Problem 25 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Maria sees the growth of her business for the upcoming year as being tied to the gross domestic product (GDP). She believes that her business will grow (or contract) at the rate of \(5 \%, 4.5 \%, 3 \%, 0 \%\), or \(-0.5 \%\) per year if the GDP grows (or contracts) at the rate of between 2 and \(2.5 \%\), between \(1.5\) and \(2 \%\), between 1 and \(1.5 \%\), between 0 and \(1 \%\), and between \(-1\) and \(0 \%\), respectively. Maria has decided to assign a probability of \(.12, .24, .40, .20\), and \(.04\), respectively, to each outcome. At what rate does Maria expect her business to grow next year?

Step-by-Step Solution

Answer

Maria expects her business to grow at a rate of 2.86% next year. This is calculated using the given growth rates and their corresponding probabilities to find the expected value which represents the weighted average of all possible outcomes.

Step 1: List the growth rates and their probabilities

We are given the growth rates as \(5\%, 4.5\%, 3\%, 0\%, -0.5\%\) along with their respective probabilities \(.12, .24, .40, .20, .04\).

Step 2: Multiply each growth rate by its probability

To find the weighted average, we must multiply each growth rate by its probability: \(5\% * .12 = 0.6\%\) \(4.5\% * .24 = 1.08\%\) \(3\% * .40 = 1.2\%\) \(0\% * .20 = 0\%\) \(-0.5\% * .04 = -0.02\%\)

Step 3: Calculate the expected growth rate

Now, we sum up the results from step 2 to find the expected growth rate: \(0.6\% + 1.08\% + 1.2\% + 0\% - 0.02\% = 2.86\% \) So, Maria expects her business to grow at a rate of 2.86% next year.

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the likelihood of different outcomes. Essentially, it quantifies uncertainty and provides a way to make predictions about future events.
When Maria assigns probabilities to each possible growth rate of her business based on changes in GDP, she is applying probability theory. These probabilities must add up to 1, as they represent all possible scenarios.
Here are some important aspects of probability theory:

• **Probability Values:** The probability of an event ranges from 0 (impossible) to 1 (certain).
• **Outcome Distinction:** Different events have different probabilities. In Maria's case, the distinct growth rates of 5%, 4.5%, 3%, 0%, and -0.5% have assigned probabilities.
• **Calculation:** These probabilities can be used to calculate the likelihood of different outcomes and help make informed predictions.

Understanding probability enables businesses to manage risks and prepare for various economic conditions.

Expected Value Calculation

Expected value is a key concept that helps predict the average outcome of a random event. It combines the different possible outcomes by weighting them according to their probabilities.
For Maria, calculating the expected growth rate of her business involves computing a weighted average of all her predicted growth rates.
Here's how to perform an expected value calculation:

• **Identify Outcomes:** List all possible outcomes and their probabilities. Maria's outcomes were the different growth percentages.
• **Multiply:** Each growth rate is multiplied by its probability to find the contribution of each scenario to the overall expectation.
• **Sum:** Add these weighted values together to get the expected value.

This method provides a singular outlook, giving Maria a clear understanding of her business's likely performance, which is an expected growth of 2.86%.

Statistical Analysis

Statistical analysis involves collecting, analyzing, and interpreting data to understand patterns and trends. For businesses, this is crucial in making informed decisions and strategies.
Taking Maria's example, she uses statistical analysis to interpret economic predictions and assess her business outlook.
Key points in statistical analysis:

• **Data Collection:** Gather numerical data relevant to the business context, such as GDP growth rates.
• **Analysis:** Use mathematical techniques to process this data. Probability and expected value calculations are part of this step.
• **Interpretation:** Draw conclusions from the data analysis. Maria's expected growth rate is a conclusion drawn from her statistical analysis of GDP rates and their impact.

Statistical analysis empowers business owners to predict trends and navigate future economic scenarios confidently.