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 $10,100 will be accepted. Assume that the competitor's bid x is a random variable that is uniformly distributed between $10,100 and %15,100.
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 $13,050. Which bid will give you the larger expected profit?
What is the expected profit for this bid (to 2 decimals)?
To solve this problem, we need to understand the concept of uniform distribution and probabilities. Let's break down each part of the question and explain the steps to find the answers.
a. To calculate the probability that your bid of $12,000 will be accepted, we need to find the fraction of the total range that falls within the acceptable bids.
1. First, calculate the range of acceptable bids: $15,100 - $10,100 = $5,000.
2. Then, determine the range between your bid and the lower limit of acceptable bids: $12,000 - $10,100 = $1,900.
3. Divide the range between your bid and the lower limit by the total range of acceptable bids: $1,900 / $5,000 = 0.38 or 38%.
Therefore, the probability that your bid of $12,000 will be accepted is 0.38 or 38%.
b. Similarly, for a bid of $14,000, repeat the steps above:
1. Range of acceptable bids: $15,100 - $10,100 = $5,000.
2. Range between your bid and the lower limit of acceptable bids: $14,000 - $10,100 = $3,900.
3. Divide the range between your bid and the lower limit by the total range of acceptable bids: $3,900 / $5,000 = 0.78 or 78%.
Therefore, the probability that your bid of $14,000 will be accepted is 0.78 or 78%.
c. To determine the amount you should bid to maximize the probability of obtaining the property, you need to find the midpoint of the range of acceptable bids.
1. Calculate the midpoint by adding the lower and upper limits of acceptable bids and dividing by 2: ($10,100 + $15,100) / 2 = $12,600.
Therefore, you should bid $12,600 to maximize the probability of obtaining the property.
d. To compare the expected profits of bidding $13,050 and the amount from part c ($12,600), we need to calculate the expected profit for each bid.
1. Expected profit is calculated by multiplying the probability of winning the bid by the profit earned in that scenario.
For bidding $13,050:
- The probability of winning is the range between the bid and lower limit divided by the total range of acceptable bids: ($13,050 - $10,100) / $5,000 = 0.59 or 59%.
- The profit earned in this scenario is the selling price minus the bid amount: $16,000 - $13,050 = $2,950.
- Multiply the probability of winning by the profit earned: 0.59 * $2,950 = $1,735.50.
For bidding the amount from part c ($12,600):
- The probability of winning is 100% since it's the midpoint of the range of acceptable bids.
- The profit earned in this scenario is the selling price minus the bid amount: $16,000 - $12,600 = $3,400.
Therefore, it is expected that bidding the amount from part c will give you a larger expected profit.
The expected profit for bidding the amount from part c is $3,400.