Suppose we are interested in bidding on a piece of land and we know one other bidder is interested. The seller announced that th

Question

2 months ago

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Suppose we are interested in bidding on a piece of land and we know one other bidder is interested. The seller announced that th

e highest bid in excess of $10,000 will be accepted. Assume that the competitor's bid x is a is a random variable that is uniformly distributed between $10,000 and $15,000.
a. Suppose you bid $12,000. What is the probability that your bid will be accepted? (please show calculations)
b. Suppose you bid $14,000. What is the probability that your bid will be accepted? (please show calculations)
c. What amount should you bid to maximize the probability that you get the property? (please show calculations)d. Suppose you know someone who is willing to pay you $16,000 for the property. Would you consider bidding less than the amount in part (c)? Why or why not?

Mathematics

1 answer:

Zina [12.3K] · Answer 1 · 2026-03-22T23:05:51+03:00

Answer:

Step-by-step reasoning:

(a)

To have an accepted bid, it needs to exceed $10,000. Let bid x be a continuous random variable uniformly distributed between

$10,000 and $15,000

The range for accepted bidding is $[ {\rm{\$ 10,000, \$ 15,000}]$ , with b = $15000 and a = $10000.

The provided bidding range is [$10,000,$12,000]. The probability is determined as follows,

$\begin{array}{c}\\P\left( {X{\rm{ < 12,000}}} \right){\rm{ = }}1 - P\left( {X > 12000} \right)\\\\ = 1 - \int\limits_{12000}^{15000} {\frac{1}{{15000 - 10000}}} dx\\\\ = 1 - \int\limits_{12000}^{15000} {\frac{1}{{5000}}} dx\\\\ = 1 - \frac{1}{{5000}}\left[ x \right]_{12000}^{15000}\\\end{array}$

$=1- \frac{[15000-12000]}{5000}\\\\=1-0.6\\\\=0.4$

(b) The accepted bidding range is [$10,000,$15,000], where b = $15,000 and a =$10,000. The given bidding range is [$10,000,$14,000].

$\begin{array}{c}\\P\left( {X{\rm{ < 14,000}}} \right){\rm{ = }}1 - P\left( {X > 14000} \right)\\\\ = 1 - \int\limits_{14000}^{15000} {\frac{1}{{15000 - 10000}}} dx\\\\ = 1 - \int\limits_{14000}^{15000} {\frac{1}{{5000}}} dx\\\\ = 1 - \frac{1}{{5000}}\left[ x \right]_{14000}^{15000}\\\end{array} P(X14000)$

$=1- \frac{[15000-14000]}{5000}\\\\=1-0.2\\\\=0.8$

(c)

The optimal amount to bid for maximizing the probability of acquiring the property is calculated as,

The accepted bidding range is [$10,000,$15,000],

where b = $15,000 and a = $10,000. The provided bidding range is [$10,000,$15,000].

$\begin{array}{c}\\f\left( {X = {\rm{15,000}}} \right){\rm{ = }}\frac{{{\rm{15000}} - {\rm{10000}}}}{{{\rm{15000}} - {\rm{10000}}}}\\\\{\rm{ = }}\frac{{{\rm{5000}}}}{{{\rm{5000}}}}\\\\{\rm{ = 1}}\\\end{array}$

(d) If you know someone willing to pay you $16,000 for the property, would you still consider bidding less than the amount mentioned in part (c)? Why or why not?