Question

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 $10,200 will be accepted. Assume that the competitor's bid x is a random variable that is uniformly distributed between $10,200 and $15,000 .

a. Suppose you bid $12,000. What is the probability that your bid will be accepted (to 2 decimals)?

_____________

b. Suppose you bid $14,000. What is the probability that your bid will be accepted (to 2 decimals)?

_____________

c. What amount should you bid to maximize the probability that you get the property?

$ _____________

d. Suppose that you know someone is willing to pay you $16,000 for the property. You are considering bidding the amount shown in part (c) but a friend suggests you bid $13,100. If your objective is to maximize the expected profit, what is your bid?

- Select your answer: Stay with your bid in part (c); it maximizes expected profit or Bid $13,100 to maximize the expected profitItem

What is the expected profit for this bid (to 2 decimals)?

Answer 1

To determine the probability of a bid being accepted and maximizing expected profit, one must consider uniform distribution, range, and resale value. The provided answers involve calculations based on the uniform distribution of the competitor's bid ranging from $10,200 to $15,000.

Step-by-step explanation:

The bidding on a piece of land involves understanding the probability and expected value within the context of uniform distribution. Let's start by calculating the probability for each bid.

Probability of $12,000 Bid Being Accepted

The competitor's bid (X) is uniformly distributed between $10,200 and $15,000. If you bid $12,000, the probability that your bid will be accepted is the probability that X is less than $12,000. The range of the distribution is $15,000 - $10,200 = $4,800. So, the probability is calculated as ($12,000 - $10,200) / $4,800 which equals 0.375 or 37.5% when converted to a percentage.

Probability of $14,000 Bid Being Accepted

If you bid $14,000, then the probability that your bid will be accepted is ($14,000 - $10,200) / $4,800 which equals 0.7917 or 79.17% when converted to a percentage.

Maximizing the Probability of Getting the Property

To maximize the probability of getting the property, you should bid more than $15,000, which is the maximum of the competitor's bid range.

Maximizing Expected Profit

Considering the resale value is $16,000, if you bid the amount in part (c), your expected profit would be based on the difference between $16,000 and your bid. Since expected profit should take into account both the probability of winning the bid and the profit margin, the decision between part (c) and $13,100 should be based on maximizing this expected value.