A consultant knows that it will cost him 10,000 to fulfill...

A consultant knows that it will cost him $$\$ 10,000$$ to fulfill a particular contract. The contract is to be put out for bids, and he believes that the lowest bid, excluding his own, can be represented by a distribution that is uniform between $$\$ 8,000$$ and $$\$ 20,000$$. Therefore, if the random variable $\mathrm{X}$ denotes the lowest of all other bids (in thousands of dollars), its probability density function is as follows: $$ f(x)= \begin{cases}1 / 12 & \text { for } 8<x<20 \\ 0 & \text { for all other values of } x\end{cases} $$ a. What is the probability that the lowest of the other bids will be less than the consultant's cost estimate of $$\$ 10,000$$ ? b. If the consultant submits a bid of $$\$ 12,000$$, what is the probability that he will secure the contract? c. The consultant decides to submit a bid of $$\$ 12,000$$. What is his expected profit from this strategy? d. If the consultant wants to submit a bid so that his expected profit is as high as possible, discuss how he should go about making this choice.

A consultant knows that it will cost him $$\$ 10,000$$ to fulfill a particular contract. The contract is to be put out for bids, and he believes that the lowest bid, excluding his own, can be represented by a distribution that is uniform between $$\$ 8,000$$ and $$\$ 20,000$$. Therefore, if the random variable $\mathrm{X}$ denotes the lowest of all other bids
(in thousands of dollars), its probability density function is as follows:
$$
f(x)= \begin{cases}1 / 12 & \text { for } 8<x<20 \\ 0 & \text { for all other values of } x\end{cases}
$$
a. What is the probability that the lowest of the other bids will be less than the consultant's cost estimate of $$\$ 10,000$$ ?
b. If the consultant submits a bid of $$\$ 12,000$$, what is the probability that he will secure the contract?
c. The consultant decides to submit a bid of $$\$ 12,000$$. What is his expected profit from this strategy?
d. If the consultant wants to submit a bid so that his expected profit is as high as possible, discuss how he should go about making this choice.

Key Concepts

Decision Making Under Uncertainty

Decision making under uncertainty involves choosing actions by weighing the probabilities and outcomes of uncertain events. In bidding and contract scenarios, this requires balancing the potential profit against the risk of losing the bid due to competitors’ lower bids, guiding the optimal strategic decision.

Expected Value and Expected Profit

Expected value is the long-run average outcome of a random variable, calculated by integrating or summing over all possible outcomes weighted by their probabilities. In scenarios involving profit, the expected profit is computed as the potential gain minus any costs, accounting for the likelihood of various outcomes. This concept is key to evaluating strategies under uncertainty.

Optimal Bidding Strategy

An optimal bidding strategy seeks to maximize expected profit by carefully selecting a bid that considers both the chance of winning the contract and the resulting profit margin if the bid is successful. This involves analyzing the distribution of competitors' bids and determining a bid value that achieves the best trade-off between winning probability and profitability.

Uniform Distribution

A uniform distribution is a probability distribution where all outcomes within a specified interval are equally likely. This concept is essential for problems that assume a constant probability over an interval, allowing for straightforward probability calculations based on the length of the interval segments relative to the total interval.

Probability Density Function

A probability density function (pdf) defines how probabilities are distributed over the values of a continuous random variable. For a uniform distribution, the pdf is constant over the defined interval and zero elsewhere, which simplifies the integration needed to find probabilities over specific ranges.

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