Question

Software. A small software company bids on two contracts. 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

The expected profit is \( \$27,000 \).

Step-by-step solution

Understand the Problem

We need to calculate the expected profit for a company bidding on two independent contracts. These contracts have different probabilities of being awarded and different profit values.

Identify the Probabilities and Profits

The larger contract offers a profit of \( \\(50,000 \) with a \( 30\% \) chance of being awarded. The smaller contract offers a profit of \( \\)20,000 \) with a \( 60\% \) chance.

Express Probabilities as Decimals

Convert the percentage probabilities to decimal forms for calculations: \( 30\% = 0.30 \) and \( 60\% = 0.60 \).

Calculate the Expected Profit for Each Contract

Calculate the expected profit for each contract by multiplying the profit by its probability.- Larger contract: Expected Profit = \( 50,000 \times 0.30 = 15,000 \).- Smaller contract: Expected Profit = \( 20,000 \times 0.60 = 12,000 \).

Compute the Total Expected Profit

Add the expected profits from both contracts to find the total expected profit.Total Expected Profit = \( 15,000 + 12,000 = 27,000 \).

Key concepts

Understanding Probability

Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 means an event will not happen, and 1 means it certainly will. In the context of the software company bidding on contracts, probability helps quantify the chance of being awarded each contract.

To work with probability in mathematical calculations, we often convert percentages to decimals. For example, a 30% chance gets converted to 0.30, and a 60% chance becomes 0.60. This transformation is crucial for multiplying probabilities with profits to compute expected values. By understanding these basic principles, determining how likely the company is to win each contract becomes straightforward.

Independent Events in Probability

Events are considered independent when the occurrence of one event does not affect the occurrence of another. In our exercise, the awarding of one contract does not influence the outcome of the other. This concept is crucial because it ensures that we can analyze each contract separately when calculating the total expected profit.

In more technical terms, if two events A and B are independent, the probability of both occurring is the product of their individual probabilities. However, since our task does not involve finding the probability of both contracts being awarded simultaneously, we apply independence to separately compute the expected profit of each contract.

For students, understanding independence simplifies complex situations by allowing them to treat each probability event individually without worrying about its impact on other events.

Profit Calculation Through Expected Value

Expected value is a fundamental concept used to determine the average outcome of a random event based on its probabilities and possible outcomes. In this case, we're calculating the expected profit the software company can anticipate.

The expected profit for each contract is computed using the formula:

• Multiply the profit associated with a contract by its probability of being awarded

This results in the expected profit for each contract:

• Larger contract: \[ \text{Expected Profit} = \\(50,000 \times 0.30 = \\)15,000\]
• Smaller contract: \[ \text{Expected Profit} = \\(20,000 \times 0.60 = \\)12,000\]

To find the total expected profit, simply sum the expected profits from both contracts: \[\text{Total Expected Profit} = \\(15,000 + \\)12,000 = \$27,000\]This calculation shows the average profit the company might expect if the situation was to be repeated many times.