SOLUTION: The population of 100,000 consumers make the following purchases in a week. 20,000 purchased brand 𝐴; 35,000 purchased brand 𝐵 and 45,000 purchased neither brand. From a ma

Question 1197518: The population of 100,000 consumers make the following purchases in a week. 20,000 purchased brand
𝐴; 35,000 purchased brand 𝐵 and 45,000 purchased neither brand.
From a market study, it is estimated that of those who purchased brand 𝐴, 80% will purchase it again next
week, 15% will purchase brand 𝐵 and 5% will purchase neither brand. Of those who purchased brand
𝐵, 85% will purchase it again, 12% will purchase brand 𝐴 and 3% will purchase neither brand and of those
who purchased neither brand, 20% will purchase brand 𝐴, 15% will purchase brand 𝐵 and 65% will purchase
neither brand.
i. Form the transition and state matrices. [2 marks]
ii. Determine how many customers will be in each brand after 2 weeks. [3 marks]
b. The output levels of machinery, electricity and oil of a small country are 3000, 5000, and 2000 respectively.
Each unit of machinery requires inputs of 0.3 units of electricity and 0.3 units of oil.
Each unit of electricity requires inputs of 0.1 units of machinery and 0.2 units of oil.
Each unit of oil requires inputs of 0.2 units of machinery and 0.1 units of electricity.
Determine the machinery, electricity and oil available for export
Answer by onyulee(41) (Show Source): You can put this solution on YOUR website!
**Part i**
**Transition Matrix**
* **Define States:**
* State 1: Purchased Brand A
* State 2: Purchased Brand B
* State 3: Purchased Neither Brand
* **Transition Probabilities:**
| State | State 1 (A) | State 2 (B) | State 3 (Neither) |
|---|---|---|---|
| **State 1 (A)** | 0.80 | 0.15 | 0.05 |
| **State 2 (B)** | 0.12 | 0.85 | 0.03 |
| **State 3 (Neither)** | 0.20 | 0.15 | 0.65 |
**State Matrix (Initial Distribution)**
* **State 1 (A):** 20,000 / 100,000 = 0.2
* **State 2 (B):** 35,000 / 100,000 = 0.35
* **State 3 (Neither):** 45,000 / 100,000 = 0.45
* **Initial State Matrix:**
[0.2 0.35 0.45]
**Part ii**
**Determine the number of customers in each brand after 2 weeks**
* **Multiply the Initial State Matrix by the Transition Matrix twice:**
* **After 1 week:**
[0.2 0.35 0.45] *
[ 0.80 0.15 0.05
0.12 0.85 0.03
0.20 0.15 0.65 ]
* **After 2 weeks:**
[Result from 1st week] *
[ 0.80 0.15 0.05
0.12 0.85 0.03
0.20 0.15 0.65 ]
* **Calculate the resulting state matrix.**
* **Multiply each state probability by the total population (100,000) to find the number of customers in each brand after 2 weeks.**
**Part b**
**Determine the machinery, electricity, and oil available for export**
Let:
* X: Machinery output
* Y: Electricity output
* Z: Oil output
We have the following system of equations:
* X = 3000 - 0.1Y - 0.2Z
* Y = 5000 - 0.3X - 0.2Z
* Z = 2000 - 0.3X - 0.1Y
**Solve this system of equations to find the values of X, Y, and Z.**
* **Method 1: Substitution or Elimination**
* Solve for one variable in one equation and substitute it into the other equations.
* Repeat until you find the values of all three variables.
* **Method 2: Matrix Method**
* Represent the system of equations in matrix form and use matrix inversion to solve for X, Y, and Z.
**Once you find the values of X, Y, and Z:**
* **Machinery available for export:** Total machinery output (3000) - Machinery used as input (value of X)
* **Electricity available for export:** Total electricity output (5000) - Electricity used as input (value of Y)
* **Oil available for export:** Total oil output (2000) - Oil used as input (value of Z)
This will give you the amounts of machinery, electricity, and oil available for export after accounting for the internal consumption within the country.
**Note:**
* This analysis assumes a closed economic system where all production and consumption occur within the country.
* In reality, there may be imports and exports of these goods, which would affect the final available quantities.

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