Problem 19 - A First Course in Probability

Question

A total of 48 percent of the women and 37 percent of the men that took a certain "quit smoking" class remained nonsmokers for at least one year after completing the class. These people then attended a success party at the end of a year. If 62 percent of the original class was male, (a) what percentage of those attending the party were women? (b) what percentage of the original class attended the party?

Step-by-Step Solution

Answer

(a) 45.71% of those attending the party were women. (b) 41.22% of the original class attended the party.

Step 1: Identify the given values in the problem

: We are given: 1. 48 percent of the women and 37 percent of the men remained nonsmokers for at least one year. 2. 62 percent of the original class was male. 3. Let total number of students in the class be X. Now, let's find the number of women and men in the class.

Step 2: Find the number of men and women in class

: Percentage of women in the class = 100% - 62% = 38% Number of women = 0.38 * X Number of men = 0.62 * X

Step 3: Calculate the number of women and men attending the party

: Number of women attending the party = 48% * Number of women = 0.48 * (0.38 * X) Number of men attending the party = 37% * Number of men = 0.37 * (0.62 * X)

Step 4: Calculate the total number and percentage of people attending the party

: Total number of attendees = Number of women attending + Number of men attending = (0.48 * (0.38 * X)) + (0.37 * (0.62 * X)) Percentage of people attending the party: = \(\dfrac{Total\, number\, of\, attendees}{Total\, number\, of\, students}\) * 100 = \(\dfrac{(0.48 * (0.38 * X)) + (0.37 * (0.62 * X))}{X}\) * 100

Step 5: Find the percentage of women attending the party

: Percentage of women attending the party: = \(\dfrac{Number\, of\, women\, attending}{Total\, number\, of\, attendees}\) * 100 = \(\dfrac{(0.48 * (0.38 * X))}{(0.48 * (0.38 * X)) + (0.37 * (0.62 * X))}\) * 100 Now we have all the expressions to calculate the required percentages. (a) what percentage of those attending the party were women? Percentage of women attending the party: \[\dfrac{(0.48 * (0.38 * X))}{(0.48 * (0.38 * X)) + (0.37 * (0.62 * X))}\] * 100 = 45.71% (b) what percentage of the original class attended the party? Percentage of people attending the party: \[\dfrac{(0.48 * (0.38 * X)) + (0.37 * (0.62 * X))}{X}\] * 100 = 41.22%

Key Concepts

Percentage Calculation

Understanding percentage calculation is crucial not just in mathematics, but in everyday life. It represents a way to express a number as a fraction of 100, which is often used to compare proportions. In the context of our problem, we dealt with calculations such as finding what 48% of women in the class represents, or 37% of the men.

The general formula for percentage problems is \[ \frac{\text{Part}}{\text{Whole}} \times 100 \]. In the exercise, the 'whole' was the total number of students, and the 'part' was the number or proportion of students that remained nonsmokers and attended the party. By applying the formula, we were able to break down complex real-world problems into simple mathematical terms, which is the essence of percentage calculation.

Probability and Statistics

The study of probability and statistics involves analyzing the likelihood of events and summarizing data sets, respectively. Although the given problem does not directly ask for probability calculations, understanding probabilities is underlying when we talk about percentages and expected outcomes. For example, when we mention that 48% of women remained nonsmokers, we implicitly refer to the probability of a woman from the class remaining a nonsmoker.

In statistics, we use the information given—like the 62% of the original class being male—to make further statistical inferences or predictions. In our scenario, by using statistical methods, we predicted the composition of the party's attendees based on the given data.

Quantitative Reasoning

Quantitative reasoning is the application of mathematical concepts to understand and analyze real-world scenarios. This problem showcases quantitative reasoning by requiring us to interpret percentages and proportions in the context of a real-life event—the success party of a 'quit smoking' class.

To solve it, one must understand the relationships between the given numbers and use mathematical operations to deduce the required answers. Quantitative reasoning goes beyond arithmetic; it involves critical thinking and logical deduction to apply mathematical concepts in everyday situations. By framing our problem-solving in terms of a story—the class and its success party—we engage in quantitative reasoning by using mathematical tools to find meaningful conclusions about the composition of the party's attendance.