Problem 5 - Algebra 2

Question

The table shows the frequency of responses to editorials. Find each probability. $$ \begin{array}{|l|l|l|l|l|l|l|}\hline \text { Number of Responses } & {0} & {1} & {2} & {3} & {4} & {5} & {6 \text { or more }} & {\text { Total }} \\ \hline \text { Number of Editorials } & {20} & {30} & {56} & {38} & {34} & {16} & {6} & {200} \\ \hline\end{array} $$ \(P(0-2 \text { responses })\)

Step-by-Step Solution

Answer

The probability of having 0 to 2 responses is \(P(0-2 \text { responses }) = \frac {106}{200}\)

Step 1: Identify the total number of outcomes

From the table, the total number of editorials, which represents the total number of outcomes, is 200.

Step 2: Identify the number of favorable outcomes

The favorable outcomes are the editorials with responses in the range of 0 to 2. We get these by adding up the number of editorials that received 0, 1, or 2 responses: 20 (for 0 responses) + 30 (for 1 response) + 56 (for 2 responses) = 106.

Step 3: Calculate the probability

The probability is calculated by dividing the number of favorable outcomes by the total number of outcomes. In mathematical terms: \(P(0-2 \text { responses }) = \frac {Number \, of \, Favorable \, Outcomes}{Total \, Number \, of \, Outcomes}\) or \(P(0-2 \text { responses }) = \frac {106}{200}\)

Key Concepts

Frequency Distribution

Frequency distribution is a helpful statistical tool that allows us to see how often different outcomes occur. In this context, we are looking at how often various numbers of responses are received for editorials.
The table presented in the exercise is a perfect example of a frequency distribution. It lists the number of editorials for each possible number of responses, ranging from 0 to 6 or more. This structured display helps us to quickly understand both the individual and overall distribution of responses.
Frequency distributions make it easy to identify patterns, compare different data points, and find relevant values such as totals or averages. In probabilities, frequency distribution is especially crucial as it provides the foundation upon which probabilities are calculated.

Editorials Response Analysis

Editorials response analysis involves examining and interpreting the data gathered from how many responses various editorials received. This analysis gives insight into audience engagement and interest trends.
With the dataset shown, we can identify that certain number of responses are more common than others. For instance, focusing on the segment with 0 to 2 responses can tell us about less engaging or rarely shared articles.
Such analysis can guide editorial strategies, helping to improve content to achieve higher readership engagement, possibly by focusing on the factors contributing to higher response counts.

Favorable Outcomes

Favorable outcomes are specific results within a probability calculation pipeline that are considered desirable or necessary for the probability in question. Here, the favorable outcomes are the editorials that received between 0 and 2 responses.
In probability terms, when we determine favorable outcomes, we define a subset of possible outcomes from the entire pool. In this case, it means counting the number of editorials that fall within the response range of interest.
Understanding and defining favorable outcomes is crucial as it directly influences the probability to be calculated. By counting these events accurately, we ensure that our probability calculations are based on precise data.

Probability Formula

The probability formula is a fundamental concept in statistics and is used to calculate the likelihood of a particular outcome occurring. In this exercise, the probability formula is expressed as:

• \(P( ext{specific event}) = \frac{ ext{Number of Favorable Outcomes}}{ ext{Total Number of Outcomes}}\)

To calculate the probability of an editorial receiving 0 to 2 responses, the formula requires us to use the number of favorable outcomes (editorials with responses from 0 to 2) divided by the total number of editorials.
By applying this step-by-step method, it simplifies the complex nature of probability, allowing students to understand and perform it effectively. Using this clear breakdown ensures that the concept remains accessible and demystified, making it a powerful tool in statistical analysis.