Question

An engineering professor acquires a new computer once every 2 years. The professor can choose from three models: M1, M1, and M3. If the present model is M1, the next model can be M2 with probability 0.2, or M3 with probability 0.15. If the present model is M2, the probabilities of switching to M1 and M3 are 0.6 and 0.25, respectively. And, if the present model is M3, then the probabilities of purchasing M1 and M2 are 0.5 and 0.1, respectively. Represent the situation as a Markov chain.

Determine the probability that the professor will purchase the current model in 4 years.

Answer 1

In 4 years, the probability of buying the same model is:

- If it starts with M1, the probability is P=0.62.

- If it starts with M2, the probability is P=0.17.

- If it starts with M3, the probability is P=0.46.

Explanation:

The transition matrix, according to the data given by the question, can be written as:

`TM=\begin{bmatrix}&M1&M2&M3\\\\M1&0.65&0.20&0.15\\\\M2&0.60&0.15&0.25\\\\M3&0.50&0.1&0.60 \end{bmatrix}`

This is the probability of the transition from t=0 to t=2 years.

To calculate the probability from t=0 to t=4 years, we can multiply the matrix by itself:

`TM^2=\begin{bmatrix}0.65&0.20&0.15\\\\0.60&0.15&0.25\\\\0.50&0.1&0.60 \end{bmatrix} \cdot \begin{bmatrix}0.65&0.20&0.15\\\\0.60&0.15&0.25\\\\0.50&0.1&0.60 \end{bmatrix}`

`TM^2=\begin{bmatrix}0.6175&0.175&0.2375\\\\0.605&0.1675&0.2775\\\\0.685&0.175&0.46\end{bmatrix}`

Then, in 4 years, the probability of buying the same model is:

- If it starts with M1, the probability is P=0.6175.

- If it starts with M2, the probability is P=0.1675.

- If it starts with M3, the probability is P=0.4600.