Problem 22 - Algebra Form and Function

Question

A city is divided into 4 voting precincts, \(A, B, C,\) and \(D\). Table 17.20 shows the results of mayoral election held for two candidates, a Republican and a Democrat.$$ \begin{array}{c|c|c|c} \hline \text { Precinct } & \text { Number voters } & \text { Republican } & \text { Democrat } \\ \hline \mathrm{A} & 10,000 & 4,200 & 5,800 \\ \mathrm{~B} & 15,000 & 7,100 & 7,900 \\ \mathrm{C} & 17,000 & 8,200 & 8,800 \\ \mathrm{D} & 18,000 & 12,400 & 5,600 \\ \hline \end{array}$$ Assuming random selection, what is the probability, given as a percentage, that a voter: (a) Lives in precinct \(B ?\) (b) Is a Republican? (c) \(\operatorname{Both}(\) a) and \((\mathrm{b})\) (d) Is Republican given that he or she lives in precinct \(B ?\) (e) Lives in precinct \(B\) given that he or she is Republican?

Step-by-Step Solution

Answer

Answer: The probability of a voter being a Republican given that he or she lives in precinct B is 47.33%.

Step 1: Calculate the total number of voters in the city

Sum the number of voters in each precinct: $$Total\_voters = 10,000 + 15,000 + 17,000 + 18,000 = 60,000$$

Step 2: Calculate the Probability of a voter living in precinct B

Divide the number of voters in precinct B by the total number of voters in the city and multiply the result by 100 to get the percentage: $$P(B) = \frac{15,000}{60,000} * 100\% = 25\%$$

Step 3: Calculate the Probability of a voter being a Republican

Sum the number of Republican voters in the city and divide it by the total number of voters. Multiply the result by 100 to get the percentage: $$P(Republican) = \frac{4,200 + 7,100 + 8,200 + 12,400}{60,000} * 100\% = 55\%$$

Step 4: Calculate the Probability of a voter both living in precinct B and being a Republican

Divide the number of Republican voters in precinct B by the total number of voters in the city and multiply the result by 100 to get the percentage: $$P(B \cap Republican) = \frac{7,100}{60,000} * 100\% = 11.83\%$$

Step 5: Calculate the Probability of a voter being a Republican given that he or she lives in precinct B

Divide the number of Republican voters in precinct B by the number of voters in precinct B, and multiply the result by 100 to get the percentage: $$P(Republican | B) = \frac{7,100}{15,000} * 100\% = 47.33\%$$

Step 6: Calculate the Probability of a voter living in precinct B given that he or she is a Republican

Divide the number of Republican voters in precinct B by the total number of Republican voters in the city, and multiply the result by 100 to get the percentage: $$P(B | Republican) = \frac{7,100}{(4,200+7,100+8,200+12,400)} * 100\% = 21.52\%$$ To summarize the results: (a) The probability of a voter living in precinct B is 25%. (b) The probability of a voter being a Republican is 55%. (c) The probability of a voter both living in precinct B and being a Republican is 11.83%. (d) The probability of a voter being a Republican given that he or she lives in precinct B is 47.33%. (e) The probability of a voter living in precinct B given that he or she is a Republican is 21.52%.

Key Concepts

Probability

In this problem, we explore the concept of probability, which refers to the likelihood of an event occurring. In essence, it is a way of quantifying uncertainty. Here, when we calculate probabilities, it is often expressed as a percentage. This helps to provide a clear understanding of how likely events, such as voters living in certain precincts, are to occur.

Understanding probability can be broken down as follows:

• **Basic Probability**: This is calculated by dividing the number of successful outcomes by the total number of possible outcomes.
• **Conditional Probability**: This is the probability of an event occurring, given that another event has already occurred. For example, finding the probability of a voter being Republican, given that they live in Precinct B, is conditional probability.

The steps taken in this analysis illustrate these concepts well, such as determining the probability of a voter residing in a specific precinct versus being affiliated with a political party.

Statistics

Statistics is essentially about turning raw data into meaningful insights. It includes collecting, analyzing, and interpreting this data to draw conclusions. In this exercise, we rely heavily on statistics to determine the likelihood of certain voter characteristics.

When calculating these probabilities, it is crucial to understand the role of different statistical measures:

• **Summation**: This involves adding up numbers to find totals, like all the voters in this city.
• **Percentage Calculation**: The total number of specific voters is often represented in terms of percentage for better understanding of proportions.

By using statistics, we gain insights into how voters in diverse precincts tend and compare their affiliations, thus allowing analysis beyond just raw numbers.

Voting Data Analysis

Voting data analysis involves examining electoral data to understand patterns and trends in voting behavior. This may include analyzing results across various precincts or demographic segments.

Key components in this analysis are:

• **Data Segmentation**: Breaking down data into smaller segments, such as each precinct's voting data, can provide deeper insights.
• **Comparative Analysis**: Understanding the differences between segments, such as comparing turnout in different precincts or between political parties.

For instance, determining which precincts have higher Republican or Democrat voters can help with campaign strategies. Analyzing such data provides a vivid picture of the political landscape within a city, thereby aiding in informed decision-making for political stakeholders.