Finite Mathematics : Markov Chains

Study concepts, example questions & explanations for Finite Mathematics

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Example Questions

Example Question #1 : Markov Chains

Use a calculator with matrix arithmetic capability for this problem.

Forman County has 2,500 voters - 1,000 registered Democrats and 1,500 registered Republicans. Every year, 1% of the Democrats become Republicans, and 2% of the Republicans become Democrats. Ignoring other factors, after five years, by how much (nearest hundred) will Republicans outnumber Democrats in Forman County?

Possible Answers:

Republicans will outnumber Democrats by about 200.

Republicans will outnumber Democrats by about 100.

Republicans will outnumber Democrats by about 300.

Republicans will outnumber Democrats by about 400.

Democrats will outnumber Republicans.

Correct answer:

Republicans will outnumber Democrats by about 300.

Explanation:

This problem can be modeled using a Markov chain matrix. If we let the rows represent the initial party affiliation, the columns represent final party affiliation, and Democrats be represented by the first row and column, the Markov matrix that represents the system is

The initial state of the Markov system can be represented by the row vector

The distribution of Democrats and Republicans after five years can be determined by evaluating the product

.

Using a calculator, we can determine that this is

The second number will be larger, so there will be more Republicans. Also, the difference will be

.

Republicans will outnumber Democrats by about 300.

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