91Ó°ÊÓ

A business executive, transferred from Chicago to Atlanta, needs to sell her house in Chicago quickly. The executive's employer has offered to buy the house for \(\$ 210,000\) but the offer expires at the end of the week. The executive does not currently have a better offer but can afford to leave the house on the market for another month. From conversations with her realtor, the executive believes the price she will get by leaving the house on the market for another month is uniformly distributed between \(\$ 200,000\) and \(\$ 225,000\) a. If she leaves the house on the market for another month, what is the mathematical expression for the probability density function of the sales price? b. If she leaves it on the market for another month, what is the probability that she will get at least \(\$ 215,000\) for the house? c. If she leaves it on the market for another month, what is the probability that she will get less than \(\$ 210,000 ?\) d. Should the executive leave the house on the market for another month? Why or why not?

Step by step solution

01

Understanding Uniform Distribution

The problem states that the selling price in a month is uniformly distributed between $200,000 and $225,000. In a uniform distribution, each outcome in a range has an equal probability of occurrence. The range here is from $200,000 to $225,000.

02

Define the Probability Density Function (PDF)

For a random variable \(X\) uniformly distributed between \(a\) and \(b\), the probability density function is defined as: \[ f(x) = \frac{1}{b-a} \text{ for } a \leq x \leq b \]Applying the values from this problem, we have: \[ f(x) = \frac{1}{225,000 - 200,000} = \frac{1}{25,000} \text{ for } 200,000 \leq x \leq 225,000 \]

03

Calculate Probability of Getting at Least $215,000

To find the probability that the selling price will be at least \(215,000, calculate the area under the PDF from \)215,000 to $225,000: \[ P(X \geq 215,000) = \int_{215,000}^{225,000} \frac{1}{25,000} \, dx \]The probability is equal to: \[ \frac{225,000 - 215,000}{25,000} = \frac{10,000}{25,000} = 0.4 \]

04

Calculate Probability of Getting Less Than $210,000

To find the probability that the selling price will be less than \(210,000, calculate the area under the PDF from \)200,000 to $210,000:\[ P(X < 210,000) = \int_{200,000}^{210,000} \frac{1}{25,000} \, dx \]The probability is:\[ \frac{210,000 - 200,000}{25,000} = \frac{10,000}{25,000} = 0.4 \]

05

Decision on Selling the House

The executive should compare the $210,000 guaranteed offer from the employer with the expected outcomes in waiting. Accepting the offer guarantees $210,000. Waiting gives a 0.6 probability of at least $210,000, but a more lucrative 0.4 probability of at least $215,000. The decision depends on risk preference, but taking the guaranteed $210,000 avoids the risk of getting less and thus might be more prudent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Density Function

In this exercise, we are dealing with a uniform distribution for the house's potential selling price. A uniform distribution means that every value within a specified range has an equal likelihood of occurring. In our scenario, this range is between \(200,000 and \)225,000.
To simplify decision-making, we utilize the probability density function (PDF), which defines how probability is distributed over a set of values. The PDF for a uniformly distributed random variable between two values, let's call them \(a\) and \(b\), is defined as:\[f(x) = \frac{1}{b-a} \quad \text{for} \quad a \leq x \leq b\]This equation tells us that the probability is uniformly spread over the interval, with each outcome equally probable.

• In our scenario, \(a = 200,000\) and \(b = 225,000\).
• This gives us: \(f(x) = \frac{1}{25,000}\).

The uniform distribution is a foundational concept in probability that helps in analyzing situations where outcomes are equally likely.

Expected Value

The expected value is a key concept when making decisions under uncertainty. It represents the average outcome if an experiment, like selling the house, were repeated many times. In simple terms, it is the mean of all possible outcomes, weighted by their probabilities.
For a continuous uniform distribution, the expected value can be calculated by:\[E(X) = \frac{a + b}{2}\]This formula is essentially the midpoint of the interval in a uniform distribution, as it strikes a balance between the highest and lowest possible values.

• In our problem, this would mean:\(E(X) = \frac{200,000 + 225,000}{2} = 212,500\).

The expected value helps us to make informed predictions about the average selling price one might achieve by leaving the house on the market. It is crucial in situations where multiple potential outcomes must be weighed against each other.

Decision Making

Making decisions under uncertainty requires evaluating potential outcomes and their probabilities. This involves weighing the risks and rewards associated with different choices. In this case, the executive needs to choose between accepting a guaranteed offer of $210,000 or waiting for a better offer.
Here are some considerations:

• The guaranteed offer of $210,000 is certain and immediate.
• By leaving the house on the market, there is a 0.4 probability of selling it for at least $215,000, and a 0.6 probability of selling it for less than $210,000.

Thus, decision-making involves evaluating these odds against one's risk tolerance. While the expected value suggests an average return of $212,500, it does not account for the guaranteed nature of the employer's offer. In scenarios like these, being risk-averse might lead to accepting the guaranteed offer to mitigate any potential losses.
Ultimately, the decision involves balancing risk and certainty in light of financial priorities and personal circumstances.