Suppose we are interested in bidding on a piece of land and we know one other bi

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 $9,600 will be accepted. Assume that the competitor's bid x is a random variable that is uniformly distributed between $9,600 and $14,500.

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 (in dollars)?
  

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 $12,800. If your objective is to maximize the expected profit, what is your bid?

e) What is the expected profit for this bid (in dollars)?

Explanation / Answer

a).P(x < 12000) = (12000 - 9600) / (14500 - 9600) = 0.49

b). P(x < 14000) = (14000 - 9600) / (14500 - 9600) = .90

c). $14,500

d).& e). Lets say you get the property. You sell it for $16,000, and thus make a profit. However, if you bid a ton, you make less profit, but have a smaller change of getting it.

This is an expected value problem: you want to maximize the following:
($16000-bid)*(chance of winning it with the bid)

If you bid $14500, you are guaranteed $1500. EV = $1500
If you bid $12800, you have a 65.71% chance of winning $3200. EV = $2089.80

Thus, you bid your friend's recommendation (and treat her out for a date) since you get an expected profit of $8089.80 for the bid.

If you bid X: then profit = (16000-X) and P(profit) = (X?9600)/4900. So expected profit = ..., and you want to find X to maximize that.

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