Question 1207252
**a) Initial Probability Matrix**

* Let's define:
    * **F:** Full-fat milk drinkers
    * **L:** Low-fat milk drinkers

* The initial probability matrix represents the initial distribution of milk drinkers:

   ```
   [P(F)  P(L)] = [0.60  0.40] 
   ```

**b) Transition Matrix**

* The transition matrix represents the probabilities of switching between milk types:

   ```
       To F   To L
   From F | 0.60  0.40 |
   From L | 0.20  0.80 |
   ```

* **Explanation:**
    * 0.60 in the first row, first column: 60% of full-fat drinkers stay with full-fat milk.
    * 0.40 in the first row, second column: 40% of full-fat drinkers switch to low-fat milk.
    * 0.20 in the second row, first column: 20% of low-fat drinkers switch to full-fat milk.
    * 0.80 in the second row, second column: 80% of low-fat drinkers stay with low-fat milk.

**c) Percentage of People Expected to Drink Full-Fat Milk One Year From Now**

* To find the distribution after one year, multiply the initial probability matrix by the transition matrix:

   ```
   [0.60  0.40] * 
   [0.60  0.40]
   [0.20  0.80] 
   = [0.44  0.56]
   ```

* After one year, 44% of people are expected to drink full-fat milk.

**d) Percentage of People Expected to Drink Full-Fat Milk Three Years From Now**

* To find the distribution after three years, multiply the initial probability matrix by the transition matrix three times:

   ```
   [0.60  0.40] * 
   [0.60  0.40]  * 
   [0.20  0.80]   * 
   [0.60  0.40]
   [0.20  0.80] 
   [0.60  0.40]
   [0.20  0.80] 
   = [0.376  0.624]
   ```

* After three years, 37.6% of people are expected to drink full-fat milk.

**Key Points:**

* Markov chains are used to model systems that transition between different states over time. 
* The transition matrix describes the probabilities of moving between states.
* By multiplying the initial probability vector by the transition matrix repeatedly, we can predict the distribution of states at future time points.