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 $15,000 will be accepted. Assume that the competitor's bid x is a random variable that is uniformly distributed between $15,000 and $20,000.

(a)

Suppose you bid $16,000. What is the probability that your bid will be accepted?

(b)

Suppose you bid $18,000. What is the probability that your bid will be accepted?

(c)

What amount should you bid in dollars to maximize the probability that you get the property?

$

(d)

Suppose you know someone who is willing to pay you $21,000 for the property.

What is the expected profit in dollars if you bid the amount given in part (c)?

$

Find a bid in dollars which produces a greater expected profit than bidding the amount given in part (c). (If an answer does not exist, enter DNE.)

$

Would you consider bidding less than the amount in part (c)? Why or why not?

Yes. There is a bid which gives a greater expected profit than the bid given in part (c), and thus a higher expected profit is possible with a bid smaller than the amount in part (c). No. The bid which maximizes the expected profit is the amount given in part (c), thus it does not make sense to place a smaller bid.    

Answer 1

Solution :-

Given data :

The seller announced that the highest bid in excess of $15,000 will be accepted.

Assume that the competitor's bid x is a random variable that is uniformly distributed between $15,000 and $20,000.

(a):-

Suppose you bid $16,000. What is the probability that your bid will be accepted?

Required probability is P( 15000 < X < 16000)

P( 15000 < X < 16000) = [15000-16000] / [20000-15000]

P( 15000 < X < 16000) = 1000 / 5000

P( 15000 < X < 16000) = 0.2

Hence,The probability that your bid will be accepted is 0.2

(b):-

Suppose you bid $18,000. What is the probability that your bid will be accepted?

Required probability is P( 15000 < X < 18000)

P( 15000 < X < 18000) = [18000-15000] / [20000-15000]

P( 15000 < X < 18000) = 3000 / 5000

P( 15000 < X < 18000) = 0.6

Hence,The probability that your bid will be accepted is 0.6

(c):-

What amount should you bid in dollars to maximize the probability that you get the property?

Answer is $20,001

$20,001 is bid in dollars to maximize the probability that you get the property.

(d):-

Suppose you know someone who is willing to pay you $21,000 for the property.

(1):-

What is the expected profit in dollars if you bid the amount given in part (c)?

Expected profit = $21,000 - $20,001

Expected profit = $999

Hence, Expected profit is $999

(2):-

Find a bid in dollars which produces a greater expected profit than bidding the amount given in part (c).

Bid in dollars = { [ 21000 - 15000 ] /2 } + 15000

Bid in dollars = {6000 / 2} + 15000

Bid in dollars = 3000 + 15000

Bid in dollars = 18000

Hence, bid in dollars which produces a greater expected profit is $18,000.

(3):-

Would you consider bidding less than the amount in part (c)? Why or why not?

Answer is,

No. The bid which maximizes the expected profit is the amount given in part (c), thus it does not make sense to place a smaller bid.