Markov Chains - Finite Mathematics

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Question

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?

Answer

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

$P= \begin{bmatrix} 0.99 & 0.01 \ 0.02 & 0.98 \end{bmatrix}$

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

$A = \begin{bmatrix} 1,000 & 1,500 \end{bmatrix}$

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

$AP^{5}$ .

Using a calculator, we can determine that this is

$AP^{5}= \begin{bmatrix} 1,000 & 1,500 \end{bmatrix}\begin{bmatrix} 0.99 & 0.01 \ 0.02 & 0.98 \end{bmatrix}^{5}$

$\approx \begin{bmatrix} 1,094 & 1,406 \end{bmatrix}$

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

.

Republicans will outnumber Democrats by about 300.

Compare your answer with the correct one above

Question

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?

Answer

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

$P= \begin{bmatrix} 0.99 & 0.01 \ 0.02 & 0.98 \end{bmatrix}$

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

$A = \begin{bmatrix} 1,000 & 1,500 \end{bmatrix}$

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

$AP^{5}$ .

Using a calculator, we can determine that this is

$AP^{5}= \begin{bmatrix} 1,000 & 1,500 \end{bmatrix}\begin{bmatrix} 0.99 & 0.01 \ 0.02 & 0.98 \end{bmatrix}^{5}$

$\approx \begin{bmatrix} 1,094 & 1,406 \end{bmatrix}$

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

.

Republicans will outnumber Democrats by about 300.

Compare your answer with the correct one above

Question

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?

Answer

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

$P= \begin{bmatrix} 0.99 & 0.01 \ 0.02 & 0.98 \end{bmatrix}$

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

$A = \begin{bmatrix} 1,000 & 1,500 \end{bmatrix}$

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

$AP^{5}$ .

Using a calculator, we can determine that this is

$AP^{5}= \begin{bmatrix} 1,000 & 1,500 \end{bmatrix}\begin{bmatrix} 0.99 & 0.01 \ 0.02 & 0.98 \end{bmatrix}^{5}$

$\approx \begin{bmatrix} 1,094 & 1,406 \end{bmatrix}$

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

.

Republicans will outnumber Democrats by about 300.

Compare your answer with the correct one above

Question

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?

Answer

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

$P= \begin{bmatrix} 0.99 & 0.01 \ 0.02 & 0.98 \end{bmatrix}$

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

$A = \begin{bmatrix} 1,000 & 1,500 \end{bmatrix}$

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

$AP^{5}$ .

Using a calculator, we can determine that this is

$AP^{5}= \begin{bmatrix} 1,000 & 1,500 \end{bmatrix}\begin{bmatrix} 0.99 & 0.01 \ 0.02 & 0.98 \end{bmatrix}^{5}$

$\approx \begin{bmatrix} 1,094 & 1,406 \end{bmatrix}$

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

.

Republicans will outnumber Democrats by about 300.

Compare your answer with the correct one above

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