Problem 40 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Electricity in the United States is generated from many sources. The following table gives the sources as well as their share in the production of electricity: $$ \begin{array}{lcccccc} \hline \text { Source } & \text { Coal } & \text { Nuclear } & \text { Natural gas } & \text { Hydropower } & \text { Oil } & \text { Other } \\ \hline \text { Share, } \% & 50.0 & 19.3 & 18.7 & 6.7 & 3.0 & 2.3 \\ \hline \end{array} $$ If a source for generating electricity is picked at random, what is the probability that it comes from a. Coal or natural gas? b. Nonnuclear sources?

Step-by-Step Solution

Answer

The probability that a source for generating electricity picked at random comes from: a. Coal or natural gas is \(0.687\) or \(68.7\%\). b. Nonnuclear sources is \(0.807\) or \(80.7\%\).

Step 1: Calculate the probability of a source picked coming from coal or natural gas

To calculate the probability that a source picked at random comes from coal or natural gas, we simply add their shares and divide by 100. Probability = (Share of Coal + Share of Natural Gas) / 100 = (50.0 + 18.7) / 100 = 68.7 / 100

Step 2: Calculate the probability of a source picked coming from nonnuclear sources

To calculate the probability that a source picked at random comes from nonnuclear sources, we subtract the share of nuclear sources from 100 and then divide by 100. Probability = (100 - Share of Nuclear) / 100 = (100 - 19.3) / 100 = 80.7 / 100

Step 3: Simplify the probabilities

Now we need to simplify the probabilities found in Step 1 and Step 2. a. Probability of source picked coming from coal or natural gas: P(Coal or Natural Gas) = 68.7 / 100 = 0.687 b. Probability of source picked coming from nonnuclear sources: P(Nonnuclear sources) = 80.7 / 100 = 0.807 Thus, the probability that a source for generating electricity picked at random comes from: a. Coal or natural gas is 0.687 or 68.7%. b. Nonnuclear sources is 0.807 or 80.7%.

Key Concepts

Energy Sources

Energy sources are the origins from which electricity is generated. In the United States, electricity is produced from various sources, including coal, natural gas, nuclear power, hydropower, oil, and other renewable energies like solar and wind. These sources contribute differently depending on their availability, cost-effectiveness, and environmental impact.

• Coal: A traditional but high-emission source, widely available, providing 50% of the U.S. electricity in the given context.


• Nuclear: A significant, low-emission source that contributes 19.3%.


• Natural Gas: A cleaner fossil fuel compared to coal, accounting for 18.7%.


• Hydropower: A renewable source harnessing water energy, making up 6.7%.


• Oil: Though less used for electricity, it contributes 3%.


• Other: Includes renewables such as wind and solar, contributing 2.3%.

Understanding the shares of each source helps in assessing their impact on the environment and planning for sustainable future energy strategies.

Mathematical Problem Solving

Mathematical problem solving involves breaking down a problem into manageable steps to find a solution. In this exercise, the goal was to calculate probabilities related to electricity sources. Probability calculation here requires identifying relevant components and performing the right operations.

• Step 1: Determine the total percentage by adding the shares of Coal and Natural Gas to find the probability of a source being one of these.


• Step 2: Calculate the nonnuclear probability by subtracting the Nuclear share from 100% to understand the other sources' probability.


• Step 3: Simplify the probabilistic fractions to standard decimal forms, providing straightforward results.

These steps require careful attention to detail and arithmetic skills, highlighting the importance of systematic approaches in mathematics.

Statistics

Statistics is a branch of mathematics dealing with data collection, analysis, interpretation, and presentation. In the scenario of the electricity sources, statistical data represent the production share of each energy type. This data is crucial for calculating probabilities, understanding trends, and influencing decision-making.

• Data Representation: The source shares are percentages, a statistical measure indicating each source's contribution.


• Probability Calculations: We apply statistics to calculate probabilities for coal or natural gas and nonnuclear sources by using their statistical data.

The statistical knowledge helps us draw meaningful conclusions from the data, such as assessing the dominance of certain energy types or the overall reliance on renewable versus nonrenewable sources.

Applied Mathematics

Applied mathematics refers to the use of mathematical techniques and principles to solve problems in real-world contexts, like energy production. By applying mathematics to the practical problems of energy source probability, we can make informed decisions and forecasts.

• Real-World Context: Energy source data is analyzed using probability to understand and optimize resource usage.


• Practical Applications: These probability outcomes can inform policy decisions, such as investing in more renewable energies or optimizing the existing energy mix.


• Link to Policy-making: Governments and organizations can use such mathematical analyses to guide sustainable energy development and transition strategies.

Hence, applied mathematics bridges theoretical knowledge with practical applications, facilitating well-informed decision-making and strategic planning in energy management.