**Educational Content: Probability and Sales Price Distribution** **Scenario Overview:** A business executive, recently transferred from Chicago to Atlanta, needs to sell her house in Chicago quickly due to relocation. The executive received an offer from a coworker to buy her house for $210,000, but seeks more competitive offers. She decides to list the house on the market for another month; the sales price she might receive (denoted as \(Y\)) is uniformly distributed between $200,000 and $250,000. **Probability Density Function of Sales Price (Y):** The uniform distribution is defined from $200,000 to $250,000, noted as \( f(y) \), where \( f(y) = \frac{1}{b-a} \) if \( a \leq y \leq b \). In this case: \[ f(y) = \frac{1}{250 - 200} = \frac{1}{50} = 0.02 \] **Probability Calculations:** 1. **Probability the Sales Price is Between $200,000 and $225,000:** \[ P(200 \leq Y \leq 225) = \int_{200}^{225} 0.02 \, dy = 0.02 \times (225 - 200) = 0.02 \times 25 = 0.5 \] 2. **Probability the Sales Price is at Least $215,000:** \[ P(Y \geq 215) = \int_{215}^{250} 0.02 \, dy = 0.02 \times (250 - 215) = 0.02 \times 35 = 0.7 \] 3. **Probability the Sales Price is Less than $210,000:** \[ P(Y < 210) = \int_{200}^{210} 0.02 \, dy = 0.02 \times (210 - 200) = 0.02 \times 10 = 0.2 \] **Discussion:** Given these calculations, the executive should analyze if leaving the house on the market for another month is financially viable based on the probabilities of achieving different sales prices. The selection should depend on whether the probability of getting a higher offer aligns with her financial needs and relocation timeline.
**Educational Content: Probability and Sales Price Distribution** **Scenario Overview:** A business executive, recently transferred from Chicago to Atlanta, needs to sell her house in Chicago quickly due to relocation. The executive received an offer from a coworker to buy her house for $210,000, but seeks more competitive offers. She decides to list the house on the market for another month; the sales price she might receive (denoted as \(Y\)) is uniformly distributed between $200,000 and $250,000. **Probability Density Function of Sales Price (Y):** The uniform distribution is defined from $200,000 to $250,000, noted as \( f(y) \), where \( f(y) = \frac{1}{b-a} \) if \( a \leq y \leq b \). In this case: \[ f(y) = \frac{1}{250 - 200} = \frac{1}{50} = 0.02 \] **Probability Calculations:** 1. **Probability the Sales Price is Between $200,000 and $225,000:** \[ P(200 \leq Y \leq 225) = \int_{200}^{225} 0.02 \, dy = 0.02 \times (225 - 200) = 0.02 \times 25 = 0.5 \] 2. **Probability the Sales Price is at Least $215,000:** \[ P(Y \geq 215) = \int_{215}^{250} 0.02 \, dy = 0.02 \times (250 - 215) = 0.02 \times 35 = 0.7 \] 3. **Probability the Sales Price is Less than $210,000:** \[ P(Y < 210) = \int_{200}^{210} 0.02 \, dy = 0.02 \times (210 - 200) = 0.02 \times 10 = 0.2 \] **Discussion:** Given these calculations, the executive should analyze if leaving the house on the market for another month is financially viable based on the probabilities of achieving different sales prices. The selection should depend on whether the probability of getting a higher offer aligns with her financial needs and relocation timeline.
Transcribed Image Text:**Educational Content: Probability and Sales Price Distribution**
**Scenario Overview:**
A business executive, recently transferred from Chicago to Atlanta, needs to sell her house in Chicago quickly due to relocation. The executive received an offer from a coworker to buy her house for $210,000, but seeks more competitive offers. She decides to list the house on the market for another month; the sales price she might receive (denoted as \(Y\)) is uniformly distributed between $200,000 and $250,000.
**Probability Density Function of Sales Price (Y):**
The uniform distribution is defined from $200,000 to $250,000, noted as \( f(y) \), where \( f(y) = \frac{1}{b-a} \) if \( a \leq y \leq b \). In this case:
\[ f(y) = \frac{1}{250 - 200} = \frac{1}{50} = 0.02 \]
**Probability Calculations:**
1. **Probability the Sales Price is Between $200,000 and $225,000:**
\[
P(200 \leq Y \leq 225) = \int_{200}^{225} 0.02 \, dy = 0.02 \times (225 - 200) = 0.02 \times 25 = 0.5
\]
2. **Probability the Sales Price is at Least $215,000:**
\[
P(Y \geq 215) = \int_{215}^{250} 0.02 \, dy = 0.02 \times (250 - 215) = 0.02 \times 35 = 0.7
\]
3. **Probability the Sales Price is Less than $210,000:**
\[
P(Y < 210) = \int_{200}^{210} 0.02 \, dy = 0.02 \times (210 - 200) = 0.02 \times 10 = 0.2
\]
**Discussion:**
Given these calculations, the executive should analyze if leaving the house on the market for another month is financially viable based on the probabilities of achieving different sales prices. The selection should depend on whether the probability of getting a higher offer aligns with her financial needs and relocation timeline.
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![**Educational Content: Probability and Sales Price Distribution**
**Scenario Overview:**
A business executive, recently transferred from Chicago to Atlanta, needs to sell her house in Chicago quickly due to relocation. The executive received an offer from a coworker to buy her house for $210,000, but seeks more competitive offers. She decides to list the house on the market for another month; the sales price she might receive (denoted as \(Y\)) is uniformly distributed between $200,000 and $250,000.
**Probability Density Function of Sales Price (Y):**
The uniform distribution is defined from $200,000 to $250,000, noted as \( f(y) \), where \( f(y) = \frac{1}{b-a} \) if \( a \leq y \leq b \). In this case:
\[ f(y) = \frac{1}{250 - 200} = \frac{1}{50} = 0.02 \]
**Probability Calculations:**
1. **Probability the Sales Price is Between $200,000 and $225,000:**
\[
P(200 \leq Y \leq 225) = \int_{200}^{225} 0.02 \, dy = 0.02 \times (225 - 200) = 0.02 \times 25 = 0.5
\]
2. **Probability the Sales Price is at Least $215,000:**
\[
P(Y \geq 215) = \int_{215}^{250} 0.02 \, dy = 0.02 \times (250 - 215) = 0.02 \times 35 = 0.7
\]
3. **Probability the Sales Price is Less than $210,000:**
\[
P(Y < 210) = \int_{200}^{210} 0.02 \, dy = 0.02 \times (210 - 200) = 0.02 \times 10 = 0.2
\]
**Discussion:**
Given these calculations, the executive should analyze if leaving the house on the market for another month is financially viable based on the probabilities of achieving different sales prices. The selection should depend on whether the probability of getting a higher offer aligns with her financial needs and relocation timeline.](https://content.bartleby.com/qna-images/question/fa11c8a8-eb56-4157-9233-facd50115ae0/5d59ece9-96fb-463f-b494-ac372e3ff7e3/2ou2ekt.jpeg)
