Question 1207252: Results of a consumer survey show that over a one-year period, 40% of full-fat milk drinkers will switch to low-fat milk, and 20% of low-fat milk drinkers will switch to full-fat milk. At present, 60% of the people surveyed drink full-fat milk.
a) Write the initial probability matrix.
b) Write the transition matrix.
c) Determine the percentage of people expected to drink full-fat milk one year from now.
d) Determine the percentage of people expected to drink full-fat milk three years from now.
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! **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.
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