Problem 20 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The proprietor of Midland Construction Company has to decide between two projects. He estimates that the first project will yield a profit of $$\$ 180,000$$ with a probability of \(.7\) or a profit of $$\$ 150,000$$ with a probability of \(.3\); the second project will yield a profit of $$\$ 220,000$$ with a probability of \(.6\) or a profit of $$\$ 80,000$$ with a probability of \(.4\). Which project should the proprietor choose if he wants to maximize his expected profit?

Step-by-Step Solution

Answer

The proprietor should choose the first project to maximize his expected profit, as its expected profit is \$171,000, while the second project's expected profit is \$164,000.

Step 1: Calculate the expected profit for the first project.

To find the expected profit for the first project, we will multiply the profit for each possible outcome by its probability and sum the results: \[E(P_1) = (\$180,000 \times 0.7) + (\$150,000 \times 0.3)\]

Step 2: Calculate the expected profit for the second project.

To find the expected profit for the second project, we will multiply the profit for each possible outcome by its probability and sum the results: \[E(P_2) = (\$220,000 \times 0.6) + (\$80,000 \times 0.4)\]

Step 3: Compare the expected profits.

Now, we will calculate the expected profits for both projects and compare them to determine which project the proprietor should choose to maximize his expected profit. For the first project, we get: \[E(P_1) = (\$180,000 \times 0.7) + (\$150,000 \times 0.3) = \$126,000 + \$45,000 = \$171,000\] For the second project, we get: \[E(P_2) = (\$220,000 \times 0.6) + (\$80,000 \times 0.4) = \$132,000 + \$32,000 = \$164,000\]

Step 4: Make a decision.

Comparing the expected profits, we see that the first project has a higher expected profit (\$171,000) than the second project (\$164,000). Therefore, the proprietor should choose the first project to maximize his expected profit.

Key Concepts

Decision Making

When faced with multiple options, decision-making becomes a critical skill. It is essential to weigh the potential outcomes of each option given the circumstances and information at hand. For instance, the proprietor of Midland Construction Company must choose between two construction projects. Each project has its own set of potential profits and respective probabilities. By making an informed decision based on expected profits, the proprietor can select the project that offers the maximum potential gain. This process involves:

• Identifying possible outcomes for each choice
• Weighing these outcomes based on their likelihoods
• Selecting the option with the most favorable long-term results

In essence, decision-making in business often revolves around maximizing expected value to ensure financial success.

Probability Theory

Probability theory plays a central role in decision-making, especially when it comes to predicting financial outcomes. It involves quantifying uncertainty and calculating how likely particular outcomes are. In the scenario from Midland Construction Company, the proprietor is given probabilities for different profit levels for each construction project. For example:

• The first project has a 70% chance of yielding $180,000 and a 30% chance of making $150,000.
• The second project boasts a 60% chance of yielding $220,000 and a 40% chance of making $80,000.

Understanding and applying these probabilities allows the proprietor to calculate expected values, providing a mathematical basis for decision-making. Incorporating probability theory helps in reducing uncertainty and clarifying the potential risks and rewards associated with each choice.

Profit Maximization

Profit maximization is a key objective for any business. Here, it means making choices that lead to the highest expected profit. For Midland Construction Company’s proprietor, this involves calculating and comparing the expected profits of each project option.By calculating the expected profit value:

• The first project's expected profit is computed as \\(171,000\
• The second project's expected profit amounts to \\)164,000\

Since \\(171,000\ exceeds \\)164,000\, the proprietor will choose the first project to achieve maximum profit.This approach ensures that decisions are grounded in solid financial reasoning, focusing not just on potential high profits but on how likely these profits are to be realized. The ultimate goal is to optimize the company’s earnings through statistically informed choices.