Problem 24 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Bob, the proprietor of Midway Lumber, bases his projections for the annual revenues of the company on the performance of the housing market. He rates the performance of the market as very strong, strong, normal, weak, or very weak. For the next year, Bob estimates that the probabilities for these outcomes are $$18, .27$$, $$.42, .10$$, and $$.03$$, respectively. He also thinks that the revenues corresponding to these outcomes are $$\$ 20, \$$ 18.8$$, $$\$ 16.2, \$$ 14$$, and $$\$ 12$$ million, respectively. What is Bob's expected revenue for next year?

Step-by-Step Solution

Answer

Bob's expected revenue for next year is approximately \$17.24 million.

Step 1: Listing the outcomes, revenues, and probabilities

First, let's list out the outcomes (market performance), their associated revenues, and probabilities: 1. Very strong: Revenue = \$20M, Probability = 0.18 2. Strong: Revenue = \$18.8M, Probability = 0.27 3. Normal: Revenue = \$16.2M, Probability = 0.42 4. Weak: Revenue = \$14M, Probability = 0.10 5. Very weak: Revenue = \$12M, Probability = 0.03

Step 2: Calculate expected revenue

To calculate the expected revenue, we will multiply the revenue for each outcome by its respective probability and then add the results together. The formula for expected revenue (E[R]) is: E[R] = (Revenue₁ × Probability₁) + (Revenue₂ × Probability₂) + ... + (Revenueₙ × Probabilityₙ) Here, n is the number of outcomes, which in our case is 5.

Step 3: Plug in the values and compute the result

Now, we can plug in the values from Step 1 into the formula: E[R] = (\$20M × 0.18) + (\$18.8M × 0.27) + (\$16.2M × 0.42) + (\$14M × 0.10) + (\$12M × 0.03) E[R] = (\$3.6M) + (\$5.076M) + (\$6.804M) + (\$1.4M) + (\$0.36M) E[R] = \$17.240M Bob's expected revenue for next year is approximately \$17.24 million.

Key Concepts

Probability and Statistics

Understanding probability and statistics is fundamental in making predictions about future events based on past data. In the context of business, such as Bob's lumber company, probabilities represent the chance of different market conditions occurring, such as a very strong or very weak housing market. Each possible outcome is assigned a probability, which is a number between 0 and 1, where a probability of 0 means the event will not occur and a probability of 1 means it will definitely occur. It's crucial that the sum of the probabilities for all possible outcomes equals 1, ensuring that one of the outcomes must happen.

These probabilities can be used for more than just predicting a single outcome; they serve as weights in calculating a more informative value known as the mathematical expectation, which in business applications can be revenue, costs, or profits. Bob's situation uses his estimated probabilities for various market conditions to forecast expected revenues, giving him a more nuanced financial projection rather than just preparing for the best or worst-case scenario.

Mathematical Expectation

The mathematical expectation or expected value of a random variable is a weighted average, where each possible value the variable can take is multiplied by its probability of occurring. In essence, it's the long-term average outcome if an experiment or business scenario was repeated many times. This concept is central in both probability and statistics and is an indispensable tool in many fields including finance, insurance, and economics.

The calculation is straightforward: each outcome's value is multiplied by its associated probability, and the results are summed up. In Bob's revenue calculation, for instance, the expected revenue is found by multiplying the revenue for each housing market condition by the probability of that condition occurring, and adding these products together. This provides Bob with an 'average' revenue figure. While actual revenue might deviate from this expected value, it gives a solid basis for planning and decision-making.

Business Mathematics

The exercise involving Bob's company touches on important concepts in business mathematics, which combines mathematical methods and commercial concepts to aid in managerial decision-making. In calculating expected revenue, Bob is practicing what many businesses do to forecast earnings under uncertainty. Business mathematics covers topics like profit maximization, cost minimization, and optimization of resources which are all important for the effective operation of a business.

For Bob's lumber company, calculating the expected revenue is a practical application of business mathematics that helps in setting budgets, strategic planning, and assessing the financial health of his business. Although expected revenue is not guaranteed, it provides a base value that can be used to analyze different business strategies, such as how much stock to order or how aggressive marketing campaigns should be.