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 $10,000 will be accepted. Assume that the competitor's bid x 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? If required, round your answer to two decimal places.
.40
 
(b)	Suppose you bid $14,000. What is the probability that your bid will be accepted? If required, round your answer to two decimal places.
.80
 
(c)	What amount should you bid to maximize the probability that you get the property?

To solve these questions, let’s analyze the competitor’s bid $ x $, which is uniformly distributed between $10,000 and $15,000.

Since $ x $ is uniformly distributed on $[10,000, 15,000]$, the probability density function (pdf) is:
\[
f(x) = \frac{1}{15,000 - 10,000} = \frac{1}{5,000} = 0.0002
\]

Given that the highest bid wins, we need to calculate the probability that your bid $ B $ is greater than $ x $, i.e., $ P(B > x) $, based on different values for $ B $.

### Part (a): Probability of Bid $12,000$ Being Accepted
If you bid $12,000$, we want the probability that $ x $ (the competitor's bid) is less than $12,000$:
\[
P(x < 12,000) = \int_{10,000}^{12,000} f(x) \, dx
\]

Since $ f(x) = 0.0002 $:
\[
P(x < 12,000) = (12,000 - 10,000) \cdot 0.0002 = 2,000 \cdot 0.0002 = 0.40
\]

So, if you bid $12,000$, the probability that your bid will be accepted is **0.40**.

### Part (b): Probability of Bid $14,000$ Being Accepted
If you bid $14,000$, we want the probability that $ x $ is less than $14,000$:
\[
P(x < 14,000) = \int_{10,000}^{14,000} f(x) \, dx
\]

\[
P(x < 14,000) = (14,000 - 10,000) \cdot 0.0002 = 4,000 \cdot 0.0002 = 0.80
\]

Thus, if you bid $14,000$, the probability that your bid will be accepted is **0.80**.

### Part (c): Optimal Bid to Maximize Probability
To maximize the probability of winning, you should bid as close as possible to $15,000, but any amount up to $15,000 would maximize your probability, as $ P(x < B) \to 1 $ as $ B \to 15,000 $. Therefore, the optimal bid to maximize the probability of acceptance is just slightly below $15,000.

**In summary:**
- **(a)** The probability your bid of $12,000 will be accepted: **0.40**
- **(b)** The probability your bid of $14,000 will be accepted: **0.80**
- **(c)** To maximize the probability of acceptance, you should bid slightly below **$15,000**.

Would you like further details on any part? 

Here are some related questions to explore:
1. What would the probability be if you bid $13,000?
2. How does the probability change if the competitor's bid distribution changes?
3. What if the competitor's bid range were increased to $10,000 to $20,000?
4. How does this scenario change if there are multiple competitors?
5. What if the lowest bid were accepted instead of the highest?

**Tip:** In uniform distribution problems, knowing the range can simplify the calculations significantly, as probabilities are linear over intervals.

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 $10,000 will be accepted. Assume that the competitor's bid x 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? If required, round your answer to two decimal places.

(b) Suppose you bid $14,000. What is the probability that your bid will be accepted? If required, round your answer to two decimal places.

(c) What amount should you bid to maximize the probability that you get the property?

Explore the probability of winning a bidding scenario where competitor bids are uniformly distributed between $10,000 and $15,000. Learn calculations for different bid amounts, including $12,000 and $14,000, and find the optimal bid to maximize winning chances.

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