to see the solution.

4. Two brands of a certain product are available. Of those who buy Brand 1, 75% will buy it the next time, and the remainder will buy Brand 2. Of those who buy Brand 2, 60% will buy it the next time, and the remainder will buy Brand 1. Suppose that curently, 30% of those using this product have purchased Brand 1. What percentage will purchase Brand 1 next time?

   1.   ?    50.5%
   2.   ?    64.5%
   3.   ?    40%
   4.   ?    None of the answers given; click to see the solution.

5. A certain community is served by 3 free local newspapers. Each month, people have a choice of receiving one of the 3 papers. Of those who receive newspaper 1 each month, 30% switch to newspaper 2 and 10% switch to newspaper 3. Of those who receive newspaper 2 each month, 20% switch to newspaper 1 and 25% switch to newspaper 3. Of those who receive newspaper 3 each month, 15% switch to newspaper 1 and 15% switch to newspaper 2. Write the transition matrix for this Markov chain. (Note: This problem will be continued in problems # 6 and #7.)

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   4.   ?    None of the answers given; click to see the solution.

6. Suppose that this month in the community discussed in problem #5, 50% of the homes in the town have received newspaper 1, 30% have received newspaper 2, and 20% have received newspaper 3. What percentage will receive newspaper 3 one month from now?

   1.   ?    26.5%
   2.   ?    39%
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   4.   ?    None of the answers given; click to see the solution.

7. Suppose that this month in the community discussed in problem #5, 50% of the homes in the town have received newspaper 1, 30% have received newspaper 2, and 20% have received newspaper 3. What percentage will receive newspaper 2 four months from now?

   1.   ?    33.42%
   2.   ?    33.05%
   3.   ?    
   4.   ?    None of the answers given; click to see the solution.