Problem 11 - A First Course in Probability

Question

A total of 28 percent of American males smoke cigarettes, 7 percent smoke cigars, and 5 percent smoke both cigars and cigarettes. (a) What percentage of males smokes neither cigars nor cigarettes? (b) What percentage smokes cigars but not cigarettes?

Step-by-Step Solution

Answer

(a) 70% of American males smoke neither cigars nor cigarettes. (b) 2% of American males smoke cigars but not cigarettes.

Step 1: Understand the principle of inclusion-exclusion

The principle of inclusion-exclusion states that in a set of objects or elements, if some elements belong to two or more categories, then to find the total elements in the set, we need to add the number of elements in each category and subtract the overlapping elements, i.e., those elements which belong to more than one category. In this case, we have three categories: males who smoke cigars, males who smoke cigarettes, and males who smoke both cigars and cigarettes. To find the total percentage of males who smoke neither cigars nor cigarettes, we need to consider all American males as our set and subtract the percentage of males who belong to each of the given categories.

Step 2: Calculate the percentage of males who smoke cigars or cigarettes

According to the principle of inclusion-exclusion, the percentage of males who smoke cigars or cigarettes is the sum of the percentage of males who smoke cigars and the percentage of males who smoke cigarettes minus the percentage of males who smoke both cigars and cigarettes. Let A = percentage of males who smoke cigars = 7% Let B = percentage of males who smoke cigarettes = 28% Let C = percentage of males who smoke both cigars and cigarettes = 5% Percentage of males who smoke cigars or cigarettes = A + B - C = 7% + 28% - 5% = 30%

Step 3: Calculate the percentage of males who smoke neither cigars nor cigarettes

Now that we have the percentage of males who smoke cigars or cigarettes, we can find the percentage of males who smoke neither cigars nor cigarettes. Since the total percentage of all American males is 100%, we can find the percentage of males who smoke neither cigars nor cigarettes by subtracting the percentage of males who smoke cigars or cigarettes from the total percentage. Percentage of males who smoke neither cigars nor cigarettes = 100% - 30% = 70% So, 70% of American males smoke neither cigars nor cigarettes. #b) Find the percentage of males who smoke cigars but not cigarettes#

Step 1: Calculate the percentage of males who smoke cigars but not cigarettes

To find the percentage of males who smoke cigars but not cigarettes, we need to subtract the percentage of males who smoke both cigars and cigarettes from the percentage of males who smoke cigars. This is because the percentage of males who smoke cigars includes both those who smoke cigars only and those who smoke both cigars and cigarettes. Percentage of males who smoke cigars but not cigarettes = A - C = 7% - 5% = 2% So, 2% of American males smoke cigars but not cigarettes.

Key Concepts

Probability

When discussing probability, we delve into the likelihood of different events occurring within a given set. It's a mathematical concept that applies to various situations, such as predicting weather patterns, gambling outcomes, or as seen in the exercise, the smoking habits of a specific population group.

In the given problem, we express the probability in percentage terms, where each percentage denotes the probability distribution of American males with specific smoking habits. By interpreting these percentages, we calculate the chances of various overlapping and non-overlapping groups within the whole.

Set Theory

Set theory serves as the foundation of modern mathematics, and it plays a crucial role in understanding the relationships between groups within a population. In the context of the problem, we consider three sets: those who smoke cigars (A), cigarettes (B), and both (C). These sets can intersect with each other, creating subsets; for example, set C is a subset where sets A and B intersect. Using set theory and visual aids like Venn diagrams, we understand complex relationships and calculate specific group sizes within a universal set (in this case, all American males).

Percentage Calculation

Percentage calculation allows us to accurately represent the proportion of a group in relation to the whole. In the exercise, we convert raw data into percentages to depict the proportion of American males engaged in particular smoking habits. This quantitative measure is critical when comparing different parts of a population or analyzing changes in data over time. To find the percentage of a subset, such as 'males who smoke cigars but not cigarettes,' we performed a simple subtraction: 7% (those who smoke cigars) minus 5% (those who smoke both), resulting in 2% (those who smoke only cigars).

Overlapping Sets

Finally, the concept of overlapping sets is crucial for understanding how different groups can share common elements, as seen in the study of males who smoke cigars, cigarettes, or both. The overlapping set, in this case, represents males smoking both cigars and cigarettes.

By mastering these concepts, students can handle complex problems involving overlapping sets more effectively. It's essential to subtract the overlapping percentage to avoid double-counting individuals in both groups, keeping our calculations accurate and our conclusions valid.