Answer:
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.
Step-by-step 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}](https://tex.z-dn.net/?f=TM%3D%5Cbegin%7Bbmatrix%7D%26M1%26M2%26M3%5C%5C%5C%5CM1%260.65%260.20%260.15%5C%5C%5C%5CM2%260.60%260.15%260.25%5C%5C%5C%5CM3%260.50%260.1%260.60%20%5Cend%7Bbmatrix%7D)
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}](https://tex.z-dn.net/?f=TM%5E2%3D%5Cbegin%7Bbmatrix%7D0.65%260.20%260.15%5C%5C%5C%5C0.60%260.15%260.25%5C%5C%5C%5C0.50%260.1%260.60%20%5Cend%7Bbmatrix%7D%20%5Ccdot%20%5Cbegin%7Bbmatrix%7D0.65%260.20%260.15%5C%5C%5C%5C0.60%260.15%260.25%5C%5C%5C%5C0.50%260.1%260.60%20%5Cend%7Bbmatrix%7D)
![TM^2=\begin{bmatrix}0.6175&0.175&0.2375\\\\0.605&0.1675&0.2775\\\\0.685&0.175&0.46\end{bmatrix}](https://tex.z-dn.net/?f=TM%5E2%3D%5Cbegin%7Bbmatrix%7D0.6175%260.175%260.2375%5C%5C%5C%5C0.605%260.1675%260.2775%5C%5C%5C%5C0.685%260.175%260.46%5Cend%7Bbmatrix%7D)
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.