91Ó°ÊÓ

Suppose we are interested in bidding on a piece of land and we know one other bidder is interested. \(^{1}\) 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

Understand the Problem

We are given that the competitor's bid, \(x\), is a random variable uniformly distributed between \(\\(10,000\) and \(\\)15,000\). We want to find the probability that a bid we place is accepted under different scenarios.

02

Determine the Probability for a $12,000 Bid

If we bid \(\\(12,000\), then our bid will be accepted if it exceeds the competitor's bid, \(x\). Therefore, \(\\)12,000 > x\). The probability of this happening is the probability that \(x\) is less than \(\\(12,000\). In a uniform distribution from \(\\)10,000\) to \(\$15,000\), this is given by:\[P(x < 12,000) = \frac{12,000 - 10,000}{15,000 - 10,000} = \frac{2,000}{5,000} = 0.4\]

03

Determine the Probability for a $14,000 Bid

If we bid \(\\(14,000\), then our bid will be accepted if it exceeds the competitor's bid, \(x\). Therefore, \(\\)14,000 > x\). The probability of this happening is the probability that \(x\) is less than \(\$14,000\). Using the uniform distribution:\[P(x < 14,000) = \frac{14,000 - 10,000}{15,000 - 10,000} = \frac{4,000}{5,000} = 0.8\]

04

Calculate Optimal Bid for Maximum Probability

To maximize the probability of winning the bid, we should aim to bid just above \(x\) as much as possible within the distribution limit. The closer our bid is to the maximum competitor's bid of \(\\(15,000\), the higher the probability we will win, since:\[P(x < 15,000) = \frac{15,000 - 10,000}{15,000 - 10,000} = 1\]Placing a bid of exactly \(\\)15,000\) guarantees winning if \(x < 15,000\) since the range doesn't include \(\\(15,000\). A bid just above \(\\)15,000\) guarantees winning (since the competitor can only bid up to \(\$15,000\)).

05

Analyze the Decision with a Potential Buyer

If someone is willing to pay \(\\(16,000\), you gain a profit if your bid is less than \(\\)16,000\) and you win. If you bid less than \(\\(15,000\) (the amount calculated in Step 4 to maximize your winning probability), there's a risk of losing even if the competitor bids at \(\\)15,000\).It's best to bid closer to \(\$15,000\) to ensure winning, maximizing your profit upon resale.

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Key Concepts

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

Uniform Distribution

When we talk about a uniform distribution, we mean that

• Every outcome in a given range is equally likely
• In our exercise, the range of the competitor's bid is between \(10,000\) and \(15,000\)

This means that any bid within this range has the same chance of being placed. If you imagine the possible bids on a number line, every point from \(10,000\) to \(15,000\) is equally likely.
The probability of an event is calculated based on how much of this range falls under the event condition. For example, the probability that a competitor bids less than \(12,000\) would be the length of the range from \(10,000\) to \(12,000\), divided by the full range from \(10,000\) to \(15,000\). Thus, each calculation involves simple subtraction and division.

Bidding Strategy

A bidding strategy involves selecting the best bid to maximize certain outcomes, often winning the auction.
In auctions, understanding your competitors and distribution of their possible bids can guide your strategy. Since the highest bid above a certain threshold wins, our goal is to place a bid that has the best probability of being higher than the competitor's.
In this scenario, since the competitor's bid is uniformly distributed up to \(15,000\), bidding slightly above \(15,000\) would guarantee a win because the competitor cannot bid more than that.
However, financial considerations are important too, and understanding both the maximum likelihood of winning and the cost involved helps in making a strong bid decision.

Random Variable

A random variable is a variable whose value is determined by the outcome of a random phenomenon. It could assume different values, each with a certain probability. In our land bidding exercise, the competitor’s bid, denoted as \(x\), is a random variable. This bid could randomly fall anywhere between \(10,000\) and \(15,000\).
Random variables can be:

• Discrete, taking distinct values
• Continuous, taking any value within a range

In our case, the bid is continuous, meaning it can have any value in the range of \(10,000\) to \(15,000\). Understanding the nature of a random variable within a given context is crucial for calculating probabilities and making informed decisions.

Expected Value

The expected value gives us the average of all possible outcomes of a random variable, weighted by their probabilities. It's essentially what you expect to happen on average if the random variable process is repeated many times.
Although not directly computed in our exercise, understanding expected value helps in making informed decisions. For example, if you could repeat the bidding process many times, the expected value would guide you on the most probable financial outcome.
In auctions, knowing the expected profit based on your bid and potential selling price can guide you in determining how much to bid. Balancing the bid amount with expected profit ensures that over the long term, your strategy maximizes gains while minimizing potential losses.