91Ó°ÊÓ

Suppose we are interested in bidding on a piece of land and we know one other bidder is interested.* The seller announced that the highest bid in excess of \(\$ 10,000\) will be accepted. Assume that the competitor's bid \(x\) is a random variable that is uniformly distributed between \(\$ 10,000\) and \(\$ 15,000\) a. Suppose you bid \(\$ 12,000 .\) What is the probability that your bid will be accepted? b. Suppose you bid \(\$ 14,000\). What is the probability that your bid will be accepted? c. What amount should you bid to maximize the probability that you get the property? d. Suppose you know someone who is willing to pay you \(\$ 16,000\) for the property. Would you consider bidding less than the amount in part (c)? Why or why not?

Step by step solution

01

Understanding the Uniform Distribution

The competitor's bid \(x\) is uniformly distributed between \(\\(10,000\) and \(\\)15,000\). This means every value within this range is equally likely. The probability density function \(f(x)\) is given by \(f(x) = \frac{1}{15,000 - 10,000} = \frac{1}{5,000}\) for \(10,000 \leq x \leq 15,000\).

02

Probability Calculation for $12,000 Bid

To determine the probability that your \(\\(12,000\) bid will be accepted, calculate the probability that the competitor bids less than \(\\)12,000\). Since \(x\) is uniformly distributed, the probability is the fraction of the interval from \(10,000\) to \(12,000\). \[P(x < 12,000) = \frac{12,000 - 10,000}{15,000 - 10,000} = \frac{2,000}{5,000} = 0.4\]

03

Probability Calculation for $14,000 Bid

Similarly, if you bid \(\\(14,000\), calculate the probability that the competitor's bid is less than \(\\)14,000\). The probability again is the fraction of the interval from \(10,000\) to \(14,000\). \[P(x < 14,000) = \frac{14,000 - 10,000}{15,000 - 10,000} = \frac{4,000}{5,000} = 0.8\]

04

Maximizing the Probability to Get the Property

To maximize the chance of winning, bid just above the maximum possible competitor bid, which is \(\\(15,000\). However, the smallest bid your competitor cannot outbid within their range is \(\\)15,001\). Therefore, bidding \(\\(15,001\) ensures you win if their maximum bid is \(\\)15,000\).

05

Considering Other Offers

You know someone is willing to pay you \(\\(16,000\) for the property. Even if you bid the maximum from Step 4, \(\\)15,001\), you still gain a \(\$999\) profit. Bidding less could increase risk, as your bid may not be the highest.

06

Conclusion on Bidding Strategy

Therefore, you should bid \(\\(15,001\) to maximize your chance of acquisition. As no competitor can outbid this within their distribution range, and you would still profit from the resale at \(\\)16,000\).

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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, understanding the probability density function (PDF) is crucial as it helps us know how the competitor's bid is spread over a specific range. A uniform distribution is particularly easy to deal with because it implies that every number within the specified range is equally likely.
For a uniform distribution, like the one where the competitor's bid is between \(10,000 and \)15,000, the PDF can be mathematically expressed as:

• The function for any value within the range is constant. Therefore:\[ f(x) = \frac{1}{15,000 - 10,000} = \frac{1}{5,000} \]
• This means that if you choose any number between \(10,000 and \)15,000, it is equally likely to occur with a probability of \(\frac{1}{5,000}\).
• Outside of this range, the probability is zero because the bid cannot fall outside \(10,000 to \)15,000, as per the problem assumption.

Simply put, understanding the uniform PDF allows you to calculate the probability of the competitor's bid being less than a certain number, which is key for determining how likely your bid is to win.

Bidding Strategy

Developing a strategic approach to bidding ensures you maximize your chances of winning the property given what you know about the competitor's bids. The competitor's bid is uniformly distributed between $10,000 and $15,000, so a rational bidding strategy involves understanding and selecting a bid that either minimizes the chance of being outbid or maximizes your potential profit.
Key considerations for your bidding strategy include:

• Maximizing Your Bid: To ensure you win, one strategy is to bid just above $15,000. In this scenario, an optimal bid would be $15,001, as the competitor cannot bid more than $15,000, ensuring your bid is always higher.
• Risk Assessment: Bidding under certain conditions might be tempting if a higher resale offer is available. However, bidding too low increases the risk of being outbid,
• Profit Maximization: If you can sell the property at $16,000 after winning, it's wise to bid $15,001, knowing that even at this high bid, you still turn a profit after the sale.

These elements help form a comprehensive bidding strategy, balancing the risks and rewards effectively.

Probability Calculation

Calculating the probability of your bid beating the competitor's involves finding the likelihood that the competitor bids less than your amount. This process uses the properties of uniform distribution to determine the chances of your bid succeeding.
Consider the following examples based on different bid amounts:

• For a \(12,000 bid, calculate as:\[ P(x < 12,000) = \frac{12,000 - 10,000}{15,000 - 10,000} = 0.4 \]This results in a 40% chance your bid is higher.
• If bidding \)14,000, calculate as:\[ P(x < 14,000) = \frac{14,000 - 10,000}{15,000 - 10,000} = 0.8 \]This results in an 80% chance your bid is higher.
• Maximizing the Bid: Bidding $15,001 assures a 100% probability as the competitor cannot place a higher bid within the given range.

This method of calculating probabilities is critical for understanding how your bid measures against potential competitor bids. It provides a mathematical basis for decision-making in a competitive bidding process.