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 $15,400.
Suppose you bid $12,000. What is the probability that your bid will be accepted (to 2 decimals)?
Suppose you bid $14,000. What is the probability that your bid will be accepted (to 2 decimals)?
What amount should you bid to maximize the probability that you get the property (in dollars)?
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. If your objective is to maximize the expected profit, what is your bid?
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What is the expected profit for this bid (in dollars)?
To answer these questions, we need to analyze the probability distributions based on the given information.
Let's first compute the probability that your bid will be accepted when you bid $12,000.
1. Calculate the range of the competitor's bid: $15,400 - $10,100 = $5,300
2. Calculate the range above your bid: $15,400 - $12,000 = $3,400
3. Calculate the probability of the competitor bidding lower than or equal to $12,000: (range above your bid) / (range of competitor's bid)
Probability = $3,400 / $5,300 = 0.6415 (rounded to 2 decimal places)
Therefore, the probability that your bid will be accepted when you bid $12,000 is 0.64 (rounded to 2 decimal places).
Now, let's calculate the probability when you bid $14,000.
1. Calculate the range of the competitor's bid: $15,400 - $10,100 = $5,300
2. Calculate the range above your bid: $15,400 - $14,000 = $1,400
3. Calculate the probability of the competitor bidding lower than or equal to $14,000: (range above your bid) / (range of competitor's bid)
Probability = $1,400 / $5,300 = 0.2642 (rounded to 2 decimal places)
Therefore, the probability that your bid will be accepted when you bid $14,000 is 0.26 (rounded to 2 decimal places).
To maximize the probability of getting the property, you need to bid an amount that minimizes the range above your bid. From the given options, you should choose the bid that is closest to $10,100, which is the lower limit. Therefore, you should bid $10,100.
Now, let's calculate the profit for the optimal bid.
1. Calculate the probability of winning the bid by bidding $10,100: It's equal to 1, as any bid above that will be accepted.
2. Calculate the profit: It's the difference between the bid and the selling price. In this case, it's $16,000 - $10,100 = $5,900.
Therefore, if your objective is to maximize the expected profit, you should bid $10,100.
The expected profit for this bid is simply the profit multiplied by the probability of winning:
Expected Profit = Probability of winning * Profit = 1 * $5,900 = $5,900.