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Chapter 14: Problem 15

Software A small software company bids on two contracts and knows it can only get one of them. It anticipates a profit of $$\$ 50,000$$ if it gets the larger contract and a profit of $$\$ 20,000$$ on the smaller contract. The company estimates there's a \(30 \%\) chance it will get the larger contract and a \(60 \%\) chance it will get the smaller contract. Assuming the contracts will be awarded independently, what's the expected profit?

Short Answer

Expert verified
The expected profit is \$15,000 from the larger contract and \$12,000 from the smaller contract, giving a total expected profit of \$27,000.

Step by step solution

01

Identify and Interpret Given Information

The given information includes the profit from the larger contract (\$50,000), the probability of getting the larger contract (30%), the profit from the smaller contract (\$20,000), and the probability of getting the smaller contract (60%). It's important to interpret these correctly for calculating the expected profit.
02

Calculate Expected Profit for Each Contract

The expected profit from a contract is given by the profit from that contract multiplied by the likelihood (probability) of receiving that contract. So for the larger contract, it's \$50,000 * 0.30, and for the smaller contract, it's \$20,000 * 0.60.
03

Sum the Expected Profits

Finally, sum the expected profits from the larger and smaller contracts to get the total expected profit. This is done by adding the two results from step 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability
Probability is a concept that measures the chance or likelihood of an event occurring. In our context, it helps us understand how likely it is for the software company to win either contract. Probability values range from 0 to 1, where 0 means an event never happens, and 1 means it always happens.

In the exercise, the probabilities given are:
  • The likelihood of winning the larger contract is 0.30 (or 30%).
  • The likelihood of winning the smaller contract is 0.60 (or 60%).
These percentages tell the company how often they might expect to receive each contract if the bidding process were to be repeated many times over.
Independent Events
In probability, independent events are those whose occurrence or outcomes do not affect each other. This means that the result of one event has no bearing on the result of another.

In this exercise, the company treats the bidding for two different contracts as independent events. This means winning or losing one contract doesn't change the probability of winning or losing the other. This assumption simplifies our calculation because we can consider each contract separately without worrying about how one impacts the other. It allows us to calculate the expected profit for each independently and then sum them up for the total expected profit.
Profit Calculation
Profit Calculation involves determining the monetary gain from a business operation. In our exercise, the profit specifically refers to the potential financial rewards from securing either of the two contracts:
  • For the larger contract, the gain is $50,000.
  • For the smaller contract, the gain is $20,000.
Calculating potential profits against probabilities of the contracts being awarded is crucial. The expected profit for each contract is calculated by multiplying the profit amount by the probability of obtaining that contract. This step ensures we align potential outcomes with their likelihoods, weighting each profit by its chance of occurring.
Expected Value
Expected Value is a fundamental concept in probability and statistics, used to anticipate the average outcome of an event over the long term. Here, it helps forecast the company's average profit from bidding on the two contracts.

To find the expected value of the profit, we calculate each contract's expected profit:
  • The expected profit from the larger contract is \(50,000 \times 0.30 = 15,000\).
  • The expected profit from the smaller contract is \(20,000 \times 0.60 = 12,000\).
After computing each expected profit, we simply sum them:
  • \(15,000 + 12,000 = 27,000\).
Thus, the company can anticipate an average profit of $27,000 from their contract bids. This method ensures the decision is based on probabilistic averages, providing a balanced expectation aligned with potential outcomes.

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