Problem 12 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The management of the Cambridge Company has projected the sales of its products (in millions of dollars) for the upcoming year, with the associated probabilities shown in the following table: $$\begin{array}{lcccccc}\hline \text { Sales } & 20 & 22 & 24 & 26 & 28 & 30 \\\\\hline \text { Probability } & .05 & .10 & .35 & .30 & .15 & .05 \\\\\hline\end{array}$$ What does the management expect the sales to be next year?

Step-by-Step Solution

Answer

The management expects the sales to be \(25\) million dollars next year.

Step 1: List the Sales and Probabilities

We'll start by listing the given sales and probabilities in two separate arrays for easier calculations: Sales: \(x = [20, 22, 24, 26, 28, 30]\) Probabilities: \(p = [0.05, 0.1, 0.35, 0.3, 0.15, 0.05]\)

Step 2: Multiply Each Sale Value by Its Probability

Now, we'll multiply each sale value by its corresponding probability to find the weighted values: Weighted Values: \([20*0.05, 22*0.1, 24*0.35, 26*0.3, 28*0.15, 30*0.05] = [1, 2.2, 8.4, 7.8, 4.2, 1.5]\)

Step 3: Calculate the Expected Value

Finally, add up the weighted values to get the Expected Value (EV): Expected Sales: \(EV = 1 + 2.2 + 8.4 + 7.8 + 4.2 + 1.5 = 25\) Thus, the management expects the sales to be 25 million dollars next year.

Key Concepts

Probability Distribution

Understanding the essence of a probability distribution is fundamental in the study of statistics and applied mathematics. It's a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In simpler terms, it's like a map that tells us how likely it is for a particular event to occur.

Take the example of the Cambridge Company’s projected sales; their management is predicting future sales figures along with the chances of achieving each specific amount. The probabilities assigned to each sales value need to sum up to 1, since one of the outcomes must happen. When we put all these projections together - the sales figures and their associated probabilities - we have what's called a discrete probability distribution since the sales values are distinct and separate.

Understanding probability distributions is critical because they help businesses and analysts to foresee outcomes and make informed decisions. For students working on such exercises, it’s key to ensure they list all the possible outcomes correctly and comprehend the significance of probabilities totaling 1, which represents certainty in the context of all possible outcomes.

Weighted Average

A weighted average, also known as a weighted mean, is an average where different values have different levels of importance, which we call weights. In mathematical terms, it's not enough to just add up numbers and divide by the count. Instead, we consider how significant each value is.

As demonstrated in the Cambridge Company's projected sales, not all sales figures have the same chance of occurring. Therefore, we multiply each sales value by its probability (the weight) to calculate its contribution to the overall expectation. The problem essentially asks for the weighted average of the sales figures.

To grasp this concept, think of it like rating a movie. Not all reviews are from critics—some are from the general audience. If a critic’s review is weighed more heavily than a regular viewer's, it has more impact on the average rating. Similarly, in our sales projection, the more likely sales figures affect the expected value more significantly. When students calculate weighted averages, it's crucial that they understand the reason behind each weight and its impact on the final result.

Applied Mathematics

Applied mathematics is all about using mathematical methods to solve practical problems that arise in various fields like science, engineering, business, and technology. It's not just about knowing the theory; it's about applying those theories to real-world scenarios to come up with tangible solutions.

The expectation calculation for the Cambridge Company is a prime example of applied mathematics in action. By combining the theories of probability distributions and weighted averages, the company is able to estimate potential future sales. This practical application aids in strategic planning and helps the company in setting goals and preparing for different possible outcomes based on data and mathematical reasoning.

For students tackling applied mathematics problems, it’s important to connect the mathematical concepts they learn to actual problems they may encounter outside the classroom. This not merely enhances understanding but also reinforces the relevance of mathematics in everyday decision-making processes.