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.